graphic presentation of the frequency distribution

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4.2: Frequency Distributions and Statistical Graphs

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• Page ID 92985

• David Lippman
• Pierce College via The OpenTextBookStore

Once we have collected data, then we need to start analyzing the data. One way to display and summarize data is to use statistical graphing techniques. The type of graph we use depends on the type of data collected. Qualitative data use graphs like bar graphs and pie graphs. Quantitative data use graphs such as histograms and frequency polygons.

In order to create graphs, we must first organize and create a summary of the individual data values in the form of a frequency distribution . A frequency distribution is a listing all of the data values (or groups of data values) and how often those data values occur.

Frequency and Frequency Distributions

Frequency is the number of times a data value or groups of data values (called classes ) occur in a data set.

A frequency distribution is a listing of each data value or class of data values along with their frequencies.

Relative frequency is the frequency divided by \(n\), the size of the sample. This gives the proportion of the entire data set represented by each value or class. Relative frequencies are expressed as fractions, decimals, or percentages.

A relative frequency distribution is a listing of each data value or class of data values along with their relative frequencies.

The method of creating a frequency distribution depends on whether we are working with qualitative data or quantitative data . We will now look at how to create each type of frequency distribution according to the type of data and the graphs that go with them.

Organizing Qualitative Data

Qualitative data are pieces of information that allow us to classify the items under investigation into various categories. We usually begin working with qualitative data by giving the frequency distribution as a frequency table.

Frequency Table

A frequency table is a table with two columns. One column lists the categories, and another column gives the frequencies with which the items in the categories occur (how many data fit into each category).

An insurance company determines vehicle insurance premiums based on known risk factors. If a person is considered a higher risk, their premiums will be higher. One potential factor is the color of your car. The insurance company believes that people with some colors of cars are more likely to ve involved in accidents. To research this, the insurance company examines police reports for recent total-loss collisions. The data is summarized in the frequency table below.

\(\begin{array}{|c|c|} \hline \textbf { Color } & \textbf { Frequency } \\ \hline \text { Blue } & 25 \\ \hline \text { Green } & 52 \\ \hline \text { Red } & 41 \\ \hline \text { White } & 36 \\ \hline \text { Black } & 39 \\ \hline \text { Grey } & 23 \\ \hline \end{array}\)

Graphing Qualitative Data in Bar Graphs and Pie Charts

Once we have organized and summarized qualitative data into a frequency table, we are ready to graph the data. There are various ways to visualize qualitative data. In this section we will consider two common graphs: bar graphs and pie graphs .

A bar graph is displays a bar for each category. The length of each bar indicates the frequency of that category.

To construct a bar graph, we need to draw a vertical axis and a horizontal axis. The vertical direction has a scale and measures the frequency of each category. The horizontal axis has no scale in this instance but lists the categories. The construction of a bar chart is most easily described by use of an example.

Using the car color data from Example 1, note the highest frequency was 52, so the vertical axis needs to go from 0 to 52. We might as well use 0 to 55 so that we can put a hash mark every 5 units:

You should notice a few things about the correct construction of this bar graph.

• The height of each bar is determined by the frequency of the corresponding color.
• Both axes are labeled clearly.
• The bars do not touch and they are the same width.

The horizontal grid lines are a nice touch, but not necessary. In practice, you will find it useful to draw bar graphs on graph paper so the grid lines will already be in place or use technology to create the graph. Instead of grid lines, we might also list the frequencies at the top of each bar, like this:

In a survey, adults were asked whether they personally worried about a variety of environmental concerns. The numbers (out of 1012 surveyed) who indicated that they worried “a great deal” about some selected concerns are summarized below.

\(\begin{array}{|c|c|} \hline \textbf { Environmental Issue } & \textbf { Frequency } \\ \hline \text { Pollution of drinking water } & 597 \\ \hline \text { Contamination of soil and water by toxic waste } & 526 \\ \hline \text { Air pollution } & 455 \\ \hline \text { Global warming } & 354 \\ \hline \end{array}\)

Display the data using a bar graph.

Try it Now 1

A questionnaire on the makes of people's vehicles showed the following responses from 30 participants. Construct a frequency table and a bar graph to represent the data. ( F = Ford, H = Honda, V = Volkswagen, M = Mazda)

F M M M V M F M F V H H F V F H H F M M V H M V V F V H M F

A class was asked for their favorite soft drink with the following results:

• Create a frequency distribution table for the data.
• Create a relative frequency distribution table for the data.
• Draw a bar graph of the frequency distribution.
• Draw a bar graph of the relative frequency distribution.
• To make a frequency distribution table, list each drink type and and then count how often each drink occurs in the data. Notice that Coke happens 9 times in the data set, Pepsi happens 10 times, and so on.
• To make a relative frequency distribution table, use the previous results and divide each frequency by 33, which is the total number of data responses.
• Along the horizontal axis you place the drinks. Space these apart equally, and allow space to draw bars above the axis. The vertical axis shows the frequencies. Make sure you create a scale along that axis in which all of the frequencies will fit. Notice that the highest frequency is 10, so you want to make sure the vertical axis goes to at least 10, and you may want to count by two for every tick mark. Here is what the graph looks like using Excel.

• A bar graph for the relative frequency distribution is similar to the bar graph for the frequency distribution except that the relative frequencies are used along the vertical axis instead. Notice that the graph does not actually change except the numbers on the vertical scale.

Let's use the last example to introduce another way of visualizing data – a pie chart also known as circle graph .

A pie chart is a graph where the "pie" represents the entire sample and the "slices" represent the categories or classes. The size of the slice of the pie corresponds to the relative frequency for that category.

To find the angle that each “slice” takes up, multiply the relative frequency of that slice by 360°.

Note: Theoretically, the percentages of all slices of a pie chart must add to 100%. In practice, the percentages may add to be slightly more or less than 100% if percentages are rounded.

To draw a pie chart, multiply the relative frequencies of each drink by 360°. Then, use a protractor to mark off the corresponding angle in a circle. Usually it is easier to use Excel or some other spreadsheet program to draw the graph.

The pie graph from Excel is shown below.

Try it Now 2

The Red Cross Blood Donor Clinic had a very successful morning collecting blood donations. Within 3 hours, many people had made donations. The table shows the frequency distribution of the blood types of the donations. Construct a pie chart to display the relative frequency distribution.

Organizing Quantitative Data

Quantitative is data that is the result of counting or measuring some aspect of items under investigation. For this reason, this type of data is also known as numerical data. Quantitative data can also be summarized in a table to show its frequency distribution.

A teacher records scores on a 20-point quiz for the 30 students in his class. The scores are

19 20 18 18 17 18 19 17 20 18 20 16 20 15 17 12 18 19 18 19 17 20 18 16 15 18 20 5 0 0

These scores could be summarized into a frequency table by counting how many times each particular data value occurs.

\(\begin{array}{|c|c|} \hline \textbf { Score } & \textbf { Frequency } \\ \hline 0 & 2 \\ \hline 5 & 1 \\ \hline 12 & 1 \\ \hline 15 & 2 \\ \hline 16 & 2 \\ \hline 17 & 4 \\ \hline 18 & 8 \\ \hline 19 & 4 \\ \hline 20 & 6 \\ \hline \end{array}\)

In the previous example, the table listed every different data value that occurred and how often each value occurred. We call this type of frequency distribution presentation ungrouped . Sometimes it is helpful to group the data into classes to observe information about the distribution of data that otherwise wouldn't be noticeable. This is particularly true if there are many different values or each value only occurs once. You can think about classes as "bins" that we create to sort the data. When we group the data into classes, we call this type of frequency distribution presentation grouped .

When data are grouped, the following guidelines about the classes should be followed

• Classes should have the same width.
• Classes should not overlap.
• Each piece of data should belong to only one class.

Let's use the data from the previous example to create a grouped frequency distribution.

Create a grouped frequency table in two ways:

• with classes of width 5 beginning at a score of 0, and
• with classes of width 6 beginning at a score of 0.
• The first class contains the scores 0, 1, 2, 3, and 4 -- if any occur. Likewise, the second class will contain scores 5, 6, 7, 8, and 9 -- if any occur. This pattern continues until classes are no longer needed.

The first two columns of the table shows the classes and the frequency of the data in each class.

In the first column, the numbers 0, 4, 10, 15, and 20 are called the lower class limits and the numbers 4, 9, 14, 19, and 24 are the upper class limits. You can see these limits increase by 5. The class width can be determined as the difference between any two consecutive lower or upper class limits. The class mark is the midpoint of the class and is determined by averaging the lower and upper limits of the class. The class marks are shown in the third column of the table.

The modal class of a frequency distribution is the class with the highest frequency. Here the modal class is 15-19 with a frequency of 20 students. This grouping of the data allows us to more clearly see the grade distribution. Always be sure that the sum of the frequencies is the number of data values.

When the data are grouped using this structure, the modal class is 18–23.

Try it Now 3

The data below indicates number of children in a sample of 16 families:

2 1 2 1 2 5 5 3 2 3 5 2 5 2 2 1

• Create a non-grouped frequency table for the data.
• Create a grouped frequency table with first class 0-2. Identify the class width, the class mark for each class, and the modal class.

There is a "sort feature" on the TI calculator that sorts data in ascending or descending order for you. This makes organizing data and counting frequencies much easier. The steps for entering data and sorting it is shown here for the data presented in Try it 3 .

Let's consider the reverse situation when we have a frequency table with grouped data and determine information about the original data. This scenario is important because you will often see grouped data due to data storage capacities.

Answer the questions using the frequency table.

• What is the total number of data values in this data distribution?

Adding the frequencies of each class, we have \(4 + 7 + 1 + 0 +3 +5 = 20\).

• What class width is used to group the data?

Subtract any two consecutive lower class limits or any two consecutive upper class limits. For example, \(16 – 9 = 7\).

• What is the class mark of the second class ?

The class mark is the midpoint of the class. Average the lower and upper class limit: \(\frac{16+22}{2} = 19\).

• What is the modal class?

The class with the highest frequency is 16-22.

• If an additional class were added to the end of the table, what would be the upper and lower class limits?

Add the class width 7 to the last lower and upper class limits to get 51-57.

Graphing Quantitative Data in Histograms and Frequency Polygons

A histogram is a statistical graph commonly used to visualize frequency distributions of quantitative data. A histogram is like a bar graph, but where the horizontal axis is a number line.

A histogram is a graph with observed values or classes of values along the horizontal axis and frequencies along the vertical axis. A bar with a height equal to the frequency (or relative frequency) is built above each observed value or class.

In a histogram, classes may be identified by their class marks (midpoints of the classes) or by their class limits. The horizontal scale may or may not begin at 0, and but the vertical scale should always start at zero. The bars generally touch in a histogram - unless the frequency is 0 for a particular data value or class of values.

Let's illustrate how a histogram is constructed with the following example.

Each member of a class is asked how many plastic beverage bottles they use and discard in a week. Suppose the following (hypothetical) data are collected.

First, we organize the data by grouping it and presenting it in a frequency table. The classes have width 2 and begin at 1.

Next, we draw a bar for each class so that its height represents the frequency of students using those numbers of bottles. We label the midpoints of each bar with the class marks along the horizontal axis.

Graphing data can get tedious and complicated, especially if there are lots of data to organize. Excel and other software can easily make graphs. So can a TI graphing calculator. The steps to creating a histogram for these data is given below.

Suppose that we have collected weights from 100 male subjects as part of a nutrition study. For our weight data, we have values ranging from a low of 121 pounds to a high of 263 pounds, giving a total span of 263-121 = 142. We could create 7 intervals with a width of around 20, 14 intervals with a width of around 10, or somewhere in between. Often time we have to experiment with a few possibilities to find something that represents the data well. Let us try using an interval width of 15. We could start at 121, or at 120 since it is a nice round number.

\(\begin{array}{|c|c|} \hline \textbf { Interval } & \textbf { Frequency } \\ \hline 120-134 & 4 \\ \hline 135-149 & 14 \\ \hline 150-164 & 16 \\ \hline 165-179 & 28 \\ \hline 180-194 & 12 \\ \hline 195-209 & 8 \\ \hline 210-224 & 7 \\ \hline 225-239 & 6 \\ \hline 240-254 & 2 \\ \hline 255-269 & 3 \\ \hline \end{array}\)

A histogram of this data would look like

You can see the modal class is 165-179. You can also conclude there is a higher frequency of males in the lower part of the distribution of weights because the bars are taller there.

Try it Now 4

Create a histogram for the data given in Example 5 using the frequency table of ungrouped data.

Another way to visualize frequency distribution data is to construct a frequency polygon .

Frequency Polygon

An alternative representation of a histogram is a frequency polygon . A frequency polygon starts like a histogram, but instead of drawing a bar, a point is placed at the midpoint of each interval at a height equal to the frequency. Typically, the points are connected with straight lines to emphasize the shape of the data distribution.

The following example illustrates the relationship between a histogram and a frequency polygon for the same data.

Ms. Winter made a histogram and frequency polygon of the science test scores from 5 th period.

From either the histogram or the frequency polygon, we can see the class width is 10 points. We can also see that the modal class is 80-89. Finally, you can conclude that there is a larger frequency of students who scored high on the test than low on the test because the bars of the histogram and peak on the frequency polygon are taller on the right side of the horizontal axis.

The data below came from a task in which the goal is to move a computer mouse to a target on the screen as fast as possible. On 20 of the trials, the target was a small rectangle; on the other 20, the target was a large rectangle. Time to reach the target was recorded on each trial.

\(\begin{array}{|c|c|c|} \hline \begin{array}{c} \textbf { Interval } \\ \textbf { (milliseconds) } \end{array} & \begin{array}{c} \textbf { Frequency } \\ \textbf { small target } \end{array} & \begin{array}{c} \textbf { Frequency } \\ \textbf { large target } \end{array} \\ \hline 300-399 & 0 & 0 \\ \hline 400-499 & 1 & 5 \\ \hline 500-599 & 3 & 10 \\ \hline 600-699 & 6 & 5 \\ \hline 700-799 & 5 & 0 \\ \hline 800-899 & 4 & 0 \\ \hline 900-999 & 0 & 0 \\ \hline 1000-1099 & 1 & 0 \\ \hline 1100-1199 & 0 & 0 \\ \hline \end{array}\)

One option to represent this data would be a comparative histogram or bar chart, in which bars for the small target group and large target group are placed next to each other.

A pair of frequency polygons in the same graph for the same two sets of data makes it easier to see that reaction times were generally shorter for the larger target, and that the reaction times for the smaller target were more spread out.

In the next section, we will begin to analyze and describe data distributions numerically rather than graphically.

Frequency Distribution

A frequency distribution shows the frequency of repeated items in a graphical form or tabular form. It gives a visual display of the frequency of items or shows the number of times they occurred. Let's learn about frequency distribution in this article in detail.

What is Frequency Distribution?

Frequency distribution is used to organize the collected data in table form. The data could be marks scored by students, temperatures of different towns, points scored in a volleyball match, etc. After data collection, we have to show data in a meaningful manner for better understanding. Organize the data in such a way that all its features are summarized in a table. This is known as frequency distribution.

Let's consider an example to understand this better. The following are the scores of 10 students in the G.K. quiz released by Mr. Chris 15, 17, 20, 15, 20, 17, 17, 14, 14, 20. Let's represent this data in frequency distribution and find out the number of students who got the same marks.

We can see that all the collected data is organized under the column quiz marks and the number of students. This makes it easier to understand the given information and we can see that the number of students who obtained the same marks. Thus, frequency distribution in statistics helps us to organize the data in an easy way to understand its features at a glance.

Frequency Distribution Graphs

There is another way to show data that is in the form of graphs and it can be done by using a frequency distribution graph. The graphs help us to understand the collected data in an easy way. The graphical representation of a frequency distribution can be shown using the following:

• Bar Graphs : Bar graphs represent data using rectangular bars of uniform width along with equal spacing between the rectangular bars.
• Histograms : A histogram is a graphical presentation of data using rectangular bars of different heights. In a histogram, there is no space between the rectangular bars.
• Pie Chart : A pie chart is a type of graph that visually displays data in a circular chart. It records data in a circular manner and then it is further divided into sectors that show a particular part of data out of the whole part.
• Frequency Polygon: A frequency polygon is drawn by joining the mid-points of the bars in a histogram.

Types of Frequency Distribution

There are four types of frequency distribution under statistics which are explained below:

• Ungrouped frequency distribution: It shows the frequency of an item in each separate data value rather than groups of data values.
• Grouped frequency distribution: In this type, the data is arranged and separated into groups called class intervals. The frequency of data belonging to each class interval is noted in a frequency distribution table. The grouped frequency table shows the distribution of frequencies in class intervals.
• Relative frequency distribution: It tells the proportion of the total number of observations associated with each category.
• Cumulative frequency distribution: It is the sum of the first frequency and all frequencies below it in a frequency distribution. You have to add a value with the next value then add the sum with the next value again and so on till the last. The last cumulative frequency will be the total sum of all frequencies.
• Frequency Distribution Table

A frequency distribution table is a chart that shows the frequency of each of the items in a data set. Let's consider an example to understand how to make a frequency distribution table using tally marks. A jar containing beads of different colors- red, green, blue, black, red, green, blue, yellow, red, red, green, green, green, yellow, red, green, yellow. To know the exact number of beads of each particular color, we need to classify the beads into categories. An easy way to find the number of beads of each color is to use tally marks . Pick the beads one by one and enter the tally marks in the respective row and column. Then, indicate the frequency for each item in the table.

Thus, the table so obtained is called a frequency distribution table .

Types of Frequency Distribution Table

There are two types of frequency distribution tables: Grouped and ungrouped frequency distribution tables.

Grouped Frequency Distribution Table: To arrange a large number of observations or data, we use grouped frequency distribution table. In this, we form class intervals to tally the frequency for the data that belongs to that particular class interval.

For example, Marks obtained by 20 students in the test are as follows. 5, 10, 20, 15, 5, 20, 20, 15, 15, 15, 10, 10, 10, 20, 15, 5, 18, 18, 18, 18. To arrange the data in grouped table we have to make class intervals. Thus, we will make class intervals of marks like 0 – 5, 6 – 10, and so on. Given below table shows two columns one is of class intervals (marks obtained in test) and the second is of frequency (no. of students). In this, we have not used tally marks as we counted the marks directly.

Ungrouped Frequency Distribution Table: In the ungrouped frequency distribution table, we don't make class intervals, we write the accurate frequency of individual data. Considering the above example, the ungrouped table will be like this. Given below table shows two columns: one is of marks obtained in the test and the second is of frequency (no. of students).

Important Notes:

Following are the important points related to frequency distribution.

• Figures or numbers collected for some definite purpose is called data.
• Frequency is the value in numbers that shows how often a particular item occurs in the given data set.
• There are two types of frequency table - Grouped Frequency Distribution and Ungrouped Frequency Distribution.
• Data can be shown using graphs like histograms, bar graphs, frequency polygons, and so on.

Related Articles on Frequency Distribution

To learn more about the frequency distribution, check the given articles.

• Data Handling

Frequency Distribution Examples

Example 1: There are 20 students in a class. The teacher, Ms. Jolly, asked the students to tell their favorite subject. The results are as follows - Mathematics, English, Science, Science, Mathematics, Science, English, Art, Mathematics, Mathematics, Science, Art, Art, Science, Mathematics, Art, Mathematics, English, English, Mathematics.

Represent this data in the form of frequency distribution and identify the most-liked subject?

Solution: 20 students have indicated their choices of preferred subjects. Let us represent this data using tally marks. The tally marks are showing the frequency of each subject.

According to the above frequency distribution, mathematics is the most liked subject.

Example 2: 100 schools decided to plant 100 tree saplings in their gardens on world environment day. Represent the given data in the form of frequency distribution and find the number of schools that are able to plant 50% of the plants or more? 95, 67, 28, 32, 65, 65, 69, 33, 98, 96, 76, 42, 32, 38, 42, 40, 40, 69, 95, 92, 75, 83, 76, 83, 85, 62, 37, 65, 63, 42, 89, 65, 73, 81, 49, 52, 64, 76, 83, 92, 93, 68, 52, 79, 81, 83, 59, 82, 75, 82, 86, 90, 44, 62, 31, 36, 38, 42, 39, 83, 87, 56, 58, 23, 35, 76, 83, 85, 30, 68, 69, 83, 86, 43, 45, 39, 83, 75, 66, 83, 92, 75, 89, 66, 91, 27, 88, 89, 93, 42, 53, 69, 90, 55, 66, 49, 52, 83, 34, 36

Solution: To include all the observations in groups, we will create various groups of equal intervals. These intervals are called class intervals. In the frequency distribution, the number of plants survived is showing the class intervals, tally marks are showing frequency, and the number of schools is the frequency in numbers.

So, according to class intervals starting from 50 – 59 to 90 – 99, the frequency of schools able to retain 50% or more plants are 8 + 18 + 10 + 23 + 12 = 71 schools. Thus, 71 schools are able to retain 50% or more plants in their garden.

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Frequency Distribution Practice Questions

Faqs on frequency distribution, what is a frequency distribution in math.

In statistics, the frequency distribution is a graph or data set organized to represent the frequency of occurrence of each possible outcome of an event that is observed a specific number of times. Frequency distribution is a tabular or graphical representation of the data that shows the frequency of all the observations.

What are the 2 Types of Frequency Distribution Table?

The 2 types of frequency distributions are:

• Ungrouped frequency distribution
• Grouped frequency distribution

Why are Frequency Distributions Important?

Frequency charts are the best way to organize data. Doctors use it to understand the frequency of diseases. Sports analysts use it to understand the performance of a sportsperson. Wherever you have a large amount of data, frequency distribution makes it easy to analyze the data.

How do you find Frequency Distribution?

Follow the steps to find frequency distribution:

• Step 1: To make a frequency chart, first, write the categories in the first column.
• Step 2: In the next step, tally the score in the second column.
• Step 3: And finally, count the tally to write the frequency of each category in the third column.

Thus, in this way, we can find the frequency distribution of an event.

What is the Difference Between Frequency Table and Frequency Distribution?

The frequency table is a tabular method where the frequency is assigned to its respective category. Whereas, a frequency distribution is known as the graphical representation of the frequency table.

What is Grouped Frequency Distribution?

A grouped frequency distribution shows the scores by grouping the observations into intervals and then lists these intervals in the frequency distribution table. The intervals in grouped frequency distribution are called class limits.

What is Ungrouped Frequency Distribution?

The ungrouped frequency distribution is a type of frequency distribution that displays the frequency of each individual data value instead of groups of data values. In this type of frequency distribution, we can directly see how often different values occurred in the table.

What are the Components of Frequency Distribution?

The components of the frequency distribution are as follows:

• Class interval
• Types of class interval
• Class boundaries
• Midpoint or classmark
• Width or size of class interval
• Class frequency
• Frequency class width

• Mathematicians
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• Frequency Distribution – Definition and Examples

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What Is a Frequency Distribution?

Frequency distribution definition, frequency distribution examples, practice problems, frequency distribution – definition and examples.

A frequency distribution is any graph or table that shows the frequency of a set of data.

Graphical displays of frequency include histograms, dot plots, and stem and leaf plots. There are also different tabular displays, including joint frequency tables and listed tables. 

Statistics employs frequency distributions for data analysis. All fields that use statistics, including every branch of science and technology, employ frequency distributions.

Before moving on with this section, it is a good idea to review statistical frequency and frequency tables .

This section covers:

• What is a Frequency Distribution?

Frequency Distribution Table

A frequency distribution is a display of data. Specifically, it shows the number of times individual events occur.

Frequency distributions can be graphs or tables. Recall that quantitative data is data that records a number. These could include variables such as height, mass, distance, age, length of time, etc. This kind of data works best as a graph.

Also, recall that qualitative data is data that records non-numerical data. These could include day of the week, color, gender, yes/no responses, subject, etc. This kind of data tends to work well as a table.

A frequency distribution table, also called a frequency table, is a tabular display of data. Though frequency tables work well for qualitative data, they can also work for quantitative data, especially when ranges are used instead of individual data points.

There are many different ways to set up a frequency distribution table. Commonly, they will include data categories in a left-hand column or a top column. Then, the corresponding number of events will be in the column directly to the right or below the category.

Tables are great for displaying multivariate data that includes both numeric and qualitative data. These are data sets that have more than one corresponding entry for each point. For example, consider a survey of people that includes their age, sex, and favorite ice cream flavor. It has three pieces of data for each person in the survey.

A frequency distribution is any graph or table that presents the frequency of events within a data set. Sometimes lists are also considered a frequency distribution.

Examples of frequency distributions include dot plots, pie charts, histograms, bar plots, and frequency tables.

Sometimes, frequency distributions are also put into other categories. For example, frequency distributions can include grouped and ungrouped data. When data is ungrouped, it provides information about specific occurrences in the data. When data is grouped, it usually included information about ranges of data.

Frequency distributions can also be relative frequency distributions or cumulative frequency distributions. A relative frequency distribution shows the number of times each event occurs. On the other hand, a cumulative frequency distribution shows the total number of events up to that point.

For example, consider the following data about the height of 10 different students in inches:

$58, 58, 60, 60, 62, 63, 65, 65, 65, 70$

An ungrouped table looks like this.

A grouped table, however, looks like this.

Both of the above examples are relative frequency tables. That is, the right column shows how often each value or range occurs. A cumulative frequency table of the group values, though, looks like this.

This section covers common examples of problems involving frequency distributions and their step-by-step solutions

Categorize the frequency distribution shown in the histogram.

This example is a grouped frequency distribution with quantitative data. It is grouped because each bar represents the number of data points in a range. The first range is $0$ to $3$, the next is $3$ to $6$, then $6$ to $9$, and finally $9$ to $12$.

It is also a relative frequency distribution rather than a cumulative frequency distribution. It is clear because the last bar is shorter than the previous one. In a cumulative frequency distribution, the frequencies always increase.

The shape of the data is roughly normal because the data points are grouped around the center with fewer points to the left and right.

A restaurant owner surveyed everyone who came to her restaurant one day. She asked each person what their favorite menu item was. She got the following results:

Chicken tenders: 55 people

Salad: 12 people

Fries: 26 people

Pie: 72 people

Burger: 35 people

Use this data to create a pie chart.

Recall that a pie chart uses relative frequency to find the center angle for each “slice.”

That is, to find the center angle for each slice, first find the relative frequency and then multiply that by $360^{\circ}$.

$55+12+26+72+35=200$.

Therefore, the relative frequency for chicken tenders is $\frac{55}{200}$. Multiplying that by $360^{\circ}$ is $99$. Therefore, the pie slice corresponding to chicken tenders has a $99^{\circ}$ angle and represents $\frac{55}{200}=\frac{11}{40}$ths of the circle.

The angle for salad is $\frac{12}{200}\times360^{\circ}=21.6^{\circ}$.

Then, the angle for fries is $\frac{26}{200}\times360^{\circ}=46.8^{\circ}$.

$\frac{72}{200}\times360^{\circ}=129.6^{\circ}$ for pie.

$\frac{35}{200}\times360^{\circ}=63^{\circ}$ for burgers.

Suppose $150$ people of various ages were interviewed about the technology they own. Based on the table, find the relative frequency of each event stated. If there is not enough information, explain why.

A. A person owns a television

B. A person is between the ages of 0 and 10 and owns a cellphone

C. The person has a tablet but not a laptop.

A. To find the probability that a person owns a television, it is necessary to find the number of people who own televisions. Since this is stratified by age, add up all numbers in the column titled “television.” This is $1+4+20+15+10=50$. Thus, the probability is $\frac{50}{150}=\frac{1}{3}$.

B. In this case, it is necessary to find the number of people who both are between 0 and 10 and own a cellphone. This corresponds to the upper-right cell, $3$. Therefore, the probability is $\frac{3}{150}=\frac{1}{50}$.

C. There is not enough information to answer this question. Although the total number of people ($150$) is known, the overlap of owning the three technologies is not known.

Create a frequency table for the given data (mix of 2 subjects and 1 numerical)

There are 10 people interviewed about their pets. They were asked how many pets they had and whether they had at least one dog and at least one cat. These were the results:

($1$, n, n); ($4$, y, y); ($3$, y, y); ($3$, n, y); ($2$, y, y); ($3$, y, y); ($0$, n, n); ($2$, n, y); ($2$, n, y); ($1$, y, n).

Where the first number is the total number of pets, the second column regards whether the person had at least one dog with a “y” meaning “yes” and “n” meaning “no.” Similarly, the last column regarded whether the person had at least one cat.

In this case, it would make sense to include columns for each of the three questions. That is, include a column for the number of pets, a column for cats, and a column for dogs.

There are multiple ways to organize this data, but one way that makes sense is to organize from least to greatest by the total number of pets.

In addition to these questions, there is other information available in the data. For example, if a person said “no” to have at least one dog and at least one cat, then they have neither. Likewise, the total number of people asked is not equal to the number of people with dogs plus the number of people with cats plus the number of people with neither because there is some overlap with having both a dog and a cat.

Therefore, the table structure that makes sense is to have the number of pets on the left, the number in that range with dogs, the number in that range with cats, the number with neither, and the total number of people in that range.

This is the resulting table.

Use the given cumulative frequency distribution to determine the size of each range.

The first range is $0-100,000$. The corresponding bar has a height of $4$. Since this is the first bar, that means there are $4$ data points in the range of $0-100,000$.

The next bar has a height of $7$. Since this is a cumulative distribution, there are $7-4=3$ data points in the range of $100,000-200,000$.

Likewise, for $200,000-300,000$ there are $12-7=5$ data points.

For $300,000-400,000$, there are $16-12=4$ data points.

Then, for $400,000-500,000$ there are $18-16=2$ data points.

Finally, the last range of $500,000-600,000$ has one data point.

• Two field workers take the temperature of a body of water in different spots on the same day. One fieldworker has a thermometer that measures to the nearest hundredth of a degree, and the other has a thermometer that measures to the nearest tenth of a degree. They collect $250$ readings between the two of them. What type of graph would be best for this data and why?
• In a survey of $90$ students, 35 liked history class the best, 30 liked science the best, 15 liked math best, and everyone else liked art best. Create a pie chart for the data.

• This display is ambiguous. Each range could have an increasing number of data points, or it could be a cumulative distribution.
• The best type of frequency distribution for this data is a histogram. There are many data points, and they are measured to different points (tenths and hundredths), so a dot plot or table would be confusing. Using ranges with a histogram would work best.

• A. $\frac{30}{97}$ B. $\frac{25}{97}$ C. $\frac{50}{97}$
• $\frac{551}{24}$

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2.2: Organizing and Graphing Quantitative Data

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For most of the work you do in this course, you will be working with quantitative data, and you will use a frequency table and frequency histogram to organize and graph the data. An advantage of a frequency table and frequency histogram is that they can be used to organize and display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more.

Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows:

5; 6; 3; 3; 2; 4; 7; 5; 2; 3; 5; 6; 5; 4; 4; 3; 5; 2; 5; 3.

Table lists the different data values in ascending order and their frequencies.

Definition: Relative Frequency

A frequency is the number of times a value of the data occurs. According to Table \(\PageIndex{1}\), there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.

Definition: Relative frequencies

A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample–in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.

The sum of the values in the relative frequency column of Table \(\PageIndex{2}\) is \(\frac{20}{20}\), or 1.

Definition: Cumulative Relative Frequency

Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row, as shown in Table \(\PageIndex{3}\).

The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.

Table \(\PageIndex{4}\) represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.

The data in this table have been grouped into the following intervals:

• 61.95 to 63.95 inches
• 63.95 to 65.95 inches
• 65.95 to 67.95 inches
• 67.95 to 69.95 inches
• 69.95 to 71.95 inches
• 71.95 to 73.95 inches
• 73.95 to 75.95 inches

In this sample, there are five players whose heights fall within the interval 59.95–61.95 inches, three players whose heights fall within the interval 61.95–63.95 inches, 15 players whose heights fall within the interval 63.95–65.95 inches, 40 players whose heights fall within the interval 65.95–67.95 inches, 17 players whose heights fall within the interval 67.95–69.95 inches, 12 players whose heights fall within the interval 69.95–71.95, seven players whose heights fall within the interval 71.95–73.95, and one player whose heights fall within the interval 73.95–75.95. All heights fall between the endpoints of an interval and not at the endpoints.

Collaborative Exercise \(\PageIndex{7}\)

In your class, have someone conduct a survey of the number of siblings (brothers and sisters) each student has. Create a frequency table. Add to it a relative frequency column and a cumulative relative frequency column. Answer the following questions:

• What percentage of the students in your class have no siblings?
• What percentage of the students have from one to three siblings?
• What percentage of the students have fewer than three siblings?

Example \(\PageIndex{7}\)

Nineteen people were asked how many miles, to the nearest mile, they commute to work each day. The data are as follows: 2; 5; 7; 3; 2; 10; 18; 15; 20; 7; 10; 18; 5; 12; 13; 12; 4; 5; 10. Table \(\PageIndex{6}\) was produced:

• Is the table correct? If it is not correct, what is wrong?
• True or False: Three percent of the people surveyed commute three miles. If the statement is not correct, what should it be? If the table is incorrect, make the corrections.
• What fraction of the people surveyed commute five or seven miles?
• What fraction of the people surveyed commute 12 miles or more? Less than 12 miles? Between five and 13 miles (not including five and 13 miles)?
• No. The frequency column sums to 18, not 19. Not all cumulative relative frequencies are correct.
• False. The frequency for three miles should be one; for two miles (left out), two. The cumulative relative frequency column should read: 0.1052, 0.1579, 0.2105, 0.3684, 0.4737, 0.6316, 0.7368, 0.7895, 0.8421, 0.9474, 1.0000.
• \(\frac{5}{19}\)
• \(\frac{7}{19}\), \(\frac{12}{19}\), \(\frac{7}{19}\)

A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data.

The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample.(Remember, frequency is defined as the number of times an answer occurs.) If:

• \(f\) is frequency
• \(n\) is total number of data values (or the sum of the individual frequencies), and
• \(RF\) is relative frequency,

\[RF=\dfrac{f}{n} \label{2.3.1}\]

For example, if three students in Mr. Ahab's English class of 40 students received from 90% to 100%, then, f = 3, n = 40, and RF = fn = 340 = 0.075. 7.5% of the students received 90–100%. 90–100% are quantitative measures.

To construct a histogram, first decide how many bars or intervals, also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is 6.1 and this is the smallest value, a convenient starting point is \(6.05 (6.1 – 0.05 = 6.05)\). We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is \(1.495 (1.5 – 0.005 = 1.495)\). If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is \(0.9995 (1.0 – 0.0005 = 0.9995)\). If all the data happen to be integers and the smallest value is two, then a convenient starting point is \(1.5 (2 - 0.5 = 1.5)\). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.

Example \(\PageIndex{1}\)

The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data, since height is measured.

60; 60.5; 61; 61; 61.5

63.5; 63.5; 63.5

64; 64; 64; 64; 64; 64; 64; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5

66; 66; 66; 66; 66; 66; 66; 66; 66; 66; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5

68; 68; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69.5; 69.5; 69.5; 69.5; 69.5

70; 70; 70; 70; 70; 70; 70.5; 70.5; 70.5; 71; 71; 71

72; 72; 72; 72.5; 72.5; 73; 73.5

The smallest data value is 60. Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, etc. are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point.

60 – 0.05 = 59.95 which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95.

The largest value is 74, so 74 + 0.05 = 74.05 is the ending value.

Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.

We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals.

The boundaries are:

• 59.95 + 2 = 61.95
• 61.95 + 2 = 63.95
• 63.95 + 2 = 65.95
• 65.95 + 2 = 67.95
• 67.95 + 2 = 69.95
• 69.95 + 2 = 71.95
• 71.95 + 2 = 73.95
• 73.95 + 2 = 75.95

The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95.

The following histogram displays the heights on the x -axis and relative frequency on the y -axis.

Example \(\PageIndex{2}\)

The following data are the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data , since books are counted.

1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1 2; 2; 2; 2; 2; 2; 2; 2; 2; 2 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3 4; 4; 4; 4; 4; 4 5; 5; 5; 5; 5 6; 6

Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.

Because the data are integers, subtract 0.5 from 1, the smallest data value and add 0.5 to 6, the largest data value. Then the starting point is 0.5 and the ending value is 6.5.

Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers 1, 2, 3, 4, 5, 6, and the starting point is 0.5, a width of one places the 1 in the middle of the interval from 0.5 to 1.5, the 2 in the middle of the interval from 1.5 to 2.5, the 3 in the middle of the interval from 2.5 to 3.5, the 4 in the middle of the interval from _______ to _______, the 5 in the middle of the interval from _______ to _______, and the _______ in the middle of the interval from _______ to _______ .

Calculate the number of bars as follows:

\(\frac{6.5 - 0.5}{\text{number of bars}}\) = 1

where 1 is the width of a bar. Therefore, bars = 6.

The following histogram displays the number of books on the x -axis and the frequency on the y -axis.

Example \(\PageIndex{3}\)

Using this data set, construct a histogram.

Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.

Frequency Polygons

Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons. To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the x -axis and y -axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.

Example \(\PageIndex{4}\)

A frequency polygon was constructed from the frequency table below.

The first label on the x -axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 54.5 represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.

One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.

Constructing a Time Series Graph

To construct a time series graph, we must look at both pieces of our paired data set . We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.

Example \(\PageIndex{6}\)

The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.

Uses of a Time Series Graph

Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.

A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually goes on y -axis with the frequency being graphed on the x -axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time.Glossary

WeBWorK Problems

Query \(\PageIndex{1}\)

Query \(\PageIndex{2}\)

Query \(\PageIndex{3}\)

Query \(\PageIndex{4}\)

• Data on annual homicides in Detroit, 1961–73, from Gunst & Mason’s book ‘Regression Analysis and its Application’, Marcel Dekker
• “Timeline: Guide to the U.S. Presidents: Information on every president’s birthplace, political party, term of office, and more.” Scholastic, 2013. Available online at www.scholastic.com/teachers/a...-us-presidents (accessed April 3, 2013).
• “Presidents.” Fact Monster. Pearson Education, 2007. Available online at http://www.factmonster.com/ipka/A0194030.html (accessed April 3, 2013).
• “Food Security Statistics.” Food and Agriculture Organization of the United Nations. Available online at http://www.fao.org/economic/ess/ess-fs/en/ (accessed April 3, 2013).
• “Consumer Price Index.” United States Department of Labor: Bureau of Labor Statistics. Available online at http://data.bls.gov/pdq/SurveyOutputServlet (accessed April 3, 2013).
• “CO2 emissions (kt).” The World Bank, 2013. Available online at http://databank.worldbank.org/data/home.aspx (accessed April 3, 2013).
• “Births Time Series Data.” General Register Office For Scotland, 2013. Available online at www.gro-scotland.gov.uk/stati...me-series.html (accessed April 3, 2013).
• “Demographics: Children under the age of 5 years underweight.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (accessed April 3, 2013).
• Gunst, Richard, Robert Mason. Regression Analysis and Its Application: A Data-Oriented Approach . CRC Press: 1980.
• “Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013).

Contributors and Attributions

Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/[email protected] .

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Goodman, L. A. (1954). Kolmogorov-Smirnov tests for psychological research. Psychological Bulletin, 51 , 160–168.

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Siegel, S. (1956). Nonparametric statistics for the behavioral sciences . New York/Toronto/London: McGraw-Hill Book Company.

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• Math Article

Graphical Representation

Graphical Representation is a way of analysing numerical data. It exhibits the relation between data, ideas, information and concepts in a diagram. It is easy to understand and it is one of the most important learning strategies. It always depends on the type of information in a particular domain. There are different types of graphical representation. Some of them are as follows:

• Line Graphs – Line graph or the linear graph is used to display the continuous data and it is useful for predicting future events over time.
• Bar Graphs – Bar Graph is used to display the category of data and it compares the data using solid bars to represent the quantities.
• Histograms – The graph that uses bars to represent the frequency of numerical data that are organised into intervals. Since all the intervals are equal and continuous, all the bars have the same width.
• Line Plot – It shows the frequency of data on a given number line. ‘ x ‘ is placed above a number line each time when that data occurs again.
• Frequency Table – The table shows the number of pieces of data that falls within the given interval.
• Circle Graph – Also known as the pie chart that shows the relationships of the parts of the whole. The circle is considered with 100% and the categories occupied is represented with that specific percentage like 15%, 56%, etc.
• Stem and Leaf Plot – In the stem and leaf plot, the data are organised from least value to the greatest value. The digits of the least place values from the leaves and the next place value digit forms the stems.
• Box and Whisker Plot – The plot diagram summarises the data by dividing into four parts. Box and whisker show the range (spread) and the middle ( median) of the data.

General Rules for Graphical Representation of Data

There are certain rules to effectively present the information in the graphical representation. They are:

• Suitable Title: Make sure that the appropriate title is given to the graph which indicates the subject of the presentation.
• Measurement Unit: Mention the measurement unit in the graph.
• Proper Scale: To represent the data in an accurate manner, choose a proper scale.
• Index: Index the appropriate colours, shades, lines, design in the graphs for better understanding.
• Data Sources: Include the source of information wherever it is necessary at the bottom of the graph.
• Keep it Simple: Construct a graph in an easy way that everyone can understand.
• Neat: Choose the correct size, fonts, colours etc in such a way that the graph should be a visual aid for the presentation of information.

Graphical Representation in Maths

In Mathematics, a graph is defined as a chart with statistical data, which are represented in the form of curves or lines drawn across the coordinate point plotted on its surface. It helps to study the relationship between two variables where it helps to measure the change in the variable amount with respect to another variable within a given interval of time. It helps to study the series distribution and frequency distribution for a given problem.  There are two types of graphs to visually depict the information. They are:

• Time Series Graphs – Example: Line Graph
• Frequency Distribution Graphs – Example: Frequency Polygon Graph

Principles of Graphical Representation

Algebraic principles are applied to all types of graphical representation of data. In graphs, it is represented using two lines called coordinate axes. The horizontal axis is denoted as the x-axis and the vertical axis is denoted as the y-axis. The point at which two lines intersect is called an origin ‘O’. Consider x-axis, the distance from the origin to the right side will take a positive value and the distance from the origin to the left side will take a negative value. Similarly, for the y-axis, the points above the origin will take a positive value, and the points below the origin will a negative value.

Generally, the frequency distribution is represented in four methods, namely

• Smoothed frequency graph
• Pie diagram
• Cumulative or ogive frequency graph
• Frequency Polygon

Merits of Using Graphs

Some of the merits of using graphs are as follows:

• The graph is easily understood by everyone without any prior knowledge.
• It saves time
• It allows us to relate and compare the data for different time periods
• It is used in statistics to determine the mean, median and mode for different data, as well as in the interpolation and the extrapolation of data.

Example for Frequency polygonGraph

Here are the steps to follow to find the frequency distribution of a frequency polygon and it is represented in a graphical way.

• Obtain the frequency distribution and find the midpoints of each class interval.
• Represent the midpoints along x-axis and frequencies along the y-axis.
• Plot the points corresponding to the frequency at each midpoint.
• Join these points, using lines in order.
• To complete the polygon, join the point at each end immediately to the lower or higher class marks on the x-axis.

Draw the frequency polygon for the following data

Mark the class interval along x-axis and frequencies along the y-axis.

Let assume that class interval 0-10 with frequency zero and 90-100 with frequency zero.

Now calculate the midpoint of the class interval.

Using the midpoint and the frequency value from the above table, plot the points A (5, 0), B (15, 4), C (25, 6), D (35, 8), E (45, 10), F (55, 12), G (65, 14), H (75, 7), I (85, 5) and J (95, 0).

To obtain the frequency polygon ABCDEFGHIJ, draw the line segments AB, BC, CD, DE, EF, FG, GH, HI, IJ, and connect all the points.

Frequently Asked Questions

What are the different types of graphical representation.

Some of the various types of graphical representation include:

• Line Graphs
• Frequency Table
• Circle Graph, etc.

Read More:  Types of Graphs

What are the Advantages of Graphical Method?

Some of the advantages of graphical representation are:

• It makes data more easily understandable.
• It saves time.
• It makes the comparison of data more efficient.

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Very useful for understand the basic concepts in simple and easy way. Its very useful to all students whether they are school students or college sudents

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GRAPHICAL REPRESENTATION OF DATA

An attractive representation of a frequency distribution is graphical representation. 

Graphical representation can be used for both the educated section and uneducated section of the society. Furthermore, any hidden trend present in the given data can be noticed only in this mode of representation.

We are going to consider the following types of graphical representation :

1.  Line diagram

2. Histogram

3.  Bar diagram

4.  Pie chart

5. Frequency polygon

6. Ogives or Cumulative frequency graphs

Line diagram

When the time series exhibit a wide range of fluctuations, we may think of logarithmic or ratio chart where "Log y" and not "y" is plotted against "t".

We use Multiple line chart for representing two or more related time series data expressed in the same unit and multiple – axis chart in somewhat similar situations, if the variables are expressed in different units.

Example 1 :

The profits in thousand of dollars of an industrial house for 2002, 2003, 2004, 2005, 2006, 2007 and 2008 are 5, 8, 9, 6, 12, 15 and 24 respectively. Represent these data using a suitable diagram.

We can represent the profits for 7 consecutive years by drawing either a line diagram as given below.

Let us consider years on horizontal axis and profits on vertical axis.

For the year 2002, the profit is 5 thousand dollars. It can be written as a point (2002, 5)

In the same manner, we can write the following points for the succeeding years.

(2003, 8), (2004, 9), (2005, 6), (2006, 12), (2007, 15) and (2008, 24)

Now, plotting all these point and joining them using ruler, we can get the line diagram.

Showing line diagram for the profit of an Industrial House during 2002 to 2008.

A two dimensional graphical representation of a continuous frequency  distribution is called a histogram.

In histogram, the bars are placed continuously side by side with no gap between  adjacent bars.

That is, in histogram rectangles are erected on the class intervals of  the distribution. The areas of rectangle are proportional to the frequencies.

Example 2 :

Draw a histogram for the following table which represent the marks obtained by 100 students in an examination :

The class intervals are all equal with length of 10 marks.

Let us denote these class intervals along the X-axis.

Denote the number of students along the Y-axis, with appropriate scale.

The histogram is given below.

Bar diagram

There are two types of bar diagrams namely, Horizontal Bar diagram and Vertical bar  diagram.

While horizontal bar diagram is used for qualitative data or data varying over  space, the vertical bar diagram is associated with quantitative data or time series data.

Bars i.e. rectangles of equal width and usually of varying lengths are drawn either  horizontally or vertically.

We consider Multiple or Grouped Bar diagrams to compare  related series. Component or sub-divided Bar diagrams are applied for representing data  divided into a number of components. Finally, we use Divided Bar charts or Percentage

Bar diagrams for comparing different components of a variable and also the relating of  the components to the whole. For this situation, we may also use Pie chart or Pie diagram  or circle diagram.

Example 3 :

The total number of runs scored by a few players in one-day match is given.

Draw bar graph for the above data.

In a pie chart, the various observations or components are represented by the sectors of a circle and the whole circle represents the sum of the value of all the components .Clearly, the total angle of 360° at the center of the circle is divided according to the values of the components .

The central angle of a component is

= [Value of the component/Total value] x 360°

Sometimes, the value of the components are expressed in percentages. In such cases,

= [Percentage value of the component/100] x 360°

Example 4 :

The number of hours spent by a school student on various activities on a working day, is given below. Construct a pie chart using the angle measurement.

Draw a pie chart to represent the above information.

We may calculate the central angles for various components as follows :

From the above table, clearly, we obtain the required pie chart as shown below.

Frequency polygon

Frequency Polygon is another method of representing frequency distribution  graphically.

Obtain the frequency distribution and compute the mid points of  each class interval.

Represent the mid points along the X-axis and the frequencies along  the Y-axis.

Plot the points corresponding to the frequency at each mid point.

Join these points, by straight lines in order.

To complete the polygon join the point at each end immediately to the lower or higher class marks (as the case may be at zero frequency) on the X-axis.

Example 5 :

Draw a frequency polygon for the following data without using histogram.

Mark the class intervals along the X-axis and the frequency along the Y-axis.

We take the imagined classes 0-10 at the beginning and 90-100 at the end, each with frequency zero.

We have tabulated which is given below. 

Using the adjacent table, plot the points A (5, 0), B (15, 4), C (25, 6), D (35, 8), E (45, 10), F (55, 12), G (65, 14), H (75, 7), I (85, 5) and J (95, 0).

We draw the line segments AB, BC, CD, DE, EF, FG, GH, HI, IJ to obtain the required frequency polygon ABCDEFGHIJ, which is given below.

Ogives or Cumulative frequency graphs

By plotting cumulative frequency against the respective class boundary, we get ogives.

As such there are two ogives – less than type ogives, obtained by taking less than cumulative frequency on the vertical axis and more than type ogives by plotting more than type cumulative frequency on the vertical axis and thereafter joining the plotted points successively by line segments.

Example 6 :

Draw ogives  for the following table which represents the frequency distribution of weights of 36 students. 

To draw ogives for the above frequency distribution, we have to write less than and more than cumulative frequency as given below.

Now, we have to write the points from less than and more than cumulative frequency as given below.

Points from less than cumulative frequency :

(43.50, 0), (48.50, 3), (53.50, 7), (58.50, 12), (63.50, 19), (68.50, 28) and (73.50, 36)

Points from more cumulative frequency :

(43.50, 36 (48.50, 33), (53.50, 29), (58.50, 24), (63.50, 17), (68.50, 8) and (73.50, 0)

Now, taking frequency on the horizontal axis, weights on vertical axis and plotting the above points, we get ogives as given below.

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Chapter Questions

A set of data consists of 38 observations. How many classes would you recommend for the frequency distribution?

A set of data consists of 45 observations between $\$ 0$ and $\$ 29 .$ What size would you recommend for the class interval?

A set of data consists of 230 observations between $\$ 235$ and $\$ 567 .$ What class interval would you recommend?

A set of data contains 53 observations. The lowest value is 42 and the largest is $129 .$ The data are to be organized into a frequency distribution. a. How many classes would you suggest? b. What would you suggest as the lower limit of the first class?

Wachesaw Manufacturing, Inc. produced the following number of units the last 16 days. $$ \begin{array}{|llllllll|} \hline 27 & 27 & 27 & 28 & 27 & 25 & 25 & 28 \\ 26 & 28 & 26 & 28 & 31 & 30 & 26 & 26 \\ \hline \end{array} $$ The information is to be organized into a frequency distribution. a. How many classes would you recommend? b. What class interval would you suggest? c. What lower limit would you recommend for the first class? d. Organize the information into a frequency distribution and determine the relative frequency distribution. e. Comment on the shape of the distribution.

The Quick Change Oil Company has a number of outlets in the metropolitan Seattle area. The numbers of oil changes at the Oak Street outlet in the past 20 days are:

The manager of the BiLo Supermarket in Mt. Pleasant, Rhode Island, gathered the following information on the number of times a customer visits the store during a month. The responses of 51 customers were: a. Starting with 0 as the lower limit of the first class and using a class interval of 3 , organize the data into a frequency distribution. b. Describe the distribution. Where do the data tend to cluster? c. Convert the distribution to a relative frequency distribution.

The food services division of Cedar River Amusement Park, Inc. is studying the amount families who visit the amusement park spend per day on food and drink. A sample of 40 families who visited the park yesterday revealed they spent the following amounts. a. Organize the data into a frequency distribution, using seven classes and 15 as the lower limit of the first class. What class interval did you select? b. Where do the data tend to cluster? c. Describe the distribution. d. Determine the relative frequency distribution.

Molly's Candle Shop has several retail stores in the coastal areas of North and South Carolina. Many of Molly's customers ask her to ship their purchases. The following chart shows the number of packages shipped per day for the last 100 days. a. What is this chart called? b. What is the total number of frequencies? c. What is the class interval? d. What is the class frequency for the 10 up to 15 class? e. What is the relative frequency of the 10 up to 15 class? f. What is the midpoint of the 10 up to 15 class? g. On how many days were there 25 or more packages shipped?

The audit staff of Southeast Fire and Casualty, Inc. recently completed a study of the settlement amount, in $\$ 000,$ of claims. The staff report included the following chart. a. What is the midpoint of the 2 up to 4 class? b. How many claims were in the 2.0 up to 4.0 class? c. Approximately how many claims were studied? d. What is the class interval? e. What is this chart called?

The following frequency distribution reports the number of frequent flier miles, reported in thousands, for employees of Brumley Statistical Consulting, Inc. during the first quarter of 2004 $$ \begin{array}{|cc|} \hline \begin{array}{c} \text { Frequent Flier Miles } \\ \text { (000) } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Employees } \end{array} \\ \hline 0 \text { up to } 3 & 5 \\ 3 \text { up to } 6 & 12 \\ 6 \text { up to } 9 & 23 \\ 9 \text { up to } 12 & 8 \\ 12 \text { up to } 15 & 2 \\ \text { Total } & \frac{2}{50} \\ \hline \end{array} $$ a. How many employees were studied? b. What is the midpoint of the first class? c. Construct a histogram. d. A frequency polygon is to be drawn. What are the coordinates of the plot for the first class? e. Construct a frequency polygon. f. Interpret the frequent flier miles accumulated using the two charts.

Ecommerce.com, a large Internet retailer, is studying the lead time (elapsed time between when an order is placed and when it is filled) for a sample of recent orders. The lead times are reported in days. $$ \begin{array}{|cc|} \hline \text { Lead Time (days) } & \text { Frequency } \\ \hline 0 \text { up to } 5 & 6 \\ 5 \text { up to } 10 & 7 \\ 10 \text { up to } 15 & 12 \\ 15 \text { up to } 20 & 8 \\ 20 \text { up to } 25 & \frac{7}{40} \\ \text { Total } & 40 \\ \hline \end{array} $$ a. How many orders were studied? b. What is the midpoint of the first class? c. What are the coordinates of the first class for a frequency polygon? d. Draw a histogram. e. Draw a frequency polygon. f. Interpret the lead times using the two charts.

The following chart shows the hourly wages of a sample of certified welders in the Atlanta, Georgia, area. a. How many welders were studied? b. What is the class interval? c. About how many welders earn less than $\$ 10.00$ per hour? d. About 75 percent of the welders make less than what amount? e. Ten of the welders studied made less than what amount? f. What percent of the welders make less than $\$ 20.00$ per hour?

The following chart shows the selling price (\$000) of houses sold in the Billings, Montana, area. a. How many homes were studied? b. What is the class interval? c. One hundred homes sold for less than what amount? d. About 75 percent of the homes sold for less than what amount? e. Estimate the number of homes in the $\$ 150,000$ up to $\$ 200,000$ class. f. About how many homes sold for less than $\$ 225.000 ?$

The frequency distribution representing the number of frequent flier miles accumulated by employees at Brumley Statistical Consulting Company is repeated from Exercise $11 .$ $$ \begin{array}{|cc|} \hline \begin{array}{c} \text { Frequent Flier Miles } \\ (000) \end{array} & \text { Frequency } \\ \hline 0 \text { up to } 3 & 5 \\ 3 \text { up to } 6 & 12 \\ 6 \text { up to } 9 & 23 \\ 9 \text { up to } 12 & 8 \\ 12 \text { up to } 15 & 2 \\ \text { Total } & \frac{2}{50} \\ \hline \end{array} $$a. How many employees accumulated less than 3,000 miles? b. Convert the frequency distribution to a cumulative frequency distribution. c. Portray the cumulative distribution in the form of a cumulative frequency polygon. d. Based on the cumulative frequency polygon, about 75 percent of the employees accumulated how many miles or less?

The frequency distribution of order lead time at Ecommerce.com from Exercise 12 is repeated below. $$ \begin{array}{|cc|} \hline \text { Lead Time (days) } & \text { Frequency } \\ \hline 0 \text { up to } 5 & 6 \\ 5 \text { up to } 10 & 7 \\ 10 \text { up to } 15 & 12 \\ 15 \text { up to } 20 & 8 \\ 20 \text { up to } 25 & 7 \\ \text { Total } & 40 \\ \hline \end{array} $$ a. How many orders were filled in less than 10 days? In less than 15 days? b. Convert the frequency distribution to a cümulative frequency distribution. c. Develop a cumulative frequency polygon. d. About 60 percent of the orders were filled in less than how many days?

A small business consultant is investigating the performance of several companies. The sales in 2004 (in thousands of dollars) for the selected companies were: $$ \begin{array}{|lr|} \hline \text { Corporation } & \begin{array}{c} \text { Fourth-Quarter Sales } \\ \text { (\$ thousands) } \end{array} \\ \hline \text { Hoden Building Products } & \$ 1,645.2 \\ \text { J \& R Printing, Inc. } & 4,757.0 \\ \text { Long Bay Concrete Construction } & 8,913.0 \\ \text { Mancell Electric and Plumbing } & 627.1 \\ \text { Maxwell Heating and Air Conditioning } & 24,612.0 \\ \text { Mizelle Roofing \& Sheet Metals } & 191.9 \\ \hline \end{array} $$ The consultant wants to include a chart in his report comparing the sales of the six companies. Use a bar chart to compare the fourth quarter sales of these corporations and write a brief report summarizing the bar chart.

The Blair Corporation, located in Warren, Pennsylvania, sells fashion apparel for men and women plus a broad range of home products (http://www.blair.com). It services its customers by mail. Listed below are the net sales for Blair from 1998 through $2003 .$ Draw a line chart depicting the net sales over the time period and write a brief report. $$ \begin{array}{|lc|} \hline \text { Year } & \begin{array}{c} \text { Net Sales } \\ \text { (\$ millions) } \end{array} \\ \hline 1998 & 506.8 \\ 1999 & 522.2 \\ 2000 & 574.6 \\ 2001 & 580.7 \\ 2002 & 568.5 \\ 2003 & 581.9 \\ \end{array} $$

A headline in a Toledo, Ohio, newspaper reported that crime was on the decline. Listed below are the number of homicides from 1986 to $2003 .$ Draw a line chart to summarize the data and write a brief summary of the homicide rates for the last 18 years. $$ \begin{array}{|ccccc|} \hline \text { Year } & \text { Homicides } & & \text { Year } & \text { Homicides } \\ \hline 1986 & 21 & & 1995 & 35 \\ 1987 & 34 & & 1996 & 30 \\ 1988 & 26 & & 1997 & 28 \\ 1989 & 42 & & 1998 & 25 \\ 1990 & 37 & & 1999 & 21 \\ 1991 & 37 & & 2000 & 19 \\ 1992 & 44 & & 2001 & 23 \\ 1993 & 45 & & 2002 & 27 \\ 1994 & 40 & & 2003 & 23 \\ \hline \end{array} $$

A report prepared for the governor of a western state indicated that 56 percent of the state's tax revenue went to education, 23 percent to the general fund, 10 percent to the counties, 9 percent to senior programs, and the remainder to other social programs. Develop a pie chart to show the breakdown of the budqet.

The following table shows the population, in millions, of the United States in 5 -year intervals from 1950 to $2000 .$ Develop a line chart depicting the population growth and write a brief report summarizing your findings. $$ \begin{array}{|ccccc|} \hline \text { Year } & \begin{array}{c} \text { Population } \\ \text { (millions) } \end{array} & \text { Year } & \begin{array}{c} \text { Population } \\ \text { (millions) } \end{array} \\ \hline 1950 & 152.3 & 1980 & 227.7 \\ 1955 & 165.9 & 1985 & 238.5 \\ 1960 & 180.7 & 1990 & 249.9 \\ 1965 & 194.3 & 1995 & 263.0 \\ 1970 & 205.1 & 2000 & 281.4 \\ 1975 & 216.0 & & \\ \hline \end{array} $$

Shown below are the military and civilian personnel expenditures for the eight largest military locations in the United States. Develop a bar chart and summarize the results in a brief report. $$ \begin{array}{|lclc|} \hline \text { Location } & \begin{array}{c} \text { Amount Spent } \\ \text { (millions) } \end{array} & \text { Location } & \begin{array}{c} \text { Amount Spent } \\ \text { (millions) } \end{array} \\ \hline \text { St. Louis, M0 } & \$ 6,087 & \text { Norfolk, VA } & \$ 3,228 \\ \text { San Diego, CA } & 4,747 & \text { Marietta, GA } & 2,828 \\ \text { Pico Rivera, CA } & 3,272 & \text { Fort Worth, TX } & 2,492 \\ \text { Arlington, VA } & 3,284 & \text { Washington, DC } & 2,347 \\ \hline \end{array} $$

A dataset consists of 83 observations. How many classes would you recommend for a frequency distribution?

A dataset consists of 145 observations that range from 56 to $490 .$ What size class interval would you recommend?

The following is the number of minutes to commute from home to work for a sample of aerospace workers in Houston, Texas. $$ \begin{array}{|lllllllllllll|} \hline 28 & 25 & 48 & 37 & 41 & 19 & 32 & 26 & 16 & 23 & 23 & 29 & 36 \\ 31 & 26 & 21 & 32 & 25 & 31 & 43 & 35 & 42 & 38 & 33 & 28 & \\ \hline \end{array} $$ a. How many classes would you recommend? b. What class interval would you suggest? c. What would you recommend as the lower limit of the first class? d. Organize the data into a frequency distribution. e. Comment on the shape of the frequency distribution.

The following data give the weekly amounts spent on groceries for a sample of households in rural Wisconsin. $$ \begin{array}{|rrrrrrrrr|} \hline \$ 271 & \$ 363 & \$ 159 & \$ 76 & \$ 227 & \$ 337 & \$ 295 & \$ 319 & \$ 250 \\ 279 & 205 & 279 & 266 & 199 & 177 & 162 & 232 & 303 \\ 192 & 181 & 321 & 309 & 246 & 278 & 50 & 41 & 335 \\ 116 & 100 & 151 & 240 & 474 & 297 & 170 & 188 & 320 \\ 429 & 294 & 570 & 342 & 279 & 235 & 434 & 123 & 325 \\ \hline \end{array} $$ a. How many classes would you recommend? b. What class interval would you suggest? c. What would you recommend as the lower limit of the first class? d. Organize the data into a frequency distribution.

The following histogram shows the scores on the first statistics exam. a. How many students took the exam? b. What is the class interval? c. What is the class midpoint for the first class? d. How many students earned a score of less than $70 ?$

The following chart summarizes the selling price of homes sold last month in the Sarasota, Florida, area. a. What is the chart called? b. How many homes were sold during the last month? c. What is the class interval? d. About 75 percent of the houses sold for less than what amount? e. One hundred seventy-five of the homes sold for less than what amount?

A chain of sport shops catering to beginning skiers, headquartered in Aspen, Colorado, plans to conduct a study of how much a beginning skier spends on his or her initial purchase of equipment and supplies. Based on these figures, they want to explore the possibility of offering combinations, such as a pair of boots and a pair of skis, to induce customers to buy more. A sample of their cash register receipts revealed these initial purchases: $$ \begin{array}{rrrrrrrrr} \hline \$ 140 & \$ 82 & \$ 265 & \$ 168 & \$ 90 & \$ 114 & \$ 172 & \$ 230 & \$ 142 \\ 86 & 125 & 235 & 212 & 171 & 149 & 156 & 162 & 118 \\ 139 & 149 & 132 & 105 & 162 & 126 & 216 & 195 & 127 \\ 161 & 135 & 172 & 220 & 229 & 129 & 87 & 128 & 126 \\ 175 & 127 & 149 & 126 & 121 & 118 & 172 & 126 & \\ \hline \end{array} $$ a. Arrive at a suggested class interval. Use five classes, and let the lower limit of the first class be $\$ 80$. b. What would be a better class interval? c. Organize the data into a frequency distribution using a lower limit of $\$ 80$. d. Interpret your findings.

The numbers of shareholders for a selected group of large companies (in thousands) are: $$ \begin{array}{|lclc|} \hline & \begin{array}{c} \text { Number of } \\ \text { Shareholders } \\ \text { (thousands) } \end{array} & \text { Company } & \begin{array}{c} \text { Number of } \\ \text { Shareholders } \\ \text { (thousands) } \end{array} \\ \text { Company } & 144 & \text { Standard Oil (Indiana) } & 173 \\ \text { Southwest Airlines } & 177 & \text { Home Depot } & 195 \\ \text { General Public Utilities } & 266 & \text { Detroit Edison } & 220 \\ \text { Occidental Petroleum } & 133 & \text { Eastman Kodak } & 251 \\ \text { Middle South Utilities } & 209 & \text { Dow Chemical } & 137 \\ \text { DaimlerChrysler } & 264 & \text { Pennsylvania Power } & 150 \\ \text { Standard Oil of California } & 160 & \text { American Electric Power } & 262 \\ \text { Bethlehem Steel } & 143 & \text { Ohio Edison } & 158 \\ \text { Long Island Lighting } & 246 & \text { Transamerica Corporation } & 162 \\ \text { RCA } & 151 & \text { Columbia Gas System } & 165 \\ \text { Greyhound Corporation } & 239 & \text { International Telephone \& } & \\ \text { Pacific Gas \& Electric } & 204 & \text { Telegraph } & 223 \\ \text { Niagara Mohawk Power } & 204 & \text { Union Electric } & 158 \\ \text { E.l: du Pont de Nemours } & 195 & \text { Virginia Electric and Power } & 162 \\ \text { Morris Knudsen Corporation } & 176 & \text { Public Service Electric \& Gas } & 225 \\ \text { Union Carbide } & 175 & \text { Consumers Power } & 161 \\ \text { BankAmerica } & 200 & & \\ \text { Northeast Utilities } & & & & \\ \hline \end{array} $$ The numbers of shareholders are to be organized into a frequency distribution and several graphs drawn to portray the distribution. a. Using seven classes and a lower limit of $130,$ construct a frequency distribution. b. Portray the distribution as a frequency polygon. c. Portray the distribution in a cumulative frequency polygon. d. According to the polygon, three out of four $(75$ percent) of the companies have how many shareholders or less? e. Write a brief analysis of the number of shareholders based on the frequency distribution and graphs.

A recent survey showed that the typical American car owner spends $\$ 2,950$ per year on operating expenses. Below is a breakdown of the various expenditure items. Draw an appropriate chart to portray the data and summarize your findings in a brief report.

The Midland National Bank selected a sample of 40 student checking accounts. Below are their end-of-the-month balances. $$ \begin{array}{rrrrrrrrrr} \hline \$ 404 & \$ 74 & \$ 234 & \$ 149 & \$ 279 & \$ 215 & \$ 123 & \$ 55 & \$ 43 & \$ 321 \\ 87 & 234 & 68 & 489 & 57 & 185 & 141 & 758 & 72 & 863 \\ 703 & 125 & 350 & 440 & 37 & 252 & 27 & 521 & 302 & 127 \\ 968 & 712 & 503 & 489 & 327 & 608 & 358 & 425 & 303 & 203 \\ \hline \end{array} $$ a. Tally the data into a frequency distribution using $\$ 100$ as a class interval and $\$ 0$ as the starting point. b. Draw a cumulative frequency polygon. c. The bank considers any student with an ending balance of $\$ 400$ or more a "preferred customer." Estimate the percentage of preferred customers. d. The bank is also considering a service charge to the lowest 10 percent of the ending bal- ances. What would you recommend as the cutoff point between those who have to pay a service charge and those who do not?

Residents of the state of South Carolina earned a total of $\$ 70.6$ billion in 2003 in adjusted gross income. Seventy-three percent of the total was in wages and salaries; 11 percent in dividends, interest, and capital gains; 8 percent in IRAs and taxable pensions; 3 percent in business income pensions; 2 percent in social security, and the remaining 3 percent was from other sources. Develop a pie chart depicting the breakdown of adjusted gross income. Write a paragraph summarizing the information.

A recent study of home technologies reported the number of hours of personal computer usage per week for a sample of 60 persons. Excluded from the study were people who worked out of their home and used the computer as a part of their work. $$ \begin{array}{|rrrrrrrrrr|} \hline 9.3 & 5.3 & 6.3 & 8.8 & 6.5 & 0.6 & 5.2 & 6.6 & 9.3 & 4.3 \\ 6.3 & 2.1 & 2.7 & 0.4 & 3.7 & 3.3 & 1.1 & 2.7 & 6.7 & 6.5 \\ 4.3 & 9.7 & 7.7 & 5.2 & 1.7 & 8.5 & 4.2 & 5.5 & 5.1 & 5.6 \\ 5.4 & 4.8 & 2.1 & 10.1 & 1.3 & 5.6 & 2.4 & 2.4 & 4.7 & 1.7 \\ 2.0 & 6.7 & 1.1 & 6.7 & 2.2 & 2.6 & 9.8 & 6.4 & 4.9 & 5.2 \\ 4.5 & 9.3 & 7.9 & 4.6 & 4.3 & 4.5 & 9.2 & 8.5 & 6.0 & 8.1 \\ \hline \end{array} $$ a. Organize the data into a frequency distribution. How many classes would you suggest? What value would you suggest for a class interval? b. Draw a histogram. Interpret your result.

Merrill Lynch recently completed a study regarding the size of on-line investment portfolios (stocks, bonds, mutual funds, and certificates of deposit) for a sample of clients in the $40-$ to 50 -year-old age group. Listed below is the value of all the investments in $\$ 000$ for the 70 participants in the study. a. Organize the data into a frequency distribution. How many classes would you suggest? What value would you suggest for a class interval? b. Draw a histogram. Interpret your result.

In May 2004,18.5 percent of the Prime Time TV viewing audience watched shows on $\mathrm{ABC}$, 25.9 percent on CBS, 18.5 percent on Fox, 18.5 percent on $\mathrm{NBC}, 7.4$ percent on Warner Brothers, and 7.4 percent on UPN. You can find the latest information on TV viewing from the following website: http://tv.zap2it.com/news/ratings/. Develop a pie chart or a bar chart to depict this information. Write a paragraph summarizing the information.

The American Heart Association reported the following percentage breakdown of expenses. Draw a pie chart depicting the information. Interpret. $$ \begin{array}{|lr|} \hline \text { Category } & \text { Percent } \\ \hline \text { Research } & 32.3 \\ \text { Public Health Education } & 23.5 \\ \text { Community Service } & 12.6 \\ \text { Fund Raising } & 12.1 \\ \text { Professional and Educational Training } & 10.9 \\ \text { Management and General } & 8.6 \\ \hline \end{array} $$

In their 2003 annual report Schering-Plough Corporation reported their income, in millions of dollars, for the years 1998 to 2003 as follows. Develop a line chart depicting the results and comment on your findings. Note that there was a $\$ 46$ million loss in 2003 .

Annual revenues, by type of tax, for the state of Georgia are as follows. Develop an appropriate chart or graph and write a brief report summarizing the information. $$ \begin{array}{|lr|} \hline \text { Type of Tax } & \text { Amount (000) } \\ \hline \text { Sales } & \$ 2,812,473 \\ \text { Income (Indlividual) } & 2,732,045 \\ \text { License } & 185,198 \\ \text { Corporate } & 525,015 \\ \text { Property } & 22,647 \\ \text { Death and Gift } & 37,326 \\ \text { Total } & \$ 6,314,704 \\ \hline \end{array} $$

Annual imports from selected Canadian trading partners are listed below for the year $2003 .$ Develop an appropriate chart or graph and write a brief report summarizing the information. $$ \begin{array}{|lr|} \hline \text { Partner } & \begin{array}{c} \text { Annual Imports } \\ \text { (million) } \end{array} \\ \hline \text { Japan } & \$ 9,550 \\ \text { United Kingdom } & 4,556 \\ \text { South Korea } & 2,441 \\ \text { China } & 1,182 \\ \text { Australia } & 618 \\ \hline \end{array} $$

Farming has changed from the early 1900 s. In the early 20 th century, machinery gradually replaced animal power. For example, in 1910 U.S. farms used 24.2 million horses and mules and only about 1,000 tractors. By 1960,4.6 million tractors were used and only 3.2 million horses and mules. In 1920 there were over 6 million farms in the United States. Today there are less than 2 million. Listed below is the number of farms, in thousands, for each of the 50 states. Write a paragraph summarizing your findings.

One of the most popular candies in the United States is M\&M's, which are produced by the Mars Company. In the beginning M\&M's were all brown; more recently they were produced in red, green, blue, orange, brown, and yellow. You can read about the history of the product, find ideas for baking, purchase the candies in the colors of your school or favorite team, and learn the percent of each color in the standard bags at http://global.mms.com/us/about/products/milkchocolate.jsp. Recently the purchase of a 14-ounce bag of M\&M's Plain had 444 candies with the following breakdown by color: 130 brown, 98 yellow, 96 red, 35 orange, 52 blue, and 33 green. Develop a chart depicting this information and write a paragraph summarizing the results.

The graph below is a combination of a line chart and a pie chart. The line chart depicts the total vehicle sales from 1993 to $2003 .$ The pie charts for each year show the percentage light-duty truck sales are of total vehicle sales. Write a brief report summarizing the results. Be sure to include whether there has been a change in truck sales over the period. Also, has the percent that light-duty trucks are of the total vehicle sales changed over time? How?

A pie chart shows the market shares of cola products. The "slice" for Pepsi-Cola has a central angle of 90 degrees. What is their market share?

The following graph shows the total wages paid by software and aircraft companies in the state of Washington from 1994 until $2002 .$ Write a brief report summarizing this information.

Monthly and year-to-date truck sales are available at the website: http://www.pickuptruck. com. Go to this site and search under News to obtain the most recent information. Make a pie chart or a bar chart showing the most recent information. What is the best selling truck? What are the four or five best selling trucks? What is their market share? You may wish to group some of the trucks into a category called "Other" to get a better picture of market share. Comment on your findings.

Refer to the Real Estate data, which reports information on homes sold in the Denver, Colorado, area during the last year. a. Select an appropriate class interval and organize the selling prices into a frequency distribution. 1. Around what values do the data tend to cluster? 2. What is the largest selling price? What is the smallest selling price? b. Draw a cumulative frequency distribution based on the frequency distribution developed in part (a). 1. How many homes sold for less than $\$ 200,000 ?$ 2. Estimate the percent of the homes that sold for more than $\$ 220,000$. 3. What percent of the homes sold for less than $\$ 125,000 ?$ c. Write a report summarizing the selling prices of the homes.

Refer to the Baseball 2003 data, which reports information on the 30 Major League Baseball teams for the 2003 season. a. Organize the information on the team salaries into a frequency distribution. Select an appropriate class interval. 1. What is a typical team salary? What is the range of salaries? 2. Comment on the shape of the distribution. Does it appear that any of the team salaries are out of line with the others? b. Draw a cumulative frequency distribution based on the frequency distribution developed in part (a). 1. Forty percent of the teams are paying less than what amount in total team salary? 2. About how many teams have total salaries of less than $\$ 80,000,000 ?$ 3. Below what amount do the lowest five teams pay in total salary? c. Organize the information on the size of the various stadiums into a frequency distribution. 1. What is a typical stadium size? Where do the stadium sizes tend to cluster? 2. Comment on the shape of the distribution. Does it appear that any of the stadium sizes are out of line with the others? d. Organize the information on the year in which the 30 major league stadiums were built into a frequency distribution. (You could also create a new variable called AGE by subtracting the year in which the stadium was built from the current year.) 1. What is the year in which the typical stadium was built? Where do these years tend to cluster? 2. Comment on the shape of the distribution. Does it appear that any of the stadium ages are out of line with the others? If so, which ones?

Refer to the Wage data, which reports information on annual wages for a sample of 100 workers. Also included are variables relating to industry, years of education, and gender for each worker. Draw a bar chart of the variable occupation. Write a brief report summarizing your findings.

Refer to the CIA data, which reports demographic and economic information on 46 countries. Develop a frequency distribution for the variable GNP per capita. Summarize your findings. What is the shape of the distribution?