AP®StatisticsCollege BoardStudy GuidesUnit 1: Exploring One-Variable Data and Collecting DataSummary StatisticsStandardized Scores

Standardized Scores (College Board AP® Statistics): Study Guide

Syllabus Edition

First teaching 2026

First exams 2027

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on 21 May 2026

Standardized z-scores

What are z-scores?

A z-score for a data value tells you the number of standard deviations it is away from the mean
A positive z-score means the value is bigger than the mean
- For example, a z-score of 2.5 means it is 2.5 standard deviations bigger than the mean
A negative z-score means the value is smaller than the mean
- For example, a z-score of -1.2 means it is 1.2 standard deviations smaller than the mean
The z-score can be calculated using $z = \frac{x - μ}{σ}$
- $μ$ is the population mean
- $σ$ is the population standard deviation
For example, suppose the mean is 50 and the standard deviation is 4
- The z-score for 56 is $\frac{56 - 50}{4} = 1.5$
- The z-score for 43 is $\frac{43 - 50}{4} = - 1.75$
If the population mean and standard deviation are unknown, then the sample mean and standard deviation can be used

Worked Example

The standardized score ( $z$ -score) for Lisa's math test percentage, as compared to the math test results for other children of her age nationwide, is -0.2. Which of the following is the best interpretation of this standardized score?

(A) Lisa's math test percentage is 20%.

(B) Lisa's math test percentage is 0.2 standard deviations below the average math test result for other children her age nationwide.

(D) Only 0.2% of children Lisa's age have a lower math test percentage than she does.

Answer:

A $z$ -score measures how many standard deviations the value of $x$ is away from the mean

This means that -0.2 is 0.2 standard deviations below the mean

The correct answer is B

How can z-scores be used to compare data?

You can use z-scores to compare relative positions of values between distributions
The data value with a z-score closer to zero is the one that is relatively closer to the mean
The data value with a z-score further away from zero is the one that is relatively more extreme

Examiner Tips and Tricks

Be sure to talk about the relative position rather than the actual position.

Consider a distribution with a mean of 100 and a standard deviation of 10. The value 110 has a z-score of 1.

Consider a second distribution with a mean of 50 and a standard deviation of 5. The value 56 has a z-score of 1.2.

The data value in the first distribution is relatively closer to the mean. However, the data value in the second distribution is actually closer to the mean.

Worked Example

Sarah recently participated in two statewide academic assessments: a Mathematics exam and a History exam.

The distribution of the scores on the Mathematics exam is approximately symmetrical with a mean of 120 points and a standard deviation of 15 points. Sarah scored a 144 on the Mathematics exam.
The distribution of the scores on the History exam is approximately symmetrical with a mean of 75 points and a standard deviation of 8 points. Sarah scored an 89 on the History exam.

(a) Calculate the standardized score (z-score) for Sarah’s performance on the Mathematics exam. Interpret this value in context.

(b) Relative to the other students who took the exams, on which exam did Sarah perform better? Justify your answer using standardized scores.

Answer:

(a)

Use the formula $z = \frac{x - μ}{σ}$

$\frac{144 - 120}{15} = 1.6$

z-score for Mathematics is 1.6

Interpret the score using the definition of a z-score

Sarah’s score on the Mathematics exam is 1.6 standard deviations above the mean score of all students who took the Mathematics exam

(b)

Calculate the z-score for History

$\frac{89 - 75}{8} = 1.75$

Justify the use of using z-scores

The scores for both exams are measured on different scales with different means and standard deviations

Therefore, z-scores provide the best way to compare her relative standing

Compare the z-scores

Sarah performed better on the History exam relative to her peers because her z-score for History (1.75) is greater than her z-score for Mathematics (1.6)

This means her History score was further above the average in terms of standard deviations

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Standardized Scores (College Board AP® Statistics): Study Guide

Standardized z-scores

What are z-scores?

Worked Example

How can z-scores be used to compare data?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

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Author: Naomi C

Reviewer: Dan Finlay