Standard Deviation and the Bell Curve, Explained Simply

Take the data 2, 4, 4, 4, 5, 5, 7, 9.

1. The mean is 40 ÷ 8 = 5.
2. Subtract the mean from each value and square the results: 9, 1, 1, 1, 0, 0, 4, 16.
3. Add them: 32.
4. Divide by n (8) for the population variance: 4.
5. Take the square root: the standard deviation is 2.

If the data are a sample from a larger group, divide by n − 1 instead, giving √(32 ÷ 7) ≈ 2.14. Check with the statistics calculator.

Many natural measurements, such as heights, measurement errors and standardised test scores, cluster around a middle value and taper off symmetrically. This is the normal distribution. For such data, the 68-95-99.7 rule says that about 68% of values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three.

If a test has a mean of 70 and a standard deviation of 10, then about 68% of students scored between 60 and 80, about 95% between 50 and 90 and almost everyone between 40 and 100. A score of 90 is two standard deviations above the mean, which is high but not unheard of.

A z-score tells you how many standard deviations a value is from the mean: z = (value − mean) ÷ SD. A score of 85 on the test above is z = (85 − 70) ÷ 10 = 1.5. Z-scores let you compare values from different scales, such as a test in maths and one in history.

The 68-95-99.7 rule is only a good guide for roughly bell-shaped data. Incomes, house prices and website visits are often heavily skewed, with a long tail on one side. There the mean and standard deviation can mislead, and the median and percentiles are often more useful. See mean, median and mode explained.

Standard deviation is a basic measure of risk in finance, of consistency in manufacturing and of variation in science. A small SD means values are close together; a large SD means they are spread out.