Distributions

Normal Distribution Calculator

Areas under the bell curve: the probability of falling below, above or between any two values — and the value at any percentile.

Updated August 2026Runs in your browser — nothing is uploadedVerified against R

How to read this result

A probability is an area. The whole curve encloses an area of 1, so the shaded region in the figure is the probability — that is the only idea on this page, and everything else is arithmetic.

Standardising is what makes one table serve every normal distribution. Subtract the mean, divide by the standard deviation, and any normal becomes the standard normal with mean 0 and standard deviation 1. The z-score is that translated value.

The empirical rule is worth memorising. About 68% of the distribution lies within one standard deviation of the mean, 95% within two, and 99.7% within three. Those three numbers turn a mean and an SD into an immediate sense of the spread.

A continuous distribution assigns zero probability to any single value, so "less than" and "at most" are the same thing here. That is not true of the binomial or the Poisson, where the difference between < and ≤ is a whole bar.

The shaded tails hold 5% of the distribution between them. A statistic landing there is what “significant at α = 0.05” means — nothing more and nothing less.

The formula

f (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{1}{2} (\frac{x - μ}{σ})^{2}}

$μ$: the mean — where the curve is centred
$σ$: the standard deviation — how wide it is

The density. Probabilities are areas under it, and there is no closed form for that area — every normal probability is computed numerically, here via the incomplete gamma function.

Standardising

z = \frac{x - μ}{σ} P (X < x) = Φ (z)

Φ is the standard normal CDF. Inverting it gives the value at a percentile.

Worked example

IQ scores above 130

IQ is scaled to a mean of 100 and a standard deviation of 15. What share of people score below 130 — and what share above?

Standardise the value.
z = (130 − 100) / 15 = 2.0
Look up the standard normal CDF at 2.
Φ(2) = 0.97725
So 97.7% score below 130.
P(X < 130) = 0.9772
The complement is the share above.
P(X > 130) = 0.0228, about 1 in 44

P(X < 130) 97.72%P(X > 130) 2.28%z 2.00roughly 1 person in 44

The same z of 2.0 gives 2.3% in the upper tail for any normal distribution — that is what standardising buys. A height, a test score and a manufacturing tolerance all share the answer.

Checked against R's pnorm(130, 100, 15).

The calculator above is loaded with these numbers by the Load the worked example button.

Assumptions, and when to use something else

The normal model assumes the quantity is continuous, unbounded in both directions, and symmetric. Real data are often none of those: heights cannot be negative, incomes are strongly right-skewed, and counts are discrete.

That does not always matter. The model is a good approximation whenever the quantity is the sum of many small independent influences, and — crucially — the mean of a sample is close to normal regardless of the population's shape, which is why the normal distribution appears in tests of visibly non-normal data.

To check whether your own data are near enough, use Shapiro-Wilk and look at its Q-Q plot.

Your sample is small and σ is estimated from the datat distribution calculatorHeavier tails, which is the correct adjustment for an estimated σ.
The outcome is a count of successes in n trialsBinomial distributionDiscrete trials need the discrete distribution.
The outcome is a count of events in an intervalPoisson distributionArrivals, defects, occurrences per unit of time or space.
You want to standardise a value rather than find a probabilityZ-score calculatorThe same arithmetic framed the other way round.

Questions people ask

What is the empirical rule?

68% of a normal distribution lies within one standard deviation of the mean, 95% within two, and 99.7% within three. The exact figures are 68.27%, 95.45% and 99.73%, and the calculator prints them for your particular mean and SD.

What is the difference between the normal and the standard normal distribution?

The standard normal has mean 0 and standard deviation 1. Every normal distribution becomes it after standardising, which is why one table of values could serve all of them in the era before computers.

How do I find the value at a given percentile?

Use the inverse mode above: give the probability and it returns the value. That is NORM.INV in a spreadsheet and qnorm in R, and it is the operation behind every critical value in hypothesis testing.

Is my data normally distributed?

Test it with Shapiro-Wilk and, more usefully, look at the Q-Q plot it draws. Bear in mind that with a large sample the test rejects on departures too small to matter, and with a small one it misses departures that do.