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Statistics: Basic Concepts  - Mean, Median, and Mode

Statistics: basic concepts  -, mean, median, and mode, statistics: basic concepts mean, median, and mode.

Statistics: Basic Concepts: Mean, Median, and Mode

Lesson 10: mean, median, and mode.

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Mean, median, and mode

If you’re trying to identify patterns or trends in a data set, you can use the three measures of central tendency –mean, median, and mode. These values can help you to see which direction the information leans in, and you can draw inferences (or make conclusions) about your data.

Check out this video to learn more:

What are mean, median, and mode?

• The mean is the average of a set of values. If you add up all of the values and then divide this sum by the number of values, this will give you the mean.
• The median refers to the central value. If you order your data set from least to greatest or vice versa, the median is the middle number in your list. (If you have an even data set, then average the two middle values to find the median). 
• The mode refers to the value that’s repeated most often. If there are two values repeated more than the others, but an equal number of times, then you have a bimodal data set. 

A real-world example

There are 15 children in a choir, and their ages vary greatly.

Let’s try to find the mean, median, and mode…

• Step 1: The mean is calculated by adding up all of their ages, and then dividing the sum by the number of kids. If the number is a decimal, let’s round up to the nearest tenth. 

• Step 2: In order to calculate the median , let’s order the values from least to greatest. Since we have an odd data set (with 15 values in total), it should be easy to identify the middle value.

In this case, we can see that the median is 12 . 

If you have an even data set , then average the two middle numbers to find the median. It’s okay if you end up with a decimal. 

• Step 3: Which number is repeated the most often? Since 12 is repeated five times, this makes it the mode . 

Remember: If you have two modes, or two values repeated an equal number of times, then you have a bimodal data set. (If you have more than two modes, this is called a multimodal data set). 

The three measures of central tendency provide different ways of summarizing and describing a set of data. In other words, they can help you to gain a more complete understanding of the data’s distribution.

Next up, we’ll learn about how to make a survey . 

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• Central Tendency | Understanding the Mean, Median & Mode

Central Tendency | Understanding the Mean, Median & Mode

Published on July 30, 2020 by Pritha Bhandari . Revised on June 21, 2023.

Measures of central tendency help you find the middle, or the average, of a dataset. The 3 most common measures of central tendency are the mode, median, and mean.

• Mode : the most frequent value.
• Median : the middle number in an ordered dataset.
• Mean : the sum of all values divided by the total number of values.

In addition to central tendency, the variability and distribution of your dataset is important to understand when performing descriptive statistics .

Table of contents

Distributions and central tendency, when should you use the mean, median or mode, other interesting articles, frequently asked questions about central tendency.

A dataset is a distribution of n number of scores or values.

Normal distribution

In a normal distribution , data is symmetrically distributed with no skew . Most values cluster around a central region, with values tapering off as they go further away from the center. The mean, mode and median are exactly the same in a normal distribution.

A histogram of your data shows the frequency of responses for each possible number of books. From looking at the chart, you see that there is a normal distribution.

Skewed distributions

In skewed distributions, more values fall on one side of the center than the other, and the mean, median and mode all differ from each other. One side has a more spread out and longer tail with fewer scores at one end than the other. The direction of this tail tells you the side of the skew

In a positively skewed distribution, there’s a cluster of lower scores and a spread out tail on the right. In a negatively skewed distribution, there’s a cluster of higher scores and a spread out tail on the left.

• Positively skewed distribution
• Negatively skewed distribution

In this histogram, your distribution is skewed to the right, and the central tendency of your dataset is on the lower end of possible scores.

In a positively skewed distribution, mode < median < mean.

In a negatively skewed distribution, mean < median < mode.

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The mode is the most frequently occurring value in the dataset. It’s possible to have no mode, one mode, or more than one mode.

To find the mode, sort your dataset numerically or categorically and select the response that occurs most frequently.

To find the mode, sort your data by category and find which response was chosen most frequently.

To make it easier, you can create a frequency table to count up the values for each category.

Mode: Liberal

The mode is easily seen in a bar graph because it is the value with the highest bar.

When to use the mode

The mode is most applicable to data from a nominal level of measurement. Nominal data is classified into mutually exclusive categories, so the mode tells you the most popular category.

For continuous variables or ratio levels of measurement, the mode may not be a helpful measure of central tendency. That’s because there are many more possible values than there are in a nominal or ordinal level of measurement. It’s unlikely for a value to repeat in a ratio level of measurement .

The median of a dataset is the value that’s exactly in the middle when it is ordered from low to high.

To find the median, you first order all values from low to high. Then, you find the value in the middle of the ordered dataset—in this case, the value in the 4th position.

Median: Medium

In larger datasets, it’s easier to use simple formulas to figure out the position of the middle value in the distribution. You use different methods to find the median of a dataset depending on whether the total number of values is even or odd.

Median of an odd-numbered dataset

That means the median is the 3rd value in your ordered dataset.

Median: 345 milliseconds

Median of an even-numbered dataset

That means the middle values are the 3rd value, which is 345 , and the 4th value, which is 357 .

To get the median, take the mean of the 2 middle values by adding them together and dividing by 2.

Median: 351 milliseconds

The arithmetic mean of a dataset (which is different from the geometric mean ) is the sum of all values divided by the total number of values. It’s the most commonly used measure of central tendency because all values are used in the calculation.

First you add up the sum of all values:

Then you calculate the mean using the formula

There are 5 values in the dataset, so n = 5.

Mean (x̄ ) : 335 milliseconds

Outlier effect on the mean

Outliers can significantly increase or decrease the mean when they are included in the calculation. Since all values are used to calculate the mean, it can be affected by extreme outliers. An outlier is a value that differs significantly from the others in a dataset.

Mean: 444 milliseconds

Population versus sample mean

A dataset contains values from a sample or a population . A population is the entire group that you are interested in researching, while a sample is only a subset of that population.

While data from a sample can help you make estimates about a population, only full population data can give you the complete picture.

In statistics, the notation of a sample mean and a population mean and their formulas are different. But the procedures for calculating the population and sample means are the same.

• x̄:  sample mean

• n: number of values in the sample dataset

• μ: population mean

• N: number of values in the population dataset

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The 3 main measures of central tendency are best used in combination with each other because they have complementary strengths and limitations. But sometimes only 1 or 2 of them are applicable to your dataset, depending on the level of measurement of the variable.

• The mode can be used for any level of measurement, but it’s most meaningful for nominal and ordinal levels.
• The median can only be used on data that can be ordered – that is, from ordinal, interval and ratio levels of measurement.
• The mean can only be used on interval and ratio levels of measurement because it requires equal spacing between adjacent values or scores in the scale.

To decide which measures of central tendency to use, you should also consider the distribution of your dataset.

For normally distributed data, all three measures of central tendency will give you the same answer so they can all be used.

In skewed distributions, the median is the best measure because it is unaffected by extreme outliers or non-symmetric distributions of scores. The mean and mode can vary in skewed distributions.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Statistical power
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• Degrees of freedom
• Statistical significance

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Research bias

• Implicit bias
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• Hindsight bias
• Overconfidence bias

Measures of central tendency help you find the middle, or the average, of a data set.

The 3 most common measures of central tendency are the mean, median and mode.

• The mode is the most frequent value.
• The median is the middle number in an ordered data set.
• The mean is the sum of all values divided by the total number of values.

The measures of central tendency you can use depends on the level of measurement of your data.

• For a nominal level, you can only use the mode to find the most frequent value.
• For an ordinal level or ranked data, you can also use the median to find the value in the middle of your data set.
• For interval or ratio levels, in addition to the mode and median, you can use the mean to find the average value.

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average.

For data from skewed distributions, the median is better than the mean because it isn’t influenced by extremely large values.

The mode is the only measure you can use for nominal or categorical data that can’t be ordered.

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Bhandari, P. (2023, June 21). Central Tendency | Understanding the Mean, Median & Mode. Scribbr. Retrieved April 2, 2024, from https://www.scribbr.com/statistics/central-tendency/

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Mean, Median, Mode and Range

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Measures Of Central Tendency, Mean, Median and Mode

Introduction to Mean, Median and Mode: Often in statistics, we tend to represent a set of data by a representative value which would approximately define the entire collection. This representative value is called the measure of central tendency , and the name suggests that it is a value around which the data is centred. These central tendencies are mean, median and mode.

We are all interested in cricket but have you ever wondered during the match why the run rate of the particular over is projected and what does the run rate mean? Or, when you get your examination result card, you mention the aggregate percentage. Again what is the meaning of aggregate? All these quantities in real life make it easy to represent a collection of data in terms of a single value. It is called Statistics.

Statistics deals with the collection of data and information for a particular purpose. The tabulation of each run for each ball in cricket gives the statistics of the game. The representation of any such data collection can be done in multiple ways, like through tables, graphs, pie-charts , bar graphs, pictorial representation etc.

Now consider a 50 over ODI match going between India and Australia. India scored 370 runs by the end of the first innings. How do you decide whether India put a good score or not? It’s pretty simple, right; you find the overall run rate, which is good for such a score. Thus, here comes the concept of mean, median and mode in the picture. Let us learn in detail each of the central tendencies.

Measures of central tendency

The measures of central tendencies are given by various parameters but the most commonly used ones are mean, median and mode. These parameters are discussed below.

What is Mean?

Mean is the most commonly used measure of central tendency. It actually represents the average of the given collection of data. It is applicable for both continuous and discrete data.

It is equal to the sum of all the values in the collection of data divided by the total number of values.

Suppose we have n values in a set of data namely as x 1 , x 2 , x 3 , …, x n,  then the mean of data is given by:

\(\begin{array}{l}\bar{x} = \frac{x_{1}+x_{2}+x_{3}+……..+x_{n}}{n}\end{array} \)

It can also be denoted as:

\(\begin{array}{l}\bar{x} = \frac{\sum_{i=1}^{n}x_{i}}{n}\end{array} \)

For grouped data, we can calculate the mean using three different methods of formula.

To learn more about the mean, visit here .

What is Median?

Generally median represents the mid-value of the given set of data when arranged in a particular order.

Median: Given that the data collection is arranged in ascending or descending order, the following method is applied:

• If number of values or observations in the given data is odd, then the median is given by [(n+1)/2] th observation.
• If in the given data set, the number of values or observations is even, then the median is given by the average of (n/2) th  and  [(n/2) +1] th observation.

The median for grouped data can be calculated using the formula,

\(\begin{array}{l}Median = l + \left ( \frac{\frac{N}{2}-cf}{f} \right )\times h\end{array} \)

To understand in detail about the median, visit here .

What is Mode?

The most frequent number occurring in the data set is known as the mode.

Consider the following data set which represents the marks obtained by different students in a subject.

The maximum frequency observation is 73 ( as three students scored 73 marks), so the mode of the given data collection is 73.

We can calculate the mode for grouped data using the below formula:

\(\begin{array}{l}Mode = l + \left ( \frac{f_1-f_0}{2f_1-f_0-f_2} \right )\times h\end{array} \)

Also, learn: Mode

Video Lessons

Mean median and mode.

Measure of Central Tendency for an Ungrouped Data

Example of Mean, Median and Mode

Let us see the difference between the mean median and mode through an example.

Example: The given table shows the scores obtained by different players in a match. What is mean, median and mode of the given data?

i)  The mean is given by:

The mean of the given data is 43.

ii) To find out the median let us first arrange the given data in ascending order

As the number of items in the data is odd. Hence, the median is [(n+1)/2] th observation.

⇒ Median = [(7+1)/2] th observation = 52

iii)  Mode is the most frequent data, which is 52.

Relation of Mean Median Mode

The relation between mean, median and mode that means the three measures of central tendency for moderately skewed distribution is given the formula:

This relation is also called an empirical relationship. This is used to find one of the measures when the other two measures are known to us for certain data. This relationship is rewritten in different forms by interchanging the LHS and RHS.

In statistics, the range is the difference between the highest and lowest data value in the set. The formula is:

Range – Highest value – Lowest value

Solved Problem

Question: Find the mean, median, mode and range for the given data:

90, 94, 53, 68, 79, 94, 53, 65, 87, 90, 70, 69, 65, 89, 85, 53, 47, 61, 27, 80

Number of observations = 20

Mean = (Sum of observations)/ Number of observations

= (90 + 94 + 53 + 68 + 79 + 94 + 53 + 65 + 87 + 90 + 70 + 69 + 65 + 89 + 85 + 53 + 47 + 61 + 27 + 80)/20

Therefore, mean is 70.95.

The ascending order of given observations is:

27, 47, 53, 53, 53, 61, 65, 65, 68, 69, 70, 79, 80, 85, 87, 89, 90, 90, 94,94

Here, n = 20

Median = 1/2 [(n/2) + (n/2 + 1)]th observation

= 1/2 [10 + 11]th observation

= 1/2 (69 + 70)

Thus, the median is 69.5.

The most frequently occurred value in the given data is 53.

Therefore, mode = 53

Range = Highest value – Lowest value

= 94 – 27

Practice Questions

• The points scored by a Kabaddi team in a series of matches are as follows: 17, 2, 7, 27, 15, 5, 14, 8, 10, 24, 48, 10, 8, 7, 18, 28 Find the mean, median and mode of the points scored by the team.
• The following observations have been arranged in ascending order. If the median of the data is 63, find the value of x. 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95

To know more about Measures of central tendency and the applications of Mean, Median and Mode with solved examples stay tuned with BYJU’S.

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Measures of Central Tendency

Introduction.

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary statistics. The mean (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode.

The mean, median and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. In the following sections, we will look at the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate to be used.

Mean (Arithmetic)

The mean (or average) is the most popular and well known measure of central tendency. It can be used with both discrete and continuous data, although its use is most often with continuous data (see our Types of Variable guide for data types). The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. So, if we have \( n \) values in a data set and they have values \( x_1, x_2, \) …\(, x_n \), the sample mean, usually denoted by \( \overline{x} \) (pronounced "x bar"), is:

$$ \overline{x} = {{x_1 + x_2 + \dots + x_n}\over{n}} $$

This formula is usually written in a slightly different manner using the Greek capitol letter, \( \sum \), pronounced "sigma", which means "sum of...":

$$ \overline{x} = {{\sum{x}}\over{n}} $$

You may have noticed that the above formula refers to the sample mean. So, why have we called it a sample mean? This is because, in statistics, samples and populations have very different meanings and these differences are very important, even if, in the case of the mean, they are calculated in the same way. To acknowledge that we are calculating the population mean and not the sample mean, we use the Greek lower case letter "mu", denoted as \( \mu \):

$$ \mu = {{\sum{x}}\over{n}} $$

The mean is essentially a model of your data set. It is the value that is most common. You will notice, however, that the mean is not often one of the actual values that you have observed in your data set. However, one of its important properties is that it minimises error in the prediction of any one value in your data set. That is, it is the value that produces the lowest amount of error from all other values in the data set.

An important property of the mean is that it includes every value in your data set as part of the calculation. In addition, the mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero.

When not to use the mean

The mean has one main disadvantage: it is particularly susceptible to the influence of outliers. These are values that are unusual compared to the rest of the data set by being especially small or large in numerical value. For example, consider the wages of staff at a factory below:

The mean salary for these ten staff is $30.7k. However, inspecting the raw data suggests that this mean value might not be the best way to accurately reflect the typical salary of a worker, as most workers have salaries in the $12k to 18k range. The mean is being skewed by the two large salaries. Therefore, in this situation, we would like to have a better measure of central tendency. As we will find out later, taking the median would be a better measure of central tendency in this situation.

Another time when we usually prefer the median over the mean (or mode) is when our data is skewed (i.e., the frequency distribution for our data is skewed). If we consider the normal distribution - as this is the most frequently assessed in statistics - when the data is perfectly normal, the mean, median and mode are identical. Moreover, they all represent the most typical value in the data set. However, as the data becomes skewed the mean loses its ability to provide the best central location for the data because the skewed data is dragging it away from the typical value. However, the median best retains this position and is not as strongly influenced by the skewed values. This is explained in more detail in the skewed distribution section later in this guide.

The median is the middle score for a set of data that has been arranged in order of magnitude. The median is less affected by outliers and skewed data. In order to calculate the median, suppose we have the data below:

We first need to rearrange that data into order of magnitude (smallest first):

Our median mark is the middle mark - in this case, 56 (highlighted in bold). It is the middle mark because there are 5 scores before it and 5 scores after it. This works fine when you have an odd number of scores, but what happens when you have an even number of scores? What if you had only 10 scores? Well, you simply have to take the middle two scores and average the result. So, if we look at the example below:

We again rearrange that data into order of magnitude (smallest first):

Only now we have to take the 5th and 6th score in our data set and average them to get a median of 55.5.

The mode is the most frequent score in our data set. On a histogram it represents the highest bar in a bar chart or histogram. You can, therefore, sometimes consider the mode as being the most popular option. An example of a mode is presented below:

Normally, the mode is used for categorical data where we wish to know which is the most common category, as illustrated below:

We can see above that the most common form of transport, in this particular data set, is the bus. However, one of the problems with the mode is that it is not unique, so it leaves us with problems when we have two or more values that share the highest frequency, such as below:

We are now stuck as to which mode best describes the central tendency of the data. This is particularly problematic when we have continuous data because we are more likely not to have any one value that is more frequent than the other. For example, consider measuring 30 peoples' weight (to the nearest 0.1 kg). How likely is it that we will find two or more people with exactly the same weight (e.g., 67.4 kg)? The answer, is probably very unlikely - many people might be close, but with such a small sample (30 people) and a large range of possible weights, you are unlikely to find two people with exactly the same weight; that is, to the nearest 0.1 kg. This is why the mode is very rarely used with continuous data.

Another problem with the mode is that it will not provide us with a very good measure of central tendency when the most common mark is far away from the rest of the data in the data set, as depicted in the diagram below:

In the above diagram the mode has a value of 2. We can clearly see, however, that the mode is not representative of the data, which is mostly concentrated around the 20 to 30 value range. To use the mode to describe the central tendency of this data set would be misleading.

Skewed Distributions and the Mean and Median

We often test whether our data is normally distributed because this is a common assumption underlying many statistical tests. An example of a normally distributed set of data is presented below:

When you have a normally distributed sample you can legitimately use both the mean or the median as your measure of central tendency. In fact, in any symmetrical distribution the mean, median and mode are equal. However, in this situation, the mean is widely preferred as the best measure of central tendency because it is the measure that includes all the values in the data set for its calculation, and any change in any of the scores will affect the value of the mean. This is not the case with the median or mode.

However, when our data is skewed, for example, as with the right-skewed data set below:

We find that the mean is being dragged in the direct of the skew. In these situations, the median is generally considered to be the best representative of the central location of the data. The more skewed the distribution, the greater the difference between the median and mean, and the greater emphasis should be placed on using the median as opposed to the mean. A classic example of the above right-skewed distribution is income (salary), where higher-earners provide a false representation of the typical income if expressed as a mean and not a median.

If dealing with a normal distribution, and tests of normality show that the data is non-normal, it is customary to use the median instead of the mean. However, this is more a rule of thumb than a strict guideline. Sometimes, researchers wish to report the mean of a skewed distribution if the median and mean are not appreciably different (a subjective assessment), and if it allows easier comparisons to previous research to be made.

Summary of when to use the mean, median and mode

Please use the following summary table to know what the best measure of central tendency is with respect to the different types of variable .

For answers to frequently asked questions about measures of central tendency, please go the next page .

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Mean, Median and Mode

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Mean, Median, and Mode are measures of the central tendency. These values are used to define the various parameters of the given data set. The measure of central tendency (Mean, Median, and Mode) gives useful insights about the data studied, these are used to study any type of data such as the average salary of employees in an organization, the median age of any class, the number of people who plays cricket in a sports club, etc. 

Let’s learn more about the Mean, Median, and Mode Formulas, Examples, and FAQs in this article.

Table of Content

Measures of Central Tendency

What is mean, what is median, what is mode, relation between mean median mode, differences between mean, median and mode.

Measure of central tendency is the representation of various values of the given data set. There are various measures of central tendency and the most important three measures of central tendency are:

• Mean (x̅ or μ): The mean, or arithmetic average, is calculated by summing all the values in a dataset and dividing by the total number of values. It’s sensitive to outliers and is commonly used when the data is symmetrically distributed.
• Median (M): The median is the middle value when the dataset is arranged in ascending or descending order. If there’s an even number of values, it’s the average of the two middle values. The median is robust to outliers and is often used when the data is skewed.
• Mode (Z): The mode is the value that occurs most frequently in the dataset. Unlike the mean and median, the mode can be applied to both numerical and categorical data. It’s useful for identifying the most common value in a dataset.

Mean is the sum of all the values in the data set divided by the number of values in the data set. It is also called the Arithmetic Average. Mean is denoted as x̅ and is read as x bar .

The formula to calculate the mean is:

Formula of Mean

Symbol of Mean

The symbol used to represent the mean, or arithmetic average, of a dataset is typically the Greek letter “μ” (mu) when referring to the population mean, and “x̄” (x-bar) when referring to the sample mean.

• Population Mean: μ (mu)
• Sample Mean: x̄ (x-bar)

These symbols are commonly used in statistical notation to represent the average value of a set of data points.

Mean Formula

Mean (x̅)  = Sum of Values / Number of Values

If x 1, x 2, x 3 ,……, x n are the values of a data set then the mean is calculated as:

x̅ =  (x 1 + x 2 + x 3 + …… + x n ) / n

Example: Find the mean of data sets 10, 30, 40, 20, and 50.

Mean of the data 10, 30, 40, 20, 50 is Mean = (sum of all values) / (number of values) Mean = (10 + 30 + 40 + 20+ 50) / 5 = 30

Read More, about Mean Formula.

Mean of Grouped Data

Mean for the grouped data can be calculated by using various methods. The most common methods used are discussed in the table below:

Read More about Mean, Median and Mode of Grouped Data .

A Median is a middle value for sorted data. The sorting of the data can be done either in ascending order or descending order. A median divides the data into two equal halves. 

The formula to calculate the median of the number of terms if the number of terms is even is shown in the image below:

Median Formula for Even Terms

The formula to calculate the median of the number of terms if the number of terms is odd is shown in the image below:

Median Formula for Odd Terms

Symbol of Median

The letter “M” is commonly used to represent the median of a dataset, whether it’s for a population or a sample. This notation simplifies the representation of statistical concepts and calculations, making it easier to understand and apply in various contexts. Therefore, in Indian statistical practice, “M” is widely accepted and understood as the symbol for the median.

Median Formula

The formula for the median is:

If the number of values (n value) in the data set is odd then the formula to calculate the median is:

Median = [(n + 1)/2] th term

If the number of values (n value) in the data set is even then the formula to calculate the median is:

Median  = [(n/2) th term + {(n/2) + 1} th term] / 2

Example: Find the median of given data set 30, 40, 10, 20, and 50.

Median of the data 30, 40, 10, 20, 50 is, Step 1: Order the given data in ascending order as: 10, 20, 30, 40, 50 Step 2: Check n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value. Step 3: Here, n = 5 (odd) Median = [(n + 1)/2] th term Median = [(5 + 1)/2] th term             = 30

Median of Grouped Data

The median of the grouped data median is calculated using the formula,

Median = l + [(n/2 – cf) / f]×h where l is lower limit of median class n is number of observations f is frequency of median class h is class size cf is cumulative frequency of class preceding the median class.

Read More about Median of Grouped Data .

A mode is the most frequent value or item of the data set. A data set can generally have one or more than one mode value. If the data set has one mode then it is called “Uni-modal”. Similarly, If the data set contains 2 modes then it is called “Bimodal” and if the data set contains 3 modes then it is known as “Trimodal”. If the data set consists of more than one mode then it is known as “multi-modal”(can be bimodal or trimodal). There is no mode for a data set if every number appears only once.

The formula to calculate the mode is shown in the image below:

Formula of Median

Symbol of Mode

In statistical notation, the symbol “Z” is commonly used to represent the mode of a dataset. It indicates the value or values that occur most frequently within the dataset. This symbol is widely utilised in statistical discourse to signify the mode, enhancing clarity and precision in statistical discussions and analyses.

Mode Formula

Mode = Highest Frequency Term

Example: Find the mode of the given data set 1, 2, 2, 2, 3, 3, 4, 5.

Given set is {1, 2, 2, 2, 3, 3, 4, 5} As the above data set is arranged in ascending order. By observing the above data set we can say that, Mode = 2 As, it has highest frequency (3)

Mode of Grouped Data

The mode of the grouped data is calculated using the formula:

Mode = l + [(f 1 + f 0 ) / (2f 1 – f 0 – f 2 )] × h Where, f 1 is the frequency of the modal class, f 0 is the frequency of the class preceding the modal class, f 2 is the frequency of the class succeeding the modal class, h is the size of class intervals, and l is the lower limit of modal class.

Read More about Mode of Grouped Data .

For any group of data, the relation between the three central tendencies mean, median, and mode is shown in the image below:

Mode = 3 Median – 2 Mean

Another name for this relationship is an empirical relationship. When we know the other two measures for a given set of data, this is used to find one of the measures. The LHS and RHS can be switched to rewrite this relationship in various ways.

Range of Data

Range is the difference between the highest value and the lowest value. It is a way to understand how the numbers are spread in a data set. The range of any data set is easily calculated by using the formula given in the image below:

Formula to Find Range

Range Formula

The formula to find the Range is:

Range = Highest value – Lowest Value

Example: Find the range of the given data set 12, 19, 6, 2, 15, 4.

Given set is {12, 19, 6, 2, 15, 4}  Here,  Lowest Value = 2 Highest Value = 19 Range = 19 − 2 = 17

Difference Between Mean and Median

The key differences between mean and median are listed in the following table:

Let’s see the following example to understand the difference.

The difference between Mean and Median is understood by the following example. In a school, there are 8 teachers whose salaries are 20000 rupees, a principal with a salary of 35000, find their mean salary and median salary.

Mean = (20000+20000+20000+20000+20000+20000+20000+20000+35000)/9 = 195000/9 = 21666.67

Therefore, the mean salary is ₹21,666.67.

For median, in ascending order: 20000, 20000, 20000, 20000, 20000, 20000, 20000, 20000, 35000.

Thus, (9 + 1)/2 = 5

Thus, the median is the 5 th observation.

Median = 20000

Therefore, the median is ₹20,000.

Note: Mean gets easily affected by extreme values.

Mean, median, and mode are measures of central tendency in statistics.

Difference Between Mean and Average

The terms “mean” and “average” are frequently used in mathematics and statistics, often interchangeably. However, they possess subtle distinctions in their meanings and applications.

Mean, in statistical terms, represents the arithmetic average of a dataset. It is calculated by summing up all the values in the dataset and dividing the sum by the total number of values. For instance, if you have the numbers 2, 4, 6, 8, and 10, the mean would be (2 + 4 + 6 + 8 + 10) / 5 = 6.

On the other hand, “average” is a broader term that can refer to various measures of central tendency, including mean, median, and mode. In common usage, however, “average” often specifically denotes the mean. Like the mean, it involves summing up a set of values and dividing by the number of values to obtain a representative value.

Read More about Difference between Mean and Average .

Mean, Median and Mode are the measure of central tendency which helps us analyze and interpret data across various fields. Mean, often used as the arithmetic average, is sensitive to extreme values. On the other hand, the median, representing the middle value of any dataset. Meanwhile, the mode, indicating the most frequently occurring value.

Solved Questions on Mean, Median, and Mode

Question 1: Study the bar graph given below and find the mean, median, and mode of the given data set.

Mean = (sum of all data values) / (number of values) Mean = (5 + 7 + 9 + 6) / 4             =  27 / 2            = 6.75 Order the given data in ascending order as: 5, 6, 7, 9 Here, n = 4 (which is even) Median  =  [(n/2) th term + {(n/2) + 1} th term] / 2 Median  = (6 + 7) / 2                 =  6.5 Mode = Most frequent value            = 9  (highest value) Range = Highest value – Lowest value  Range = 9 – 5             = 4

Question 2: Find the mean, median, mode, and range for the given data

190, 153, 168, 179, 194, 153, 165, 187, 190, 170, 165, 189, 185, 153, 147, 161, 127, 180

For Mean: 190, 153, 168, 179, 194, 153, 165, 187, 190, 170, 165, 189, 185, 153, 147, 161, 127, 180 Number of observations = 18 Mean = (Sum of observations) / (Number of observations)           = (190+153+168+179+194+153+165+187+190+170+165+189+185+153+147 +161+127+180) / 18           = 2871/18           = 159.5 Therefore, the mean is 159.5 For Median: The ascending order of given observations is, 127, 147, 153, 153, 153, 161, 165, 165, 168, 170, 179, 180, 185, 187, 189, 190, 190, 194 Here, n = 18 Median = 1/2 [(n/2) + (n/2 + 1)] th observation              = 1/2 [9 + 10] th observation              = 1/2 (168 + 170)              = 338/2              = 169 Thus, the median is 169 For Mode: The number with the highest frequency = 153 Thus, mode = 53 For Range: Range = Highest value – Lowest value            = 194 – 127            = 67

Question 3: Find the Median of the data 25, 12, 5, 24, 15, 22, 23, 25

25, 12, 5, 24, 15, 22, 23, 25 Step 1: Order the given data in ascending order as:  5, 12, 15, 22, 23, 24, 25, 25  Step 2: Check n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value. Step 3: Here, n = 8 (even) then, Median = [(n/2) th term + {(n/2) + 1) th term] / 2 Median = [(8/2) th term + {(8/2) + 1} th term] / 2              = (22+23) / 2              = 22.5

Question 4: Find the mode of given data 15, 42, 65, 65, 95.

Given data set 15, 42, 65, 65, 95 The number with highest frequency = 65 Mode = 65

FAQs on Mean, Median, and Mode

What are the mean, median, and mode.

Mean, Median and Mode are the measures of central tendency. These three measures of central tendency are used to get an overview of the data. They represent the true essence of the given data set.

What is the relation between mean, median, and mode?

The relationship between mean median and mode is: Mode = 3 Median – 2 Mean

How to find mean, median, and mode?

Mean, Median, and Mode of any given data set is calculated using the suitable formulas which are discussed above in the articles.

How to find the mean?

Mean is also called the average, it is calculated for ungrouped data using the formula: Mean = (Sum of observations)/(Number of observations) In case of Grouped Data, the mean is calculated by the three methods Direct method Assumed mean method Step deviation method

How to find the median?

Median is the middle term of the data when it is arranged in either ascending or descending order. It is calculated using the formula: Median = (n + 1)/2 th observation {when n is odd} Median = Average of (n/2) th and [(n/2) + 1] th observations {when n is even}

How to find the mode?

The value with the highest frequency is called the mode. Mode is calculated by observation first the given set of values is arranged in either ascending or descending order then the value with the highest frequency is noted as Mode.

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Which Countries Are Really the Richest?

Mean vs. median wealth.

Ranking countries by the size of their economies and their overall net wealth, the U.S. is usually at the top of the list, followed by countries like China, Japan or Germany. But sorting countries for their per-capita average wealth – or even for their median per-capita wealth – other countries come out on top. Using the two metrics, Switzerland was the richest country in the world with the highest average per-capita wealth of around US$685,000 per adult. Looking at median per-capita wealth - the wealth of the person that shares their country with an equal number of richer and poorer people - Iceland tops the ranking with around $413,000 in wealth being held by this (imaginary) person.

Per-capita assets arguably show a more balanced picture of a country’s wealth by acknowledging that smaller countries with less citizens will of course accumulate less wealth in total. Yet, calculating averages does not take into account how wealth is distributed in a society. Median wealth, on the other hand, increases the more equal a country’s assets are allocated. Iceland and other Scandinavian countries are known for their more equal wealth distribution and data by Credit Suisse reflects this to a degree. Denmark comes in rank 7 and Norway in rank 10 for per-capita median wealth.

The U.S. is the third-wealthiest country on a per-capita average basis, yet Americans are only in rank 15 for median wealth. The situation in Belgium is the other way round: It is listed 13th for average wealth, but third for median wealth, showing that it is a more egalitarian country in terms of wealth distribution.

Looking at the size of the gap between mean and median wealth, the U.S. comes in rank 7 with an average wealth more than five times or 512% as high as the median wealth. This is exceeded by no major country in the world except Brazil, where this number stands at 517%. Some of the smallest relative gaps between mean and median wealth were registered in the countries topping the median wealth list: Iceland, Luxembourg and Belgium. Other countries which might not have the highest mean wealth rates but do have some of the smallest gaps are Eastern European nations Slovakia and Slovenia. Poorer countries which nevertheless have big gaps between average and median wealth include the aforementioned Brazil as well as South Africa, Russia and Nigeria.

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Mean, Median, and Mode

Jul 31, 2014

470 likes | 782 Views

Mean, Median, and Mode. By: Mr. Menjviar. 09/22/11. 09/22/11. Observe, Question, Comment . Mean, Median, and Mode. Mean, Median, and Mode Reflection. Warm-Up:. 13L. 13R. Brain Stimulation. ‘Check Skills You’ll Need’ 3, 4, 5, 6, 6, 8, 9 68, 69, 71, 72, 72 98, 101, 101, 112, 120

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Mean, Median, and Mode By: Mr. Menjviar

09/22/11 09/22/11 Observe, Question, Comment Mean, Median, and Mode Mean, Median, and Mode Reflection Warm-Up: 13L 13R

Brain Stimulation • ‘Check Skills You’ll Need’ • 3, 4, 5, 6, 6, 8, 9 • 68, 69, 71, 72, 72 • 98, 101, 101, 112, 120 • 3, 3.3, 3.7, 3.74, 37

09/22/11 09/22/11 Observe, Question, Comment Mean, Median, and Mode Mean, Median, and Mode Reflection Warm-Up: Standard: 13L 13R

CA Content Standards • Statistics Data Analysis and Probability 1.3: Understanding the meaning of, and be able to compute, the median of a data set

09/22/11 09/22/11 Observe, Question, Comment Mean, Median, and Mode Mean, Median, and Mode Reflection Warm-Up: Standard: Objective: 13L 13R

Objective: • To find the mean, median, mode and range of a set of data

09/22/11 09/22/11 Observe, Question, Comment Mean, Median, and Mode Mean, Median, and Mode Reflection Warm-Up: Standard: Objective: Vocabulary: 13L 13R

Vocabulary:Mean The sum of the data values divided by the number of data values

Vocabulary:Median The middle number when data values are written in order

Vocabulary:Mode The data item that occurs most often. There can be one mode, more than one, or no mode.

Vocabulary:Range The difference between the greatest and least values

09/22/11 09/22/11 Observe, Question, Comment Mean, Median, and Mode Mean, Median, and Mode Reflection Warm-Up: Standard: Objective: Vocabulary: Notes/Examples: 13L 13R

The BASICSYou Must Memorize • Mean = • Median = Middle of the data set • Mode = Value that repeats the most • Range = Highest Value — Lowest Value

Example!!… Let’s take a look at these students that did a Readathon at their school. Find the • Mean • Median • Mode • Range

You Do!!… Let’s take a look at these T-shirt prices that have different colors. Find the • Mean • Median • Mode • Range

You Do!!… Let’s take a look at these T-shirt prices that have different colors. Find the • Mean is $24.40 • Median is $26 • Mode (there is no mode) • Range is $8

09/22/11 09/22/11 Observe, Question, Comment Mean, Median, and Mode Mean, Median, and Mode Reflection Warm-Up: Standard: Objective: Independent Practice: Vocabulary: Notes/Examples: 13L 13R

Independent Practice • Textbook, Pg. 609, #5-13

09/22/11 09/22/11 Observe, Question, Comment Mean, Median, and Mode Mean, Median, and Mode Reflection Warm-Up: Standard: Objective: Independent Practice: Homework: Vocabulary: Notes/Examples: 13L 13R

Homework • Student Workbook, pg. 481 [1a, 2, 4, 6, 7, 8] [Due: Next Meeting]

09/22/11 09/22/11 Observe, Question, Comment Frequency Table and Histogram Frequency Table and Histogram Reflection Standard: 14L 14R

CA Content Standards Statistics Data Analysis and Probability 1.1 Know various forms of display for data sets

09/22/11 09/22/11 Observe, Question, Comment Frequency Table and Histogram Frequency Table and Histogram Reflection Standard: Objective: 14L 14R

Objective: • To display data in frequency tables and histograms

09/22/11 09/22/11 Observe, Question, Comment Frequency Table and Histogram Frequency Table and Histogram Reflection Standard: Objective: Vocabulary: 14L 14R

Vocabulary:Frequency Table It is a table that displays data, which lists each data item with the number of time it occurs.

Vocabulary:Histogram It is a graph that shows the frequencies of data items. The data is often grouped into intervals.

09/22/11 09/22/11 Observe, Question, Comment Frequency Table and Histogram Frequency Table and Histogram Reflection Standard: Objective: Vocabulary: Notes/Examples: 14L 14R

Notes/Examples:Frequency [Classroom Birth Month]

Notes/Examples:Histogram [Classroom Birth Month]

09/22/11 09/22/11 Observe, Question, Comment Frequency Table and Histogram Frequency Table and Histogram Reflection Independent Practice: Standard: Objective: Vocabulary: Notes/Examples: 14L 14R

Independent Practice • Textbook,Pg. 614: #’s 1-8, 10, 12-14

09/22/11 09/22/11 Observe, Question, Comment Frequency Table and Histogram Frequency Table and Histogram Reflection Independent Practice: Standard: Homework: Objective: Vocabulary: Notes/Examples: 14L 14R

Homework • Student Workbook, 12-2, pg. 483 [1, 2, 5]; • pg. 484 [1-6] [Due next meeting]

Exit Slip • Create a Data Set that will have the mean, median and mode to be the same answer • Meaning This: • Mean = Median = Mode

Vocabulary • Mean → the average • Median → the middle • Mode→ the most • Range→ big minus small

Vocabulary (Shout Outs) • Bar Graph → bars to show comparison • Frequency Table → list each item • Histogram→ bar graph with no spaces

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BY ZACH ELSAWY. Range, Median, Mode, and Mean. HOME RUNS HIT IN 2010 REGULAR SEASON. 3 statements. The Nationals hit 52 home runs less than the Yankees. * The Braves hit 19 more home runs than the Dodgers. * The Rays hit 2 home runs less than the Giants. RANGE. MODE. 211 - 104

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Mean, Median, and Mode. Measures of Central Tendency. Acres of Farmland. Use the data to the left to find the answer to each question below: Find the average farm acreage in the southernmost states: Texas, New Mexico, and Kansas.

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MEAN, MEDIAN, MODE

MEAN, MEDIAN, MODE. FREQUENCY DISTRIBUTION TABLE. A table containing array data have been grouped by class or specific categories. How to make it?. 2. SEE This Data. 3. FIRST STEPS ARE MAKE A FREQUENCY DISTRIBUTION TABLE. 4. STEP 01 :. SET NUMBER OF CLASSES ( Jumlah Kelas JK ).

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Mean, Median, Mode. These are measures of central tendency All three give us information about a sample But some are more meaningful depending on the level of measurement. Which measure of central tendency do you use with…. Nominal level measure? Ordinal level measure Interval level measure

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Mean, Median, Mode. Tuesday 9/18/12 Theme, sports. What you will learn (Content purpose). Perform common calculations required for data analysis. Language Purpose (What you will say). Use the following terms M ean , Median mode. Range. What you will do with your group (PGW Outcome).

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Divide the sum by the number of terms. 502 + 477 + 593 + 481 + 735 + 614 6. 3,402 6. = 567. Simplify. The mean is 567. COURSE 2 LESSON 1-10. Mean, Median, and Mode. Find the average of 502, 477, 593, 481, 735, and 614. 1-10. COURSE 2 LESSON 1-10. Mean, Median, and Mode.

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Mean, Median, Mode and Range. The Basics of Statistics. Nekela Macon Summer 2009. Did You Know…. That you probably use statistics such as mean, median, mode and range almost every day without even realizing it?!?. Today We Will Learn… . Mean Median Mode Range

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Mean, Median and Mode. Mean, Median, and Mode. Measures of Central Tendency. Mean, Median and Mode. What is the mean?. Mean, Median, and Mode. The SUM of the data, divided by the number of items in the set of data ….. commonly called “the average”. Mean, Median and Mode.

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Mean, Median, Mode and Range. These are Abby’s science test scores. 86. 84. 97. 73. 88. 63. 97. 95. 100. What can you tell us about these numbers?. 86. 84. 97. 73. 63. 88. 97. 100. 95. What is the MEAN? How do we find it?. The mean is the numerical average of the data set.

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Mean, Median, and Mode. An Introduction to Data Management: Measures of Central Tendencies. Why Analyze Data?.

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Mean, Median, Mode, and Range. Mean. The mean is the average of a set of numbers. Add the numbers and divide the sum by the amount of numbers you have. 12, 15, 11, 11, 7, 13. First, find the sum of the numbers. 12 + 15 +11 + 11 + 7 + 13 = 69. Then divide by the amount of numbers.

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Mean, Median, Mode. Social Statistics. This week. Mean Median Mode. How do we decide which is “best”?. The overall goal of central tendency is to find the single score that is most representative for the distribution. Measures of Central Tendency. Mean: Arithmetic average

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Connecting a Verbal Description to Table &amp; Graph PART 2. Common Core #4 &amp; #13. Mean, Median, and Mode. Mean, Median, and Mode. (8) How much will Genaro earn for 17 correct problems? (9) How much will Genaro earn for 111 correct problems?

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