1. How large is the sum of squares between if the group means are equal?
2. How is the sum of squares within related to how spread out the points are within
the groups? Explain why.
3. Why are the squared differences between sample means and the grand mean
multiplied by n when computing the sum of squares between.
4. Compare and contrast the analysis of sample datasets 3 and 4.
For each distribution, a Chi Square test is performed to test
whether the observed frequencies differ significantly from the
expected frequencies. Computationally, this is done by
computing (E-O)2/E for each interval and summing the results
(E is the expected frequency and O is the observed frequency).
Most likely, the test will be not show a significant difference for
the normal distribution and will show a highly significant
difference for the uniform distribution. Of course, by chance,
your results may vary. Exercises
1. Compute the Chi Square test using a hand calculator and see if your result matches the
2. The intervals for the normal distribution frequencies are:
Note that since a normal distribution is continuous, no scores will fall exactly on the
Compute the expected frequencies based on the normal distribution and see if they match
those shown in the applet.
3. Set the actual distribution to "uniform" and keep sampling until you find a sample with
a probability value of about 0.50 (for deviations from a uniform distribution). Record
these results and show them to a friend who hasn't taken statistics (Don't show the
statistical analysis). Ask the friend how often they would expect to find such large
deviations from a uniform distribution if the numbers were really sampled from a
4. What are the assumption(s) of the Chi Square Test?
5. In the Texas lottery, balls with numbers ranging from 1-50 are put in a bin and 6 are
drawn without replacement. To see if some numbers came up more frequently than
expected by chance, you could take all the numbers selected in the last year and count the
frequencies of each of the 50 possible outcomes. Would any assumption of the Chi
Square test be violated if you used it to test for deviations from a uniform distribution? If
so, would the violation tend to increase or decrease the value of Chi Square?
6. There were 104 drawings in the Texas lottery in 1998. For each of the first 100
drawings, select one of the 6 numbers at random and count the number that fall in each of
the following intervals: 1-5, 6-10, 11-15, 16-20, 21-25, 26-30, 31-35, 36-40, 41-45,
46-50. Test whether the numbers deviate significantly from a uniform distribution (for
which the expected frequency for each interval is 10 since there are 100 numbers and 10
7. Compute the mean of the six numbers drawn for each of the first 100 drawings of
1998. Test whether the numbers deviate from a normal distribution. Hint: Standardize the
100 means and use the expected values shown in the simulation with the intervals shown
in Question 2 on this page.