What Percentage Is Within 1, 2, or 3 Standard Deviations?

For a normal distribution, approximately 68% of data lies within one standard deviation of the mean, 95% within two, and 99.7% within three. This is called the 68-95-99.7 rule or empirical rule.

If the mean is $\mu$ and the standard deviation is $\sigma$, the intervals are:

Why the percentages work

A normal distribution is symmetric about its mean. Therefore, the percentage outside each central interval is divided equally between the two tails.

For example, suppose examination scores are normally distributed with $\mu=70$ and $\sigma=5$. Approximately 95% of scores fall within two standard deviations:

$$70-2(5) \leq X \leq 70+2(5),$$

so approximately 95% lie between 60 and 80. If 1000 students take the examination, the expected number in this interval is

$$0.95(1000)=950.$$

The rule also lets you estimate percentages in bands. On one side of the mean, the region from one to two standard deviations contains approximately $(95-68)/2=13.5\%$, while the region from two to three contains approximately $(99.7-95)/2=2.35\%$. To find the proportion above $$, use the upper tail: . These results follow from symmetry and subtraction of nested central areas. This avoids memorizing each band separately.

A common misconception is that this rule applies to every data set. It applies only when the distribution is normal or approximately normal; the percentages are approximations, not exact values for arbitrary distributions.

IB exam technique

This is IB Mathematics AA SL 4.9: Normal distribution and calculations, usually assessed on Paper 2 using a GDC. Sketch and label the normal curve, identify the required interval or tail, and use technology when an exact normal probability is required rather than relying only on the empirical rule.