What is the 68-95-99.7 rule for the normal distribution?
In a normal distribution, about 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. It gives a fast way to judge how unusual a value is.
Explain it simply
Think of the heights of all students in a large college. Most are close to the average, fewer are very tall or very short, and almost nobody is extreme. The bell shape makes the spread predictable: the distance from the mean in standard deviations tells you the share of people inside it.
- Mean +/- 1 SD: about 68%
- Mean +/- 2 SD: about 95%
- Mean +/- 3 SD: about 99.7%
Example
JEE-style mock test scores are roughly normal with mean 500 and SD 100.
- About 68% of students score between 400 and 600.
- About 95% score between 300 and 700.
- A score above 700 happens to roughly 2.5% of students (half of the 5% outside the 2 SD band).
from scipy.stats import norm
print(round(norm.cdf(1) - norm.cdf(-1), 4)) # 0.6827
print(round(norm.cdf(2) - norm.cdf(-2), 4)) # 0.9545What to say in the interview
"The empirical rule says roughly 68, 95 and 99.7 percent of a normal distribution lies within one, two and three standard deviations of the mean. I use it to sanity check outliers: a point beyond 3 SD is rare, about 3 in 1000, though the rule only holds when the data is actually close to normal."
Common mistakes
- Applying it to skewed data such as salaries, where the percentages are wrong.