Empirical Rule Calculator

one sigma: mean ± 1σ · two sigma: mean ± 2σ · three sigma: mean ± 3σ · share within kσ = 2Φ(k) − 1, where Φ is the standard normal distribution function

mean

The centre of the distribution. The six endpoints are symmetric around it, so moving it slides the whole set of ranges without changing any of the three shares

standardDeviation

How spread out the distribution is, and the unit the rule is counted in — one sigma, two sigma, three sigma. It must be positive: a normal distribution with a standard deviation of zero is a single point, and the rule has nothing to say about it

kσ

The number of standard deviations — 1, 2 or 3. The share inside a range depends only on k, which is why the three percentages are the same on every page of results regardless of the mean and the standard deviation

2Φ(k) − 1

The exact share between −kσ and +kσ. The empirical rule rounds this to 68 / 95 / 99.7 for reading; the page prints the unrounded value, which is what the rounded one is short for

Use it to turn a mean and a standard deviation into ranges you can actually point at. The empirical rule is usually met as a sentence about percentages, which is hard to apply to a particular measurement; the same statement in the form of "anything above 130 is more than two standard deviations out" is not. It is also the quick sanity check on a standard deviation itself: if one standard deviation either side of the mean covers a range that cannot contain 68% of the data — because it runs past a hard limit, or because the data is visibly lopsided — then the normal model is the wrong one for that data, and the check costs nothing. The page assumes a normal distribution throughout; it does not fit one to your data, and it does not tell you whether the assumption holds.

1. A mean of 100 and a standard deviation of 15

   1. One standard deviation either side of 100 is 100 − 15 = 85 to 100 + 15 = 115
   2. Two standard deviations is 100 − 30 = 70 to 100 + 30 = 130
   3. Three standard deviations is 100 − 45 = 55 to 100 + 45 = 145
   4. The shares do not use the mean or the standard deviation at all: 68.27% within 1σ, 95.45% within 2σ, 99.73% within 3σ

   The three percentages are identical on every example on this page, and that is the content of the rule rather than a coincidence of this one. A normal distribution is fully described by its centre and its spread, and the share of it inside k standard deviations of the centre depends on k alone — so the mean and the standard deviation move the endpoints and never touch the shares. If a result ever showed a percentage that moved when the mean did, the percentages would be being computed from the wrong quantity, and printing them on the same three rows is what makes that visible.

2. The standard normal: mean 0, standard deviation 1

   1. One standard deviation either side of 0 is −1 to 1
   2. Two standard deviations is −2 to 2, and three is −3 to 3
   3. The shares are unchanged at 68.27%, 95.45% and 99.73%

   With a mean of zero and a standard deviation of one the endpoints are the sigma counts themselves, which makes this the version of the rule worth memorising: the numbers −1, −2, −3 and their positives are the z-scores, and every other normal distribution is this one stretched and shifted. It also shows why the shares cannot depend on the inputs — here the endpoints are literally the values of k, and the percentages beside them are the same as they were at a mean of 100.

3. A small standard deviation: mean 500, standard deviation 0.5

   1. One standard deviation is half a unit, so the first range is 499.5 to 500.5
   2. Two standard deviations is 499 to 501, and three is 498.5 to 501.5
   3. The shares are still 68.27%, 95.45% and 99.73%

   Nothing about the rule is tied to the standard deviation being a round number or a large one, and a narrow distribution is where the ranges stop being readable at a glance. A tolerance band on a machined part is usually written this way — a target and a spread much smaller than the target — and the useful output is the endpoint row rather than the percentage row. It is also a reminder that the 99.7 range covers nearly everything only when the distribution really is normal: three standard deviations is a small interval in absolute terms here, and any real process with a heavier tail would put far more than 0.27% outside it.

4. A negative mean: −2.5 with a standard deviation of 1.25

   1. One standard deviation either side of −2.5 is −3.75 to −1.25
   2. Two standard deviations is −5 to 0 — the upper end lands exactly on zero
   3. Three standard deviations is −6.25 to 1.25, which crosses zero
   4. The shares are unchanged at 68.27%, 95.45% and 99.73%

   A mean below zero is ordinary — a temperature anomaly, a log return, a difference between two measurements — and it is the case where the ranges cross zero, so an implementation with a sign error can still look right on every positive example. The two-sigma upper end landing exactly on zero here is the kind of coincidence that makes a wrong answer look deliberate, which is why the case is pinned as a test rather than left to chance. The standard deviation stays positive regardless: it is a width, not a coordinate, and only the mean carries the sign.

The rule belongs to the normal distribution and to nothing else. It is often repeated as though 68% were a fact about data in general, but it is a property of one particular curve — a distribution with a different shape has different shares at one, two and three standard deviations, and a lopsided one may have very different shares on either side of the mean. So the page assumes normality rather than checking it, and a result here is only as meaningful as that assumption. Three further limits. The standard deviation must be positive: at zero the distribution collapses to a single point and the question of what share lies within zero of the mean has no useful answer, so the page refuses the input rather than picking one. The three shares are exact values of the normal distribution function, so they will differ in the second decimal from the 68, 95 and 99.7 the rule is usually quoted with — the page does not round them to match the name, because the rounding is the shorthand and the printed number is the thing it stands for. And the page does not fit a distribution to your data or test whether one fits; it takes a mean and a standard deviation as given and reports what follows.

Why does the page say 68.27% when the rule says 68%?

Because the 68 in the name is a rounded figure and 68.27% is the value it rounds. The empirical rule is a way of remembering three numbers, and roughly 68%, 95% and 99.7% is what most people can hold in their heads; the exact shares inside one, two and three standard deviations of the mean of a normal distribution are 68.27%, 95.45% and 99.73%, and that is what the page computes rather than quoting. If a calculation is going to be built on one of these numbers, the unrounded one is the one to build on — a rounded figure used as an input produces an answer that is off in a way nobody can trace back to the rounding.

Do the percentages change if I change the mean or the standard deviation?

No, and that is the whole content of the rule. A normal distribution is fixed by two numbers, its centre and its spread, and the share of it falling within k standard deviations of the centre depends only on k — one, two or three. So changing the mean slides all six endpoints along the number line and changing the standard deviation stretches or squeezes them, while the three shares stay exactly where they were. If a result ever showed a share moving with the inputs, the shares would be being computed from the endpoints rather than from the number of standard deviations, and that would be an error rather than a property of the data.

What does the range actually tell me — is it a probability?

It is a probability, read as a share of the distribution. If a quantity really is normally distributed with the mean and standard deviation you entered, then the chance that a single value drawn from it lands inside the one-sigma range is 68.27%, and the ranges translate standard deviations into the units you are measuring in so that the statement can be applied to a specific number. What it is not is a statement about your particular sample: a distribution can cover 68% of its probability in a range while the data in front of you has a different proportion inside it, and the page has no data — it has two parameters you typed.

How do I know whether the normal distribution is the right model?

This page cannot tell you, because it never sees the data. What it does give you is a quick check on the model's plausibility: if one standard deviation either side of the mean covers a range that cannot plausibly hold about two thirds of the values — because the range runs past a hard floor or ceiling, because the values are counts that cannot go below zero, or because the distribution is visibly lopsided — then the normal assumption is doing real damage and the percentages on this page describe a curve the data does not follow. A histogram or a normal probability plot is the usual way to answer the question properly; the arithmetic here is downstream of that answer, not a substitute for it.

Why can the standard deviation not be zero?

Because a normal distribution with a standard deviation of zero has collapsed to a single point, and the rule stops meaning anything there. Every value would sit exactly on the mean, so the ranges one, two and three standard deviations wide would all have zero width, and asking what share of the distribution they contain has two defensible answers — all of it, since everything is at the mean, or none, since the interval has no extent — with no way to choose between them. Rather than pick one silently, the page rejects the input and says the standard deviation must be positive. A standard deviation that came out as zero in a real calculation almost always means a single observation or a data-entry problem rather than a genuine distribution.

Is this the same as the calculator that gives probabilities from z-scores?

They are two entry points into the same normal distribution. The z-score calculator takes a value and tells you where it falls and what probability lies below it; this page takes the mean and the standard deviation and lays out the three standard ranges with the share each one covers. The direction of the question is what differs — one starts from a measurement, the other starts from the rule itself. The inputs are deliberately identical, the mean and the standard deviation in the same two fields, so that moving between the two pages does not mean retyping anything, and the two can never disagree about the underlying distribution because they compute it the same way.

• 68-95-99.7 Rule — from Wolfram MathWorld (the rule stated as approximately 68%, 95% and 99.7%, the more precise probabilities, and the note that the rule is specific to the normal distribution rather than a general statement about arbitrary distributions) — Wolfram MathWorld
• 6.1 The Standard Normal Distribution — Introductory Statistics 2e (the empirical rule stated for a normally distributed variable: about 68% of values within one standard deviation of the mean, 95% within two, 99.7% within three) — OpenStax (Rice University)
• 1.3.6.6.1. Normal Distribution — e-Handbook of Statistical Methods (the normal distribution its parameters and its cumulative distribution function, which is what the three shares on this page are computed from) — National Institute of Standards and Technology (NIST)