How this instrument works
The empirical rule, also called the 68-95-99.7 rule, is a shortcut for describing how data clusters around the mean in a normal (bell-curve) distribution. It states that approximately 68% of values fall within 1 standard deviation of the mean, approximately 95% fall within 2 standard deviations, and approximately 99.7% fall within 3 — three fixed checkpoints that come directly from evaluating the normal distribution's cumulative curve at those exact distances.
Those percentages aren't rounded guesses; they follow from the mathematical shape of the normal curve itself, and they only hold when the underlying data is actually close to normally distributed. Apply the empirical rule to heavily skewed data — income figures, response times, anything with a long tail — and the real percentages within 1, 2 or 3 standard deviations can differ substantially from 68, 95 and 99.7, sometimes by a lot.
That's the key trade-off against this site's Chebyshev's theorem instrument, which computes a weaker guarantee (at least 75% within 2 standard deviations, not 95%) but one that holds for absolutely any distribution, shape unknown or unverified. The empirical rule trades that universality for precision: it gives you the sharper, more useful number, but only once you've confirmed — or are willing to assume — that your data is roughly bell-shaped.
- Choose how many standard deviations out you want from k — standard deviations: 1 (±1σ), 2 (±2σ), or 3 (±3σ).
- Read the percentage of data within k standard deviations (normal distribution) for the standard empirical-rule figure at that distance.
- This rule assumes your data is roughly normally distributed — for skewed or unusual-shaped data, the true percentage can differ noticeably from 68%, 95% or 99.7%.
- For a distance-agnostic guarantee that works on any distribution, shape unknown, this site's Chebyshev's-theorem instrument gives a weaker but universally valid minimum instead.
Worked example — the standard 95% checkpoint at k = 2
Select k = 2 (±2σ). The instrument looks up the standard checkpoint for that distance and % of data within k standard deviations (normal distribution) reads 95.0%.
For a normally distributed measurement — many IQ test scales, for instance, are constructed with a mean of 100 and a standard deviation of 15 — this means about 95% of scores fall between 100 - 2(15) = 70 and 100 + 2(15) = 130. That leaves only about 5% of scores falling outside that 70-to-130 band, split roughly evenly between the two tails.
Questions
How is this different from Chebyshev's theorem?
The empirical rule gives sharper, more useful percentages — 68%, 95%, 99.7% — but only applies when the data is roughly normally distributed. Chebyshev's theorem gives a weaker guarantee (at least 75% within 2 standard deviations, not 95%) but one that holds for any distribution whatsoever, shape unknown. Use the empirical rule once you've confirmed or can reasonably assume normality; use Chebyshev's theorem when you can't.
Why doesn't the empirical rule work for skewed data?
Because the 68%, 95% and 99.7% figures are derived specifically from the symmetric, bell-shaped normal curve — they describe how that particular curve's area is distributed around its centre. A skewed distribution, with a long tail on one side, packs its probability differently, so the real percentage of data within 1, 2 or 3 standard deviations can be noticeably higher or lower than the empirical rule's figures.
What percentage of data falls beyond 3 standard deviations?
About 0.3% — the complement of the 99.7% figure. That's why a value more than 3 standard deviations from the mean is often flagged as a genuine outlier in quality control and 'six sigma' style process monitoring: under a normal distribution, such an extreme value is expected in only about 3 out of every 1,000 observations.
Where do the 68%, 95% and 99.7% figures actually come from?
They come from evaluating the normal distribution's cumulative distribution function at exactly 1, 2 and 3 standard deviations from the mean: the area under the bell curve between -1σ and +1σ works out to about 68.27%, between -2σ and +2σ to about 95.45%, and between -3σ and +3σ to about 99.73% — rounded to the familiar 68, 95 and 99.7 for everyday use.
Can I get the percentage for a value in between, like 2.5 standard deviations?
Not from this instrument directly — it's restricted to the three standard empirical-rule checkpoints (1, 2 and 3). For any other distance, you need the full normal cumulative distribution function rather than the simplified empirical rule, since the rule is specifically a memorable shortcut at those three fixed points, not a general formula for arbitrary k.