How to compare test scores using z scores?

Why raw test scores can deceive you

A raw score tells you only the number of points you earned, not how well you did compared to everyone else. If you score 85 on an easy chemistry exam where the class average is 90, you scored below average. If you score 75 on a difficult physics exam where the class average is 60, your performance was strong.

To make a fair comparison across tests with different averages and difficulty spreads, you can evaluate your performance using z scores. A z score, also called a standard score, converts any raw score into a standard unit: the number of standard deviations the score sits above or below the mean.

The three-step formula for finding a z score

To find a z score, you need three numbers: the student score (x), the class mean (mu), and the standard deviation (sigma). Subtract the mean from the raw score, then divide that difference by the standard deviation. The formula is z = (x - mu) / sigma.

The sign and magnitude tell the story. A score of z = 0 means the result equals the exact mean. A positive z score means the result is above average, while a negative z score means it fell below average. A score of z = 1.5 indicates the score is 1.5 standard deviations above the class average.

Comparing two tests: a concrete example

Suppose Maya scores 82 on a biology exam and 78 on a history exam. In biology, the class mean is 74 with a standard deviation of 8. Her biology z score is (82 - 74) / 8 = 8 / 8 = 1.00.

In history, the class mean is 66 with a standard deviation of 6. Her history z score is (78 - 66) / 6 = 12 / 6 = 2.00. Even though her raw biology score was higher, Maya performed relatively better in history because her history score was two full standard deviations above the class average.

Finding your percentile rank using a standard normal table

Once you have a z score, you can look it up in a standard normal distribution table (or z table) to find the percentage of students who scored below you. A z score of 1.00 corresponds to approximately 0.8413, meaning about 84.1% of test-takers scored below that mark. Maya's history score with z = 2.00 corresponds to 0.9772, placing her in roughly the 98th percentile.