− 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 of scores above the mean will always equal the distance below the mean.
8 This person would be considered an outlier because he or she falls so far away from the mean.
9 The range would not be affected at all by adding scores to the center of the distribution. However, the variance and standard deviation would become smaller.
10 This means that the values all fall on the same scale. The standard deviation is on the scale of the normal curve, which has unique properties that we will encounter in later modules.
11 The standard deviation represents the standardized average deviation that is applicable to the normal curve. The mean absolute deviation uses the absolute value of the deviation scores.
12 The mean absolute deviation does not fall within known places on the normal curve, which is extremely useful in the practice of statistics. Because of this, the mean absolute deviation is not commonly used.
Multiple-Choice Questions
The coach of a basketball team is interested in assessing how well his team shoots free throws. Here is the number of successful free throws that each member of his team made during their last practice session. Use this information for Questions 1 to 5.
1 The coach is asked to provide a quick measure of the dispersion for his team’s free throws. What is the range for his team?011112
2 The coach has more time on his hands and determines the variance. What is it?11.29 (using N), 12.32 (using n − 1)15.52 (using N), 16.92 (using n − 1)18.45 (using N), 13.21 (using n − 1)4.18 (using N), 4.56 (using n − 1)
3 The coach realizes that the variance is difficult to interpret and now wants to know the standard deviation for his team. What is the team’s standard deviation?5.09 (using N), 5.32 (using n − 1)1.18 (using N), 1.23 (using n − 1)4.29 (using N), (using n − 1)3.94 (using N), 4.11 (using n − 1)
4 Shortly after obtaining these data, a new player is added to the team, who makes 10 free throws. If you were to incorporate this new player’s score, which measure of dispersion would not be affected?VarianceStandard deviationRangeMean
5 The coach is interested in using the data from his team to learn about the number of free throws made by all of the teams in the league. This is an example ofdescriptive statistics.inferential statistics.central tendency.dispersion.
6 A distribution of scores is discovered to be highly leptokurtic. How much dispersion would you expect?A small amount because many scores are close to the meanA large amount because many scores are close to the meanA small amount because many scores are distant from the meanA large amount because many scores are distant from the mean
7 How much dispersion
