Skill: Compare data values from different distributions

To compare a value from one distribution with a value from another, compare how far each value is from its own distribution’s mean. A useful measure is the z-score:

$$z=\frac{x-\text{mean}}{\text{standard deviation}}$$

A larger z-score means the value is relatively higher within its distribution.

Example

A student receives:

• 82 on a mathematics test. The class mean is 75, with a standard deviation of 5.
• 91 on a history test. The class mean is 85, with a standard deviation of 10.

Which score is higher relative to its class?

Step 1: Find the mathematics z-score

$$z_{\text{math}}=\frac{82-75}{5} =\frac{7}{5} =1.4$$

The mathematics score is $1.4$ standard deviations above the class mean.

Step 2: Find the history z-score

$$z_{\text{history}}=\frac{91-85}{10} =\frac{6}{10} =0.6$$

The history score is $0.6$ standard deviations above the class mean.

Step 3: Compare the z-scores

$$1.4>0.6$$

Therefore, the mathematics score is higher relative to its class, even though the raw history score, $91$, is greater than the mathematics score, $82$.

The mathematics class has less variability, so being $7$ points above its mean represents a stronger relative performance than being $6$ points above the history mean. This comparison assumes that z-scores are reasonable summaries for the distributions, such as when the distributions are not extremely skewed or affected by unusual outliers.