Distributions Updated November 2026

Empirical rule (68-95-99.7)

For a normal distribution, about 68% of values fall within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. Use it for quick estimates: the range mean minus 2 sd to mean plus 2 sd covers roughly 95% of the data, leaving 2.5% in each tail.

Syntax

mean +/- 1 sd: 68%;  mean +/- 2 sd: 95%;  mean +/- 3 sd: 99.7%

Part	What it is
`mean`	the center of the normal distribution
`sd`	the standard deviation

Examples

Exam scores: mean 70, sd 5
68% of scores lie between 70 - 5 and 70 + 5

Result: 65 to 75

95% of scores lie between 70 - 10 and 70 + 10

Result: 60 to 80

Share of scores above 80 (2 sd above the mean)
(100% - 95%) / 2

Result: 2.5%

Share between 75 and 80 (between 1 and 2 sd above)
(95% - 68%) / 2

Result: 13.5%

Common mistakes

Applying the rule to skewed data; it only holds for roughly normal distributions, use Chebyshev's inequality for anything else.
Splitting a tail wrong; 5% lies outside 2 sd in total, so one tail is 2.5%, not 5%.
Using 95% for exactly 2 sd in precise work; 2 sd actually covers 95.45%, and 1.96 sd covers 95.00%.

Practice Empirical rule (68-95-99.7)

These drills check your answer on live data, free, no account needed.

Normal: between two valuesThe fill weight of cereal boxes is normally distributed with mean 482 grams and standard deviation 12 grams. What proportion falls between 478.4 and 495.2 grams? Round to 4 decimal places.
Normal: above a valueThe number of kilowatt-hours a household uses per week is approximately normal with mean 60 and standard deviation 6. What is the probability that a randomly chosen household is above 75? Round to 4 decimal places.

Questions

Where do the numbers 68, 95 and 99.7 come from?

They are the areas under the standard normal curve between z = -1 and 1 (0.6827), -2 and 2 (0.9545) and -3 and 3 (0.9973), rounded.

What is Chebyshev's rule?

For any distribution, at least 1 - 1/k^2 of values lie within k standard deviations of the mean: at least 75% within 2 sd and 89% within 3 sd. It is weaker than the empirical rule but always true.

Normal distributionHow to find normal distribution probabilities with z-scores, read the standard normal table, find a percentile cutoff, and use NORM.DIST and NORM.INV.
Standard deviationThe standard deviation formula, sample versus population versions, a worked example with exact numbers and the Excel functions STDEV.S and STDEV.P.
z-scoreThe z-score formula z = (x - mean) / sd, worked examples, how to read a z-score as a percentile with the normal table, and the Excel function STANDARDIZE.
OutliersHow to detect outliers with the 1.5 IQR fence rule and the z-score rule, worked examples with exact fences, and what to do with outliers once found.