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The Goodness-of-Fit Test

Project: The Frequency of First Digits

In this project students will see the results of Benford’s Law. Benford’s Law is a popular mathematical curiosity that the students will enjoy. It states that the first digits of numbers are not uniformly distributed. There are far more ones than nines. What is happening is that it is the logarithm of the first digits that is distributed uniformly. This project will give students the opportunity to use the chi-square goodness-of-fit test, and it will expose them to a counterintuitive mathematical law that may catch their interest and prompt them to explore some mathematics independently.

Objective: Perform a goodness-of-fit chi-square hypothesis test on a set of data to determine if the first digits are uniformly distributed over the set \left \{1, 2, 3, 4, 5, 6, 7, 8, 9 \right \}.


  1. Collect data and record the frequency of each possible first digit 1 to 9. Any source of data with a large range can be used. An atlas or almanac will make a good source of data. Populations of countries or cities, lengths of rivers, atomic weights, addresses, or every number in an addition of the newspaper are all good examples of sets of data that can be used. Collect a data set with at least 100 elements. Cite the source of your data.
  2. Calculate the expected frequency of each fist digit. One would expect the first digits to be evenly distributed among the numbers 1 to 9.
  3. State the null and alternative hypotheses for your data.
  4. Use the chi-square distribution table to write a rule for rejecting the null hypothesis at the 0.05 significance level.
  5. Calculate the chi-square statistic to compare the observed and expected frequencies.
  6. Determine whether to reject or to fail to reject the null hypothesis.
  7. Write a summary statement based on the results of your test.

Test of Independence

Extension: Statistics in the Social Sciences

The chi-square test of independence is frequently used in the social sciences to access if two factors are related. Being exposed to some of the many useful and widespread applications of this test and to applications of statistics in general, motivates students to learn and remember the material in this course.


Students can look for examples of research done in the social sciences that make use of the chi-square test of independence. These can be found in advanced text books, scientific journal, or by searching the internet. The students can focus on the social science that most interests them. Some of the possible areas that can be explored are psychology, politics, education, or cultural studies.

Many college majors require a basic statistics course. Statistics is, in fact, a more common requirement than calculus. Students can choose a college or university that interests them and look at which degrees and majors require a basic statistics class and which require more advanced work in statistics. This could be done in conjunction with the counseling department. Students can make a display or short presentation to share what they found with the other students at their school. This will encourage career planning and goal setting in the student population and increase student interest in the school’s statistics program.


Students can design and conduct an experiment or survey in a social science area and use the chi-square test of independence to analyze the results. Psychological experiments are often of interest to students, but any area of social science can be used. Proper experimental and survey techniques can be reviewed in chapter six of this text. Excel can be used to perform calculations and display data. Students can write a report and/or give a presentation to the class explaining their experiment or survey and interpreting what the results of the chi-squared test of independence implies for their particular topic of inquiry.

Test One Variance

Extension: The Chi-Squared Tests in Biology

The three chi-squared tests studied in this chapter are often used in biological studies. This would be a good time to do a cross-curricular project with the biology department. Biostatistics is an emerging field of study. Here are some ideas of how statistics and biology could be studied together.


Students can look for examples of biological studies that make use of any of these chi-squared tests. These might be found in an advance biology or biostatistics texts, scientific journals, or by searching the internet. The students could write a short analysis of the study and/or bring it in to share with the rest of the class.

Students can find universities that offer degrees in biostatistics and examine the programs that they offer. They should look at the following areas:

  • Who are the professors and what are their areas of expertise?
  • What classes are required for the degree? What are some of the electives?
  • What are the prerequisites of the program?
  • What are the typical standardized test scores and grade point averages of students admitted to the program?
  • What are students that completed the degree doing now?


Students can design and conduct a biological study that makes use of some, or all, of the chi-squared tests studied in this chapter. This would be best done in conjunction with a biology class. Usually students currently taking statistics have already completed a basic biology class, but may currently be in an advanced or AP biology course.

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