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Welcome to statmagic, a statistics calculator app. With 11 unique calculators, memorable example problems, and help content developed from years of one-on-one tutoring, statmagic will help you learn — and get your homework done — faster.

Looking to learn stats from the ground up? Watch our introductory statistics video series.

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Features Included:

Every one of statmagic's calculators validates your inputs, to help you prevent simple mistakes. You can also choose your final answers' rounding.

Automatic Dataset Import (iOS PRO Only)

An exceedingly easy dataset import feature. Just open a spreadsheet in a spreadsheet app, copy the data onto your device's clipboard, and statmagic PRO can read it! Statmagic PRO can then help you select the correct test for your 1-sample, 2-sample, or paired data.

Descriptive Statistics

The basics. Count, sum, min and max, mean, median, quartiles, variance, and standard deviation. Learn more.

The Normal Distribution

Calculate Z or T scores and probabilities on the normal distribution, for a random individual (x), or even a random sample mean (x-bar). Learn more.

Hypothesis Testing

Individual calculators for:

• 1 sample test of mean (Z or T). Learn more.
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Every calculator returns the 'reject Ho?' conclusion, p-value, critical test value, test score, standard error, and confidence interval.

The tests of mean also feature automatic graphics, so you can visualize what's going on.

Confidence Intervals

Create confidence intervals for means or proportions at different levels of confidence.

Learn more about CIs of a mean or CIs of a proportion.

Enter an x-y paired dataset, and this calculator automatically generates the scatterplot (and marks outliers (according to the 1.5 x IQR method)). Calculate, and it draws regression line and shows the "y = a + bx" formula.

It also returns r and R², F, RMSE, and SE, T and the p-value of the slope. Finally, you use the regression model to make a prediction: enter a chosen x* value, and this calculator returns (and plots) the predicted y-hat value (PRO only). Learn more.

Chi-Squared (PRO only)

Run a Chi-squared test on up to a 7x7 table. Easily toggle between the table's observed, expected, and Chi-squared values. Returns the 'reject Ho?' conclusion, p-value, Chi-square test value, and critical Chi-square value (with degrees of freedom). Learn more.

Calculate the required minimum sample size n for a given confidence level. Learn more.

1.1 Definitions of Statistics, Probability, and Key Terms

For each of the following eight exercises, identify: a. the population, b. the sample, c. the parameter, d. the statistic, e. the variable, and f. the data. Give examples where appropriate.

A fitness center is interested in the mean amount of time a client exercises in the center each week.

Ski resorts are interested in the mean age that children take their first ski and snowboard lessons. They need this information to plan their ski classes optimally.

A cardiologist is interested in the mean recovery period of her patients who have had heart attacks.

Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance.

A politician is interested in the proportion of voters in his district who think he is doing a good job.

A marriage counselor is interested in the proportion of clients she counsels who stay married.

Political pollsters may be interested in the proportion of people who will vote for a particular cause.

A marketing company is interested in the proportion of people who will buy a particular product.

Use the following information to answer the next three exercises: A Lake Tahoe Community College instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter.

What is the population she is interested in?

• all Lake Tahoe Community College students
• all Lake Tahoe Community College English students
• all Lake Tahoe Community College students in her classes
• all Lake Tahoe Community College math students

Consider the following:

X X = number of days a Lake Tahoe Community College math student is absent

In this case, X is an example of a:

• population.

The instructor’s sample produces a mean number of days absent of 3.5 days. This value is an example of a:

1.2 Data, Sampling, and Variation in Data and Sampling

For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous, or qualitative), and give an example of the data.

number of tickets sold to a concert

percent of body fat

favorite baseball team

time in line to buy groceries

number of students enrolled at Evergreen Valley College

most-watched television show

brand of toothpaste

distance to the closest movie theatre

age of executives in Fortune 500 companies

number of competing computer spreadsheet software packages

Use the following information to answer the next two exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of resident use of a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every 8th house in the neighborhood around the park was interviewed.

“Number of times per week” is what type of data?

• qualitative
• quantitative discrete
• quantitative continuous

“Duration (amount of time)” is what type of data?

Airline companies are interested in the consistency of the number of babies on each flight, so that they have adequate safety equipment. Suppose an airline conducts a survey. Over Thanksgiving weekend, it surveys six flights from Boston to Salt Lake City to determine the number of babies on the flights. It determines the amount of safety equipment needed by the result of that study.

• Using complete sentences, list three things wrong with the way the survey was conducted.
• Using complete sentences, list three ways that you would improve the survey if it were to be repeated.

Suppose you want to determine the mean number of students per statistics class in your state. Describe a possible sampling method in three to five complete sentences. Make the description detailed.

Suppose you want to determine the mean number of cans of soda drunk each month by students in their twenties at your school. Describe a possible sampling method in three to five complete sentences. Make the description detailed.

List some practical difficulties involved in getting accurate results from a telephone survey.

List some practical difficulties involved in getting accurate results from a mailed survey.

With your classmates, brainstorm some ways you could overcome these problems if you needed to conduct a phone or mail survey.

The instructor takes her sample by gathering data on five randomly selected students from each Lake Tahoe Community College math class. The type of sampling she used is

• cluster sampling
• stratified sampling
• simple random sampling
• convenience sampling

A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every eighth house in the neighborhood around the park was interviewed. The sampling method was:

• simple random

Name the sampling method used in each of the following situations:

• A woman in the airport is handing out questionnaires to travelers asking them to evaluate the airport’s service. She does not ask travelers who are hurrying through the airport with their hands full of luggage, but instead asks all travelers who are sitting near gates and not taking naps while they wait.
• A teacher wants to know if her students are doing homework, so she randomly selects rows two and five and then calls on all students in row two and all students in row five to present the solutions to homework problems to the class.
• The marketing manager for an electronics chain store wants information about the ages of its customers. Over the next two weeks, at each store location, 100 randomly selected customers are given questionnaires to fill out asking for information about age, as well as about other variables of interest.
• The librarian at a public library wants to determine what proportion of the library users are children. The librarian has a tally sheet on which she marks whether books are checked out by an adult or a child. She records this data for every fourth patron who checks out books.
• A political party wants to know the reaction of voters to a debate between the candidates. The day after the debate, the party’s polling staff calls 1,200 randomly selected phone numbers. If a registered voter answers the phone or is available to come to the phone, that registered voter is asked whom he or she intends to vote for and whether the debate changed his or her opinion of the candidates.

A “random survey” was conducted of 3,274 people of the “microprocessor generation” (people born since 1971, the year the microprocessor was invented). It was reported that 48% of those individuals surveyed stated that if they had $2,000 to spend, they would use it for computer equipment. Also, 66% of those surveyed considered themselves relatively savvy computer users.

• Do you consider the sample size large enough for a study of this type? Why or why not?
• Based on your “gut feeling,” do you believe the percents accurately reflect the U.S. population for those individuals born since 1971? If not, do you think the percents of the population are actually higher or lower than the sample statistics? Why? Additional information: The survey, reported by Intel Corporation, was filled out by individuals who visited the Los Angeles Convention Center to see the Smithsonian Institute's road show called “America’s Smithsonian.”
• With this additional information, do you feel that all demographic and ethnic groups were equally represented at the event? Why or why not?
• With the additional information, comment on how accurately you think the sample statistics reflect the population parameters.

The Well-Being Index is a survey that follows trends of U.S. residents on a regular basis. There are six areas of health and wellness covered in the survey: Life Evaluation, Emotional Health, Physical Health, Healthy Behavior, Work Environment, and Basic Access. Some of the questions used to measure the Index are listed below.

Identify the type of data obtained from each question used in this survey: qualitative, quantitative discrete, or quantitative continuous.

• Do you have any health problems that prevent you from doing any of the things people your age can normally do?
• During the past 30 days, for about how many days did poor health keep you from doing your usual activities?
• In the last seven days, on how many days did you exercise for 30 minutes or more?
• Do you have health insurance coverage?

In advance of the 1936 Presidential Election, a magazine titled Literary Digest released the results of an opinion poll predicting that the republican candidate Alf Landon would win by a large margin. The magazine sent post cards to approximately 10,000,000 prospective voters. These prospective voters were selected from the subscription list of the magazine, from automobile registration lists, from phone lists, and from club membership lists. Approximately 2,300,000 people returned the postcards.

• Think about the state of the United States in 1936. Explain why a sample chosen from magazine subscription lists, automobile registration lists, phone books, and club membership lists was not representative of the population of the United States at that time.
• What effect does the low response rate have on the reliability of the sample?
• Are these problems examples of sampling error or nonsampling error?
• During the same year, George Gallup conducted his own poll of 30,000 prospective voters. These researchers used a method they called "quota sampling" to obtain survey answers from specific subsets of the population. Quota sampling is an example of which sampling method described in this module?

Crime-related and demographic statistics for 47 US states in 1960 were collected from government agencies, including the FBI's Uniform Crime Report . One analysis of this data found a strong connection between education and crime indicating that higher levels of education in a community correspond to higher crime rates.

Which of the potential problems with samples discussed in 1.2 Data, Sampling, and Variation in Data and Sampling could explain this connection?

YouPolls is a website that allows anyone to create and respond to polls. One question posted April 15 asks:

“Do you feel happy paying your taxes when members of the Obama administration are allowed to ignore their tax liabilities?” (lastbaldeagle. 2013. On Tax Day, House to Call for Firing Federal Workers Who Owe Back Taxes. Opinion poll posted online at: http://www.youpolls.com/details.aspx?id=12328 (accessed May 1, 2013).)

As of April 25, 11 people responded to this question. Each participant answered “NO!”

Which of the potential problems with samples discussed in this module could explain this connection?

A scholarly article about response rates begins with the following quote:

“Declining contact and cooperation rates in random digit dial (RDD) national telephone surveys raise serious concerns about the validity of estimates drawn from such research.”(Scott Keeter et al., “Gauging the Impact of Growing Nonresponse on Estimates from a National RDD Telephone Survey,” Public Opinion Quarterly 70 no. 5 (2006), http://poq.oxfordjournals.org/content/70/5/759.full (accessed May 1, 2013).)

The Pew Research Center for People and the Press admits:

“The percentage of people we interview – out of all we try to interview – has been declining over the past decade or more.” (Frequently Asked Questions, Pew Research Center for the People & the Press, http://www.people-press.org/methodology/frequently-asked-questions/#dont-you-have-trouble-getting-people-to-answer-your-polls (accessed May 1, 2013).)

• What are some reasons for the decline in response rate over the past decade?
• Explain why researchers are concerned with the impact of the declining response rate on public opinion polls.

1.3 Frequency, Frequency Tables, and Levels of Measurement

Fifty part-time students were asked how many courses they were taking this term. The (incomplete) results are shown below:

• Fill in the blanks in Table 1.33 .
• What percent of students take exactly two courses?
• What percent of students take one or two courses?

Sixty adults with gum disease were asked the number of times per week they used to floss before their diagnosis. The (incomplete) results are shown in Table 1.34 .

• Fill in the blanks in Table 1.34 .
• What percent of adults flossed six times per week?
• What percent flossed at most three times per week?

Nineteen immigrants to the U.S were asked how many years, to the nearest year, they have lived in the U.S. The data are as follows: 2 ; 5 ; 7 ; 2 ; 2 ; 10 ; 20 ; 15 ; 0 ; 7 ; 0 ; 20 ; 5 ; 12 ; 15 ; 12 ; 4 ; 5 ; 10 .

Table 1.35 was produced.

• Fix the errors in Table 1.35 . Also, explain how someone might have arrived at the incorrect number(s).
• Explain what is wrong with this statement: “47 percent of the people surveyed have lived in the U.S. for 5 years.”
• Fix the statement in b to make it correct.
• What fraction of the people surveyed have lived in the U.S. five or seven years?
• What fraction of the people surveyed have lived in the U.S. at most 12 years?
• What fraction of the people surveyed have lived in the U.S. fewer than 12 years?
• What fraction of the people surveyed have lived in the U.S. from five to 20 years, inclusive?

How much time does it take to travel to work? Table 1.36 shows the mean commute time by state for workers at least 16 years old who are not working at home. Find the mean travel time, and round off the answer properly.

Forbes magazine published data on the best small firms in 2012. These were firms which had been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion. Table 1.37 shows the ages of the chief executive officers for the first 60 ranked firms.

• What is the frequency for CEO ages between 54 and 65?
• What percentage of CEOs are 65 years or older?
• What is the relative frequency of ages under 50?
• What is the cumulative relative frequency for CEOs younger than 55?
• Which graph shows the relative frequency and which shows the cumulative relative frequency?

Use the following information to answer the next two exercises: Table 1.38 contains data on hurricanes that have made direct hits on the U.S. Between 1851 and 2004. A hurricane is given a strength category rating based on the minimum wind speed generated by the storm.

What is the relative frequency of direct hits that were category 4 hurricanes?

• Not enough information to calculate

What is the relative frequency of direct hits that were AT MOST a category 3 storm?

1.4 Experimental Design and Ethics

How does sleep deprivation affect your ability to drive? A recent study measured the effects on 19 professional drivers. Each driver participated in two experimental sessions: one after normal sleep and one after 27 hours of total sleep deprivation. The treatments were assigned in random order. In each session, performance was measured on a variety of tasks including a driving simulation.

Use key terms from this module to describe the design of this experiment.

An advertisement for Acme Investments displays the two graphs in Figure 1.14 to show the value of Acme’s product in comparison with the Other Guy’s product. Describe the potentially misleading visual effect of these comparison graphs. How can this be corrected?

The graph in Figure 1.15 shows the number of complaints for six different airlines as reported to the US Department of Transportation in February 2013. Alaska, Pinnacle, and Airtran Airlines have far fewer complaints reported than American, Delta, and United. Can we conclude that American, Delta, and United are the worst airline carriers since they have the most complaints?

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National Statistical

News and insight from the office for national statistics, understanding the new income statistics for local areas: different measures for better insights.

• Ainslie Woods
• April 30, 2024

The ONS has been busy producing faster, more detailed statistics using new data sources. Providing clearer insights at the local level is an important aspect of this work.  Ainslie Woods explains how our  local income statistics compare and why they offer a complementary picture of local areas.  

At the ONS we have continued to deliver our ambitious project to transform our sub-national statistics and produce more timely, granular and harmonised figures. In March we published gross disposable household income (GDHI) statistics broken down to very local areas for the first time, another major success for the project.   

Today we have published an analysis article comparing our admin-based local income statistics to our official survey-based statistics .   

But why are there three different local income statistics, and how do they relate to each other?  

To help users understand the differences between them, and the wider income landscape, we have published updated coherence and comparison guidance , which is available as a related download alongside our existing Income and earning statistics guide .   

The key differences between our local income statistics include the frameworks used to produce them, whether we’re measuring household, individual or household sector income and their geographic coverage.   

The frameworks for income statistics  

Income statistics can be produced according to different frameworks. A framework is essentially a set of internationally agreed standards used to produce statistics. The frameworks for income statistics are either a “micro-framework” or a “macro-framework”. Statistics using the micro-framework are collected for individual people or households and aggregated to the area of interest. These statistics are often very rich as they can contain detailed information about participants. Both the survey-based and admin-based local income statistics are produced according to this framework.  

By contrast, the macro-framework is used to compile measures of economic activity. Statistics using the macro-framework are produced according to the internationally agreed System of National Accounts . These are compiled by the ONS at the UK level using a multitude of different sources, including household and business surveys, as well as administrative records. This includes our GDHI figures.   

Individual and household income, or income of the households sector?   

Micro-framework income statistics can either be published for individuals or households (and sometimes occupied addresses). Those living in residential institutions, such as prisons and care homes, are not included. The survey-based measure uses household income, whereas the admin-based measure uses individual and occupied address income.  

On the other hand, GDHI figures (which are produced according to the macro-framework) measure the total income of the households sector. This is a concept within the System of National Accounts, which measures the total income of all people living in the UK, including those in residential institutions.  

Geographic coverage    

Statistics can be produced at many different levels of geography, covering different countries of the UK. There are specific small area geographies that are often used for reporting official statistics. Our admin-based local income statistics and GDHI statistics are available at a ‘statistical building block’ level (called a Lower layer Super Output Area), and many other higher aggregates.   

However, our survey-based local income statistics are only available for a slightly larger small area geography (called a Middle layer Super Output Area). More information can be found on our Statistical geographies webpage.  

There are several other differences between the local income statistics which are described further in the Coherence and comparison guidance for income statistics .  

Conclusion    

There are rich sets of income statistics for local areas, which we are continuing to develop and improve to meet user needs. Explaining the coherence and comparability of these statistics is fundamental to ensuring that they continue to serve the public good.   

If you would like to get in touch, or have any suggestions for how we can further improve the coherence, or explain the nuances and differences, of income statistics, please email us at [email protected]  

Ainslie Woods is the income and earnings coherence lead at the Office for National Statistics

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