Where the three numbers land
Give the calculator a mean and a standard deviation and it marks off the three bands. The percentages are fixed properties of the bell curve, so they never change.
To place a single value, use its z-score: z = (x − μ) ÷ σ. A z of 1 sits at the edge of the 68% band, a z of 2 at the 95% band. Only about 0.3% of data lies beyond ±3σ.
How the bands stack up
Each band contains the one inside it. Widening from two to three standard deviations only adds the last few percent — the rare tails.
Going from ±2σ to ±3σ captures just 4.7% more data. That last sliver is where outliers live.
Worked example — SAT scores
With a mean of 1000 and a standard deviation of 200, the rule reads straight off the mean:
| Range | Scores | Share of students |
|---|---|---|
| μ ± 1σ | 800 – 1200 | 68% |
| μ ± 2σ | 600 – 1400 | 95% |
| μ ± 3σ | 400 – 1600 | 99.7% |
Manufacturing uses the same idea for tolerances: bolts at 10 mm mean and 0.1 mm spread put 99.7% of parts in 9.7–10.3 mm.
The mistakes to avoid
- Check normality first. The rule is meaningless on skewed or multi-peaked data. Plot it before you apply it.
- Percentages, not counts. 68% means 68 out of every 100 values, not a fixed head count.
- The tails still matter. The 0.3% beyond ±3σ is rare but can dominate in finance or safety, where one extreme event is the whole story.
Common questions
What is the 68-95-99.7 rule?
For a normal (bell-shaped) distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. It is a fast way to see how spread out data is without any tables.
When does the empirical rule not work?
Only when the data is roughly normal. If it is skewed, has more than one peak, or has heavy tails, the percentages are wrong. Check with a histogram or Q-Q plot before trusting it.
How is the empirical rule different from Chebyshev's inequality?
The empirical rule gives exact percentages but only for normal data. Chebyshev works for any distribution but is weaker: it guarantees at least 75% within two standard deviations, not exactly 95%.
