Student Study Guide to Accompany Statistics Alive!. Wendy J. Steinberg

store’s business hours, three people run in, stating that they are in a shoe emergency, and ask to purchase shoes. The shoe sizes of these three new customers are 8, 6, and 10. If they each purchase a pair of shoes, what will be the new standard deviation?

       9. How many of these 13 (including those added in Question 8) people fall within 1 SD of the mean?

       10. What is the mean absolute deviation of the original shoe sizes? How does this compare with the standard deviation?

       11. If you were to estimate the population variance in the shoe sizes of people who shop at this store from this sample, what formula would you use (using the original 10 scores)? What would be the population variance estimated from this sample?

       12. The store manager collects similar data for the following day and finds that the mean is substantially higher but the standard deviation is now 7.9. Which mean is more representative of the average shoe size of those who shop at this store?

      Computational Answers

      1 

      2 The sum of the deviations is 0.00. Variance = 126.99 (using N) or 133.67 (using n − 1); standard deviation = 11.27 (using N) or 11.56 (using n − 1).

      3 The highest grade is approximately 1.20 standard deviations above the mean. The lowest grade is approximately 1.7 SD below the mean.

      4 The new variance = 145.64 (using N) or 153.31 (using n − 1). The new standard deviation = 12.07 (using N) or 123.38 (using n − 1).

      5 The variance was the most affected by this change.

      6 Three scores: 98, 99, and 99.

      7 The variance = 6.60 (using N) or 7.33 (using n − 1). The standard deviation = 2.57 (using N) or 2.71 (using n − 1).

      8 The new standard deviation would be 2.42 (using N) or 2.52 (using n − 1).

      9 There are seven scores that fall within 1 SD above and below the mean.

      10 You would use the formula with n − 1 in the denominator because the sample size is less than 30. The estimated population variance would be 7.33.

      11 The first mean is more representative because the scores are more tightly clustered about the mean.

      True/False Questions

      1 The range is a very sensitive measure of central tendency.

      2 Adding 25 different scores to the center of a data set will affect the range.

      3 A deviation score provides a measure of the score’s distance from the mean.

      4 The sign (+ or −) of a deviation score indicates its location in relationship to the mean.

      5 Deviation scores always sum to 1.

      6 The variance is the square of the average distance of scores from the mean, measured in squared distance units.

      7 Standard deviations from different samples of the same population can be compared.

      8 The standard deviation is calculated from the square root of the variance to revert the measure to linear units.

      9 The standard deviation is the most commonly used measure of dispersion because of its interpretability and applicability to the normal curve.

      10 The mean absolute deviation is used more frequently than the standard deviation.

      11 Descriptive statistics are used to describe the characteristics of a sample.

      12 The mean and standard deviation are examples of inferential statistics.

      13 When using a sample to infer about a population, you should use n − 1 in the denominator.

      14 Using n − 1 in the denominator corrects for the bias of sample variance being larger than the population variance.

      15 There is a debate in the social sciences regarding the appropriate use of N as opposed to n − 1 in the denominator of sample variances.

      True/False Answers

      1 False

      2 False

      3 True

      4 True

      5 False

      6 True

      7 True

      8 True

      9 True

      10 False

      11 True

      12 False

      13 True

      14 False

      15 True

      Short-Answer Questions

      1 Why are measures of dispersion important when describing a sample?

      2 What aspects of the range make it a poor measure of dispersion?

      3 What are the steps involved in finding the variance?

      4 Why is it necessary to square the deviation scores when finding the variance?

      5 What unit of measurement is used for variance? What unit of measurement is used for standard deviation? How are these different?

      6 How are area units changed to linear units?

      7 Why can we always expect deviation scores to sum to 0?

      8 When running a marathon, there is a runner who runs at a time that is 4 SD below the mean. Would you consider this person an outlier? Why or why not?

      9 How would the three measures of dispersion (variance, range, and standard deviation) be affected by adding a large number of scores close to the mean of any given distribution?

      10 What does it mean for a measure to be a standardized one?

      11 What is the difference between a standard deviation and a mean absolute deviation?

      12 Why is the mean absolute deviation not commonly used?

      Answers

      1 Dispersion is important because it indicates the extent to which scores cluster around the mean or are very distant from the mean. This will help you determine how viable the mean is as a single descriptor of the data set. For example, a 0 to 10 scale with a mean of 5 and a standard deviation of 1 indicates that, on average, a score will deviate 1 point from the mean. This suggests that the majority of scores will fall between 4 and 6, making the mean a very good descriptor. However, if the mean is 5 and the standard deviation is 5, it means that the average deviation from the mean is 5 points, indicating that the scores fall everywhere on the scale.

      2 The range does not consider all of the scores in the distribution. Also, it can be heavily influenced by extreme scores (drastically different upper or lower scores), and it is unaffected by the addition of nonextreme scores (scores that are not the upper or lower limits).

      3 First, the deviation scores must be found. Then these scores must be squared. The squared deviation scores are then summed. Finally, this summed squared deviation score is divided by N or by n − 1.

      4 This is because the deviation scores will sum to 0, which would indicate that there is no variability in the sample. Squaring the deviation scores creates a positive sum.

      5 Variance is in area units. The standard deviation is in linear units. The difference is that the area units are squared, meaning one cannot apply them directly to the original scale of measurement. The linear units are in the same metric as the original scale.

      6 Area units are changed to linear units by taking the square root of the area unit.

      7 Deviation scores always sum to 0 because they represent each score’s numerical distance from the mean. Because the mean is the numerical center of the distribution, the distance