Skill: Calculate and interpret z-scores

A z-score tells you how many standard deviations a data value is from the mean.

Use the formula

$$z=\frac{x-\mu}{\sigma}$$

where:

• $x$ is the data value,
• $\mu$ is the mean,
• $\sigma$ is the standard deviation.

Worked example

The scores on a test have a mean of $70$ and a standard deviation of $8$. A student scores $82$. Find and interpret the student’s z-score.

Step 1: Identify the values.

$$x=82,\qquad \mu=70,\qquad \sigma=8$$

Step 2: Substitute into the formula.

$$z=\frac{82-70}{8}$$

Step 3: Calculate.

$$z=\frac{12}{8}=1.5$$

So, the student’s z-score is

$$\boxed{z=1.5}$$

Step 4: Interpret the z-score.

The positive sign means the score is above the mean. The value $1.5$ means the score is 1.5 standard deviations above the mean.

If the test scores are approximately normally distributed, a z-score of $1.5$ corresponds to about the 93rd percentile. This means the student scored higher than approximately $93\%$ of the scores.

Remember:

• $z>0$: the value is above the mean.
• $z<0$: the value is below the mean.
• $z=0$: the value equals the mean.
• A larger absolute value, such as $\vert z \vert =2.5$, means the value is farther from the mean.