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6. Descriptive Statistics
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Integrated Math 3

6.08 Z-scores

Lesson
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Lesson

Concept summary

To directly compare multiple normally distributed data sets, we need a common unit of measurement. In statistics involving the normal distribution, we use the number of standard deviations away from the mean as a standardized unit of measurement called a z-score.

z-score

A measurement that describes the position of a data point, measured in standard deviations, relative to the mean.

\displaystyle z=\dfrac{x-\mu}{\sigma}
\bm{z}
The z-score
\bm{x}
The data value
\bm{\mu}
The population mean
\bm{\sigma}
The population standard deviation
  • A positive z-score indicates the score was above the mean.
  • A z-score of 0 indicates the score was equal to the mean.
  • A negative z-score indicates the score was below the mean.
  • The larger the magnitude of the z-score, the further the score is from the mean

We can use z-scores along with the standard normal curve to compare values from different sets of data.

The standard normal distribution

A normal distribution with a mean of 0 and a standard deviation of 1

Worked examples

Example 1

Brock is applying to different colleges across America and needs to decide if he should emphasize his SAT score, ACT score, or both. The test scores for both the SAT and ACT are normally distributed. The data is summarized in the table provided.

Brock's scoreMeanStandard Deviation
SAT14501051211
ACT3020.85.7
a

Calculate and interpret the z-score for Brock's SAT score.

Approach

The formula for z-score is z=\dfrac{x-\mu}{\sigma}.

Solution

\displaystyle z\displaystyle =\displaystyle \dfrac{x-\mu}{\sigma}Formula
\displaystyle z\displaystyle =\displaystyle \dfrac{1450-1051}{211}Substitute
\displaystyle z\displaystyle =\displaystyle 1.89Evaluate

Brock's z-score for his SAT test is z=1.89 which means Brock scored 1.89 standard deviations above the mean.

Reflection

Remember that the z-score should be positive if the data value is above the mean, and negative if the data value is below the mean.

b

Calculate and interpret the z-score for Brock's ACT score.

Approach

The formula for z-score is z=\dfrac{x-\mu}{\sigma}.

Solution

\displaystyle z\displaystyle =\displaystyle \dfrac{x-\mu}{\sigma}Formula
\displaystyle z\displaystyle =\displaystyle \dfrac{30-20.8}{5.7}Substitute
\displaystyle z\displaystyle =\displaystyle 1.61Evaluate

Brock's z-score for his ACT test is z=1.61 which means Brock scored 1.61 standard deviations above the mean.

c

Determine which test Brock did better on relative to all other SAT and ACT test takers.

Approach

Use the z-scores from parts (a) and (b). The higher Brock's z-score, the better he did relative to the other test takers.

Solution

Brock did slightly better on his SAT test than he did on his ACT test since his z-score was higher.

Outcomes

M3.N.Q.A.1

Use units as a way to understand real-world problems.*

M3.N.Q.A.1.D

Choose an appropriate level of accuracy when reporting quantities.

M3.S.ID.A.5

Compute, interpret, and compare z-scores for normally distributed data in a real-world context.*

M3.MP2

Reason abstractly and quantitatively.

M3.MP3

Construct viable arguments and critique the reasoning of others.

M3.MP4

Model with mathematics.

M3.MP5

Use appropriate tools strategically.

M3.MP6

Attend to precision.

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