Calculate and interpret z-scores

A z-score measures how far a data value lies from the mean in standard-deviation units, calculated as $z=(x-\mu)/\sigma$ using the relevant mean and standard deviation. Its sign indicates whether the value is below or above the mean, its magnitude indicates relative distance, and $z=0$ identifies the mean; for approximately normal data, z-scores also support comparison with standardized normal probabilities and percentiles. This scope excludes advanced inferential procedures such as z-tests and confidence intervals.

Detailed Explanation: 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.

Learn by doing: Calculate and interpret z-scores

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Statistics - Standard Deviation - Graph to Z-Score

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