Step by Step
1σ
68% within 1 standard deviation
In any normal distribution, approximately 68% of all data falls within 1 standard deviation of the mean (in both directions combined).
2σ
95% within 2 standard deviations
Approximately 95% of all data falls within 2 standard deviations of the mean.
3σ
99.7% within 3 standard deviations
Approximately 99.7% of all data falls within 3 standard deviations of the mean — meaning data beyond 3 standard deviations is quite rare.
Use
When this rule applies
The Empirical Rule only applies to data that is approximately normally distributed (bell-shaped and symmetric) — it doesn't apply to skewed or otherwise non-normal distributions.
Applied Walkthrough
1
A distribution of adult heights is approximately normal, with a mean of 66 inches and a standard deviation of 3 inches.
2
Using the Empirical Rule, about 68% of people have heights between 63 and 69 inches (within 1 standard deviation of the mean).
3
About 95% have heights between 60 and 72 inches (within 2 standard deviations).
4
About 99.7% have heights between 57 and 75 inches (within 3 standard deviations) — heights outside this range would be quite unusual.
Exam Application
Exams test whether you can apply the correct percentage (68%, 95%, or 99.7%) to a given number of standard deviations from the mean, and whether you recognize the Empirical Rule only applies to approximately normal distributions.
⚠ Common Trap
The most common trap is applying the Empirical Rule to a distribution that isn't approximately normal — the 68-95-99.7 percentages only hold for bell-shaped, symmetric distributions.
✓ Quick Self-Check
1. What percentage of data falls within 1 standard deviation of the mean in a normal distribution?
68%.
Tap to reveal / hide
2. What percentage falls within 2 standard deviations?
95%.
Tap to reveal / hide
3. What percentage falls within 3 standard deviations?
99.7%.
Tap to reveal / hide
4. For a normal distribution with mean 66 and standard deviation 3, what range contains about 95% of the data?
60 to 72 (mean ± 2 standard deviations).
Tap to reveal / hide
5. Does the Empirical Rule apply to all distributions?
No — only to distributions that are approximately normal (bell-shaped and symmetric).
Tap to reveal / hide