2. What is the problem with using the sum of squares as a measure of variability, that is, why is variance a better measure?

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1 1. Identify and define the three main measures of variability: 2. What is the problem with using the sum of squares as a measure of variability, that is, why is variance a better measure? 3. What is the standard deviation our best measure of variability? 4. If the variance of a set of scores is what is the standard deviation? 5. If the standard deviation of a set of scores is 9.00 what is the variance? 6. Without calculating it, what must the variance of the following scores equal: 4, 4, 4, 4? What must the standard deviation equal? What must the sum of squares equal? Why? 7. The following scores were obtained on a quiz: { } a. Calculate the mean of the scores. b. Calculate the sum of squares. c. Calculate the sample variance. d. Calculate the standard deviation. 8. The following scores were obtained on another quiz { } a. Calculate the mean of the scores. b. Calculate the sum of squares. c. Calculate the sample variance. d. Calculate the standard deviation. 9. For the sets of data in exercises 7 and 8, what descriptive statistics are the same (if any) and which are different (if any)? What does this tell you about the measures of central tendency and the measures of variance? 10. Use the data below to answer the questions that follow. Here are the numbers of wins the New York Yankees for each of the last 10 seasons: Year Wins

2 a. Calculate the sum of squares. b. Calculate the sample variance. d. Calculate the estimate of the population variance. e. Calculate the estimate of the population standard deviation. 11. Use the data below to answer the questions that follow. Here are the numbers of wins the New York Yankees for each of the 10 seasons that preceded those in #10: Year Wins a. Calculate the sum of squares. b. Calculate the sample variance d. Calculate the estimate of the population variance. e. Calculate the estimate of the population standard deviation. 12. Use the data below to answer the questions that follow a. Calculate the sum of squares. b. Calculate the sample variance. 13. Using the data in #12, add a constant of 5 to each value and then answer each of the following questions. a. Calculate the sum of squares. b. Calculate the sample variance.

3 14. Using the data in #12, multiple each value by a constant of 2 and then answer each of the following questions. a. Calculate the sample mean. b. Calculate the sum of squares. c. Calculate the sample variance. d. Calculate the sample standard deviation. 15. How does the standard deviation help you interpret a set of data? Use the following to answer #16 19: A political scientist studied how satisfied people were with the President s job performance and Congress job performance. Each person was given a job performance satisfaction test on which scores ranged from 1 to 7, with higher scores indicating more satisfaction. The scores were: Individual President s Job Performance Congress Job Performance A 2 1 B 5 3 C 6 3 D 7 5 E Compute the mean of the satisfaction ratings for the President s and Congress job performance. 17. Compute the standard deviation for the satisfaction ratings for the President s and Congress job performance. Individual President s Job Performance Congress Job Performance X (X M) (X M) 2 X (X M) (X M) 2 A B C D E Based on the results, whose job performance are people more satisfied with? Why? 19. Based on the results, which job performance ratings are more consistent? Why 20. How accurate are people ate estimating length? In an experiment, people are presented with a length that was 25-cm in length for five seconds and asked to estimate the length of the rope. Ten people gave the following estimates (in cm): Compute the mean and standard deviation for these data. How accurate are the estimated considering the mean score across all participants? How does the standard deviation help to interpret the mean?

4 21. Following up on #20, in a second experiment, people are presented with the same 25-cm piece of rope, but for one minute before making their estimates of the ropes length. Ten people gave the following estimates (in cm): Compute the mean and standard deviation for these data. How accurate are the estimated considering the mean score across all participants? How does the standard deviation help to interpret the mean? 22. An organizational psychologist studied how satisfied employees were in two different companies. All employees were given a job satisfaction test on which scores could range from 1 to 7, with higher scores indicating greater satisfaction. The scores were as follows: Company A: Company B Compute the mean and standard deviation for each company. Based on the results, compare employee satisfaction in the two companies. How do the standard deviations help to interpret the means? 23. Suppose scores on a math achievement test approximate a normal distribution with M = 500, min = 350, and max = 650. Estimate the standard deviation of this distribution. 24. Two students who took a statistics class received the following Quiz scores (each Quiz is out of 100 points): Student A: 60, 90, 80, 60, 80 and Student B: 40, 100, 100, 40, 90. If you had an upcoming statistics test, who would you have rather had as a study partner, A or B? Support your answer. 25. Consider the two histograms displayed below. The histogram labeled X1 has a mean of 54 and the histogram labeled X2 has a mean of 53. Please indicate which one has a larger standard deviation and WHY that histogram has the larger standard deviation. (It is not necessary to do any calculations.) 6 X1 6 X2 5 5 Frequency Frequency Score Score

5 ANSWERS 1. Sum of Squares: The sum of the squared deviations from the mean. Variance: The average sum of the squared deviations from the mean, or the average sum of squares). Standard Deviation: The positive square root of the variance, which is a measure of the average deviation from the mean. 2. The sum of squares does not take into account the number of scores in the set of data. That is, you could have two sets of scores (as on p. 84) where one has a larger SS than the other, but there is less variability among the scores because the number of scores. 3. The standard deviation is an average difference between a score and the mean, and it is also in the original unit of measure, unlike the variance and the sum of squares, which are in units-squared Standard deviation = 0; Variance = 0; Sum of Squares = 0. All three measures of variability will be equal to zero, because there is no variation among the scores. 7. a. 6.5 b c d a. 6.5 b c d The sample mean and the range are the same. The sums of squares, sample variance, and sample standard deviation differ. This suggests that measures of central tendency are not affected by the variability in a set of data, and vice versa. 10. a b c. 3.9 d e a b c d e a. 22 b. 2.2 c a. 22 b. 2.2 c a. 16 b. 88 c. 8.8 d The standard deviation tells you the average difference you should expect to find between the mean of a set of data and a raw score. Thus, it can tell you how close a raw score is to the mean.

6 16. President = 5; Congress = President = 1.673; Congress = People are more satisfied with the president s job performance, because the mean job satisfaction rating is higher. 19. The ratings are more consistent for Congress, because the standard deviation is smaller. 20. The mean (26) suggests that the sample of people was accurate in estimated the length of the rope; however, the standard deviation (6.512) suggests individuals were not all that accurate. 21. The mean (24.5) suggests that the sample of people was accurate in estimated the length of the rope; and the the standard deviation (1.628) suggests individuals were not all that accurate. 22. Company A Mean = 4; Company A Standard Deviation = 1.414; Company B Mean = 4; Company B Standard Deviation = 0. The means for the companies are equal (both are 4) suggesting the overall satisfaction with each company is equal. However, there is no variability among the scores for company B (standard deviation is 0), hence, all individuals have the same level of satisfaction. In contrast, individuals in company B differ in their level of satisfaction, because the standard deviation is not Because there are approximately six standard deviations in a range of data (three above the mean and three below the mean) and the Range of these data is = 300, the standard deviation is about Both students have the same mean Quiz score (74); however, Student A s standard deviation for the quiz scores is 12 whereas the standard deviation of quiz scores for Student B is 28/ Because there is less variability among the quiz scores for Student A it would be better to study with Student A than with Student B. 25. The distribution in X2 would have a larger standard deviation, because the range is greater than in X1; hence, there is more variability among the scores.