68–95–99.7 rule diagram on the IQ scale

What it shows

A normal curve with brackets showing the share of the distribution within 1, 2 and 3 standard deviations of the mean: 68.27%, 95.45% and 99.73%. The bands are shaded by distance from the mean.

The axis carries two scales, the z-score and IQ on the common scale with mean 100 and standard deviation 15, so the rule can be read in units students recognize.

How to read it

1. 1

   The innermost bracket covers 85 to 115 on the IQ scale, z = −1 to +1: 68.27% of the distribution.

2. 2

   The next covers 70 to 130, within 2 standard deviations: 95.45%.

3. 3

   The outermost covers 55 to 145, within 3: 99.73%. Only 0.27% lies beyond it.

The numbers behind it

• Shares within k standard deviations, computed as 2Φ(k) − 1: 68.27% (k = 1), 95.45% (k = 2), 99.73% (k = 3).
• IQ = 100 + 15z, so the axis marks read 55, 70, 85, 100, 115, 130 and 145.
• The curve is drawn from −3.5 to +3.5 SD; 0.27% of the distribution lies beyond 3 SD.

Ways to use it in teaching

• Ask what share of people score above 130 on this scale, and have students work it out from the brackets.

• Translate the rule to a variable from students’ own field with a known mean and standard deviation.

• For a rough check on normality, compare the share of a real dataset within one standard deviation with the 68.27% shown here.

• The percentages hold only for normal data; for a skewed variable, use the data’s own percentiles instead.

For the summary statistics behind this kind of chart, see our plain-language guide to descriptive statistics.

Sources

• 68–95–99.7 rule. (n.d.). In Wikipedia. Retrieved October 18, 2026, from https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule
• Illowsky, B., & Dean, S. (2023). Introductory statistics 2e. OpenStax. https://openstax.org/details/books/introductory-statistics-2e
• Intelligence quotient. (n.d.). In Wikipedia. Retrieved October 18, 2026, from https://en.wikipedia.org/wiki/Intelligence_quotient