Guide

The 68–95–99.7 rule

If data are roughly bell-shaped, about 68% fall within μ ± 1σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ. That is the empirical rule, also called the 68–95–99.7 rule.

What it is for

It turns a mean and a standard deviation into a rough picture of where most values sit. It is a sketch, not a proof. Skewed or tiny datasets will not follow it closely.

A small example drawn as if it were normal

The worked example (μ = 26, σ ≈ 16.75) can be drawn as a curve so you can see the intervals even with only five points. That drawing is a teaching overlay. The five actual values still belong on a bar or box plot.

Tips

Check a histogram, bars, or a box plot before quoting the rule as a fact about your list.
Tap a band on a curve to see the numeric interval (mean ± k × SD).

Common mistakes

Applying the rule to five homework numbers as if 68% of them must sit inside ±1σ.
Forgetting that 99.7% still leaves room for real outliers in a large dataset.

How Standard Deviation Calculator helps

The bell-curve graph lets you switch 68 / 95 / 99.7 bands. Dataset bars and the box plot stay available so you are not stuck with a normal overlay.

Open the bell-curve graph on Android

68, 95, and 99.7 percent bands on the standard deviation bell curve