Standardized z-scores (College Board AP® Statistics)

Revision Note

Author

Naomi C

Expertise

Maths

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:
- $Z = \frac{X - μ}{σ}$
- where $X$ is a normal distribution with mean $μ$ and standard deviation $σ$
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
- $z = \frac{x - μ}{σ}$
- 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.

(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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