Chapter 5: Z-Scores

If your exam score is X=60, which set of parameters would give you the best grade?

a. μ = 65 and σ = 5

b. μ = 65 and σ = 2

c. μ = 70 and σ = 5

d. μ = 70 and σ = 2

a. μ=65, and σ=5

For a distribution of exam scores with μ=70, which value for the standard deviation would give the highest grade to a score of X=75?

a. σ = 1

b. σ = 2

c. σ = 5

d. σ = 10

a. σ = 1

Last week Sarah had a score of X = 43 on a Spanish exam and a score of X = 75 on an English exam. For which exam should Sarah expect the better grade?

a. Spanish

b. English

c. The two grades should be identical.

d. Impossible to determine without more information

d. Impossible to determine without more information

Of the following z-score values, which one represents the most extreme location on the left-hand side of the distribution?

a. z = +1.00

b. z = +2.00

c. z = −1.00

d. z = −2.00

d. z = −2.00

Of the following z-score values, which one represents the location closest to the mean?

a. z = +0.50

b. z = +1.00

c. z = −1.00

d. z = −2.00

a. z = +0.50

For a population with μ = 100 and σ = 20, what is the z-score corresponding to X = 105?

a. +0.25

A population of scores has μ = 44. In this population, an X value of 40 corresponds to z = −0.50. What is the population standard deviation?

a. 2

b. 4

c. 6

d. 8

d. 8

A population of scores has σ = 4. In this population, an X value of 58 corresponds to z = 2.00. What is the population mean?

a. 54

b. 50

c. 62

d. 66

b. 50

A population of scores has σ = 10. In this population, a score of X = 60 corresponds to z = −1.50. What is the population mean?

a. −30

b. 45

c. 75

d. 90

c. 75

A population with μ = 85 and σ = 12 is transformed into z-scores. After the transformation, the population of z-scores will have a standard deviation of _____

a. σ = 12

b. σ = 1.00

c. σ = 0

d. cannot be determined from the information given

b. σ = 1.00

A population has μ = 50 and σ = 10. If these scores are transformed into z-scores, the population of z-scores will have a mean of ____ and a standard deviation of ____.

a. 50 and 10

b. 50 and 1

c. 0 and 10

d. 0 and 1

d. 0 and 1

Which of the following is an advantage of transforming X values into z-scores?

a. All negative numbers are eliminated.

b. The distribution is transformed to a normal shape.

c. All scores are moved closer to the mean.

d. None of the other options is an advantage

d. None of the other options is an advantage

A distribution with μ = 47 and σ = 6 is being standardized so that the new mean and standard deviation will be μ = 100 and σ = 20. What is the standardized score for a person with X = 56 in the original distribution?

a. 110

b. 115

c. 120

d. 130

d. 130

A distribution with μ = 35 and σ = 8 is being standardized so that the new mean and standard deviation will be μ = 50 and σ = 10. In the new, standardized distribution your score is X = 60. What was your score in the original distribution?

a. X = 45

b. X = 43

c. X = 1.00

d. impossible to determine without more information

b. X = 43

Using z-scores, a population with μ = 37 and σ = 6 is standardized so that the new mean is μ = 50 and σ = 10. How does an individual’s z-score in the new distribution compare with his/her z-score in the original population?

a. new z = old z + 13

b. new z = (10/6)(old z)

c. new z = old z

d. cannot be determined with the information given

c. new z = old z

For a sample with M = 60 and s = 8, what is the z-score corresponding to X = 62?

a. 2

b. 4

c. 0.25

d. 0.50

c. 0.25

For a sample with a standard deviation of s = 5, what is the z-score corresponding to a score that is located 10 points below the mean?

a. −10

b. +2

c. −2

d. cannot answer without knowing the mean

c. -2

If a sample with M = 60 and s = 8 is transformed into z-scores, then the resulting distribution of z-scores will have a mean of ______ and a standard deviation of _____.

a. 0 and 1

b. 60 and 1

c. 0 and 8

d. 60 and 8 (unchanged)

a. 0 and 1

In N = 25 games last season, the college basketball team averaged μ = 74 points with a standard deviation of σ = 6. In their final game of the season, the team scored 90 points. Based on this information, the number of points scored in the final game was _____.

a. a little above average

b. far above average

c. above average, but it is impossible to describe how much above average

d. There is not enough information to compare last year with the average

b. far above average

Under what circumstances would a score that is 15 points above the mean be considered to be near the center of the distribution?

a. when the population mean is much larger than 15

b. when the population standard deviation is much larger than 15

c. when the population mean is much smaller than 15

d. when the population standard deviation is much smaller than 15

b. when the population standard deviation is much larger than 15

Under what circumstances would a score that is 20 points above the mean be considered to be an extreme, unrepresentative value?

a. when the population mean is much larger than 20

b. when the population standard deviation is much larger than 20

c. when the population mean is much smaller than 20

d. when the population standard deviation is much smaller than 20

d. when the population standard deviation is much smaller than 20

Z-Score

Specifies the precise location of each X value within a distribution. The sign of the z-score (+ or −) signifies whether the score is above the mean (positive) or below the mean (negative). The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and μ.

Standardized Distribution

Composed of scores that have been transformed to create predetermined values for μ and σ. Standardized distributions are used to make dissimilar distributions comparable.

Raw Score

The most basic, original score on a test or assessment, representing the direct number of points earned by a participant without any statistical adjustment or conversion

Deviation Score

The difference between an individual data point and the mean (average) of a dataset, showing how far a specific value is from the typical value. Z = Score - Mean / Standard Deviation

Z-score Transformation

A statistical process that converts a raw score into a standardized score, or z-score, indicating how many standard deviations a data point is from the mean of its distribution. The formula for a z-score is Z = (X - μ) / σ

Standardized Score

A raw score transformed to a common scale by subtracting the mean and dividing by the standard deviation of the data set.