What Percentage of Data Falls Within 1, 2, and 3 Standard Deviations of the Mean?

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 empirical rule, or the 68-95-99.7 rule.

How the Rule Works

If a random variable is normally distributed as $X \sim N(\mu,\sigma^2)$, then $\mu$ is the mean and $\sigma$ is the standard deviation. Because the normal curve is symmetric about $\mu$, equal proportions lie on either side of the mean.

The symmetry also lets you estimate bands between successive boundaries. About $68\%$ is inside one standard deviation, so $34\%$ lies between $\mu$ and $\mu+\sigma$. The proportion between one and two standard deviations on both sides is approximately $95\%-68\%=27\%$, or on each side. These are approximations, not exact probabilities for arbitrary cut-offs. Always check normality first.

For example, suppose examination scores are normally distributed with mean $70$ and standard deviation $5$. Approximately 95% of scores lie within two standard deviations:

$$70-2(5)=60, \qquad 70+2(5)=80.$$

Therefore, approximately 95% of students score between $60$ and $80$. About 2.5% score below $60$, and about 2.5% score above $80$.

A common misconception is that this rule applies to every dataset. It applies only when the distribution is normal or is being modelled as approximately normal. For exact probabilities involving other boundaries, use the GDC normal-distribution function rather than estimating with this rule.

IB Exam Technique

This is part of SL 4.9 Normal distribution. In an IB Mathematics: Applications and Interpretation response, identify the relevant interval, substitute the mean and standard deviation, and interpret the percentage in context. At SL, normal probabilities and inverse-normal values are found directly using technology; standardizing to a $z$-score is not required.