The normal distribution and z-scores

The normal distribution

Many measurements, such as heights, test scores and measurement errors, follow a normal distribution: a symmetric bell-shaped curve. It is fully described by its mean μ (the centre) and standard deviation σ (the spread). The mean, median and mode coincide at the peak.

The empirical rule (68-95-99.7)

For a normal distribution:

• About 68 percent of values lie within 1 SD of the mean.
• About 95 percent lie within 2 SDs.
• About 99.7 percent lie within 3 SDs.

So values more than 3 SDs from the mean are very rare (well under 1 percent).

z-scores

A z-score tells you how many SDs a value is from the mean:

z = (x − μ) / σ

• Positive z: above the mean. Negative z: below it. z = 0: exactly at the mean (the 50th percentile).
• The standard normal distribution has μ = 0 and σ = 1. Converting any normal variable to z lets us use one table (or calculator) for every normal problem.

Some useful values: z = 1 is about the 84th percentile; z = 1.645 about the 95th; z = 1.96 about the 97.5th, which leaves 2.5 percent in each tail for a 95 percent central region.

Worked example 1

Exam scores are approximately normal with μ = 70 and σ = 8.

1. A score of 86 has z = (86 − 70)/8 = 2. By the rule, about 95 percent are within 2 SDs, so about 2.5 percent score above 86 and the score is near the 97.5th percentile.
2. What percent score between 62 and 78? These are 70 ± 8, which is within 1 SD, so about 68 percent.
3. A score of 82 has z = (82 − 70)/8 = 1.5.

Worked example 2: comparing across tests

Anna scores 78 on a test with mean 70 and SD 5. Ben scores 90 on a different test with mean 80 and SD 10.

• Anna: z = (78 − 70)/5 = 1.6
• Ben: z = (90 − 80)/10 = 1.0

Anna did better relative to her class, even though her raw score is lower. z-scores make unlike scales comparable.

Finding a value from a z-score

Reverse the formula: x = μ + zσ. For z = −0.5 with μ = 60 and σ = 10, x = 60 − 5 = 55; conversely 55 has z = (55 − 60)/10 = −0.5.

Checking normality

A histogram that is roughly symmetric and bell-shaped, with no extreme outliers, suggests normal methods are fine. Strongly skewed data (incomes, for example) is not normal.

Common mistakes

• Using the empirical rule on a distribution that is not bell-shaped.
• Forgetting to divide by σ when computing z.
• Mixing up percentile (percent below) with percent above.