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AP®StatisticsCollege BoardStudy GuidesUnit 1: Exploring One-Variable DataThe Normal DistributionStandardized z-scores

Standardized z-scores (College Board AP® Statistics)

Study Guide

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Standardized z-scores

What is the standard normal distribution? 

  • The standard normal distribution is a normal distribution where the mean is 0 and the standard deviation is 1

    • It is denoted by Z

Why is the standard normal distribution important?

  • Any normal distribution curve can be transformed to the standard normal distribution curve by a horizontal translation and a horizontal stretch

  • We have the relationship:

    • begin mathsize 16px style Z equals fraction numerator X minus mu over denominator sigma end fraction end style

    • where X is a normal distribution with mean mu and standard deviation sigma

  • For any value of x, a z-score (z-value) can be calculated

    • This measures how many standard deviations the value of x is away from the mean

    • z space equals space fraction numerator x minus mu over denominator sigma end fraction

    • If a value of x is less than the mean then the z-score will be negative

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.

(C) Lisa's math test percentage is 20% below the average math test result for other children her age nationwide.

(D) Lisa's math test percentage is 0.2 times the average math test percentage for other children her age nationwide.

(E) 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

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    Author
    Reviewer
    Naomi C

    Author: Naomi C

    Expertise: Maths

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.