score was 94, and Harry’s was 85.6.
Module 8 z Scores
Learning Objectives
Define the term standard score
Determine the advantages of a standard score relative to a regular score
Calculate a z score
Use z scores for comparisons based on the percentage above, below, or between given z scores
Module Summary
Standard scores are scores that have been converted to a standard scale of measurement. This means that they can be compared with other scores that have been placed on the same scale. In other words, you can compare a person’s height in feet with their weight in pounds by converting both values to standard scores. This is in contrast to raw scores, which are scores on their original scale.
Using z scores standardizes scores on the scale of standard deviation units. A z score of 1 means that the raw score fell 1 SD above the mean. The sign of a z score (+ or −) indicates the location of the score with regard to the mean (− = below the mean, + = above the mean). The formula for finding a z score is
The z score formula is merely a conversion. First, you determine how far your score is from the mean in raw units (X – M). Then, you determine how many standard deviations this distance is by dividing by s. This is similar to converting feet to inches or yards to feet.
After converting a score to a z score, you are able to use it in conjunction with your knowledge of the normal curve because z scores are normally distributed. You can find the exact percent location of your z score in a normal distribution. In other words, you can find the percentage of scores above your z score and the percentage of scores below your z score. These percentages are found using a Normal Curve Table, which can be found in Appendix A of your text. The percentages on the Normal Curve Table indicate the percentage of scores at or below a particular z-score value. To determine the percentage of scores above a z-score value, subtract the percentage from 1. Thus, a z score of 1.33 will be greater than or equal to 90.82% of the scores of the distribution and less than 9.18% of the scores of the distribution. Finally, you can use the normal table to determine how many scores fall between a particular z score and the mean by subtracting 0.5 from the percentage found in the table.
An important aspect of a z score is that it includes a distribution’s central tendency (mean) and dispersion (standard deviation) in its calculation. This enables you to compare scores from completely different scales (e.g., tests in different classes) once they have been converted to z scores. Yes, this means you can finally compare apples with oranges!
Computational Exercises
1 Using a distribution with a mean of 25 and a standard deviation of 2, find the z scores for the following scores:3010401
2 Sana just took a very difficult cognitive psychology test and a very easy calculus test. On the psychology test, she earned a grade of 78 and found out that the class had a mean of 72 with a standard deviation of 4. In contrast, she earned a grade of 87 on the calculus test and found out that the class had a mean of 89 and a standard deviation of 6. How would you explain to Sana that she should feel good about her psychology test grade?
3 Here are the scores obtained by each of the five members of Tim’s bowling team during last night’s tournament. Convert each of the scores to z scores:133, 159, 112, 131, 169
4 Two of the local little league baseball teams want to have a contest to determine which team can catch more fly balls. Here are the means and standard deviations for the number of fly balls caught by all the members of each team.Team A: mean = 7.9, SD = 2.3Team B: mean = 8.3, SD = 3.9Saul, Team A’s best player, caught 10 fly balls. Ari, Team B’s best player, also caught 10 fly balls. Which of these two players is better relative to the performance of his team?
5 Here is the number of court cases a particular judge saw per day in the past week.7, 5, 9, 3, 4What is the z score for the day in which he saw the most cases? What percentage falls between this score and the mean?
6 Based on the information in Question 3, what percentage of the normal curve is equal to or below the lowest z score? What about the highest z score?
7 In Question 3, the person who bowled a 159 thinks that she is better than the average bowler. How much percentage above the mean is this person’s z score?
8 See the information given in Question 4. Jose caught four fly balls and is on Team A. What proportion of the normal curve falls between Jose’s z score and the mean of Team A?
Computational Answers
1 a. 2.5, b. −7.5, c. 7.5, d. −12
2 You should tell her that she did much better on the psychology test relative to everyone else as she had a z score of 1.5. However, on the calculus test, she didn’t do as well compared with everyone else as she had a z score of −0.33.
3 M = 140.8, s = 20.56
4 Saul’s z score = 0.91. Ari’s z score = 0.44. Saul is better relative to his team than Ari is.
5 The z score for 9 (Day 3) was 1.58 (M = 5.6, s = 2.15). Approximately 44% of the normal curve falls between this score and the mean.
6 8% (.08); 91% (.91).
7 Thirty-one percent of the normal distribution falls between a z score of 0.89 and the mean of 0.
8 Approximately 46% of the normal curve falls between Jose’s z score and the mean of Team A.
True/False Questions
1 Standardized scores tell you the position of a score in reference to all of the other scores in a distribution.
2 A z score of 1.5 corresponds to a score that is 1.5 standard deviations above the mean.
3 A z score tells you the location of a score in relation to the mean.
4 The scale of z scores is the scale of variance units.
5 The proportion of the normal curve below a z score of 0.12 is .45.
6 You can compare the scores from entirely different groups after converting them to z scores.
7 Brand A sells an average amount of 5.4 units per day, with a standard deviation of 1.2. Brand B sells an average amount of 3.2 units per day, with a standard deviation of 2.5. A z score of 1 for these two brands would represent different raw scores but indicate a score that fell in the same place on the normal curve.
True/False Answers
1 True
2 True
3 True
4 False
5 False
6 True
7 True
