By Chris Deziel
Updated Aug 30, 2022

Darkdiamond67/iStock/GettyImages

Imagine scoring 80 % on a test while the class average is 50 %. That tells you you did better than most, but it doesn’t reveal how far above the mean you truly are. A Z‑score gives you that deeper insight by factoring in the spread of all scores. It is calculated by subtracting the mean score from your individual score and dividing the result by the standard deviation. You can even convert the Z‑score into a percentile to see exactly where you stand among your peers.

Why Z‑Scores Matter

Known as a standard score, the Z‑score is a cornerstone of statistical analysis because it normalises data across different distributions. For instance, if your test score is 80 and the mean is 50, you’re above average, but you still need to know how many classmates performed as well as you. A high Z‑score indicates you belong to a select group of top performers, while a low Z‑score signals you’re closer to the bottom of the curve. The same principle applies to other measurements such as weight, height, or test scores in any field.

How to Compute a Z‑Score

For any dataset with a mean (M) and a standard deviation (SD), the Z‑score for a specific observation (D) is calculated as:

(D – M) / SD = Z‑score

Before applying the formula, you must first determine the mean and the standard deviation:

Mean = (sum of all scores) / (number of respondents)

To find the standard deviation, subtract the mean from each score, square the difference, sum all squared differences, divide by the number of respondents, and finally take the square root:

SD = √[(Σ (score – mean)²) / N]

Example: Calculating a Z‑Score

Consider a test with a maximum score of 100 taken by ten students, including Tom. The scores are:

• Tom – 75
• 67, 42, 82, 55, 72, 68, 75, 53, 78

1. Compute the mean: (75 + 67 + 42 + 82 + 55 + 72 + 68 + 75 + 53 + 78) / 10 = 66.7.

2. Find the standard deviation:

• Subtract the mean from each score and square the result:
• (75 – 66.7)² = 69.89
• (67 – 66.7)² = 0.09
• (42 – 66.7)² = 605.29
• (82 – 66.7)² = 234.49
• (55 – 66.7)² = 137.29
• (72 – 66.7)² = 28.09
• (68 – 66.7)² = 1.69
• (75 – 66.7)² = 69.89
• (53 – 66.7)² = 181.69
• (78 – 66.7)² = 127.69

Sum of squared differences = 1,536.6. Divide by 10 to get 153.66, then take the square root: SD ≈ 12.4.

3. Calculate Tom’s Z‑score:

Z = (75 – 66.7) / 12.4 ≈ 0.669.

A Z‑score of 0.669 corresponds to the 75th percentile on the standard normal distribution, meaning Tom outperformed about 75 % of his peers and was surpassed by roughly 25 %.