How to use the Normal Distribution
- Enter the mean and standard deviation of your distribution.
- Enter the lower and upper bounds of the range you care about.
- Read P(a < X < b) for the probability of falling inside that range.
- Use the tail rows for one-sided questions such as 'what fraction exceeds this threshold'.
- Check the central 95% range for a quick sense of where almost all the distribution lies.
How the calculation works
The normal distribution is defined by just two parameters, mean and standard deviation, and everything else follows. It shows up everywhere because of the central limit theorem: sums and averages of many independent contributions tend toward normality almost regardless of the shape of the individual contributions. That is why measurement error, aggregated demand and sample means are so often near-normal, and why individual incomes and file sizes are not.
Probabilities come from the area under the density curve, which has no elementary antiderivative. Each bound is standardised to a z-score and evaluated with a high-accuracy rational approximation to Φ, then subtracted to get the interval probability. Percentiles run the process backwards by numerical inversion, since there is no closed-form quantile function either.
The empirical rule follows directly from these areas: about 68% within one standard deviation, 95.4% within two, 99.7% within three. The 1.96 multiplier used for 95% intervals is the exact figure that the round '2' approximates. Beyond three standard deviations the normal model's thin tails become unreliable for real data, which is exactly where financial and insurance models that assume normality fail hardest.
f(x) = (1/(σ√(2π)))·e^(−(x − μ)²/(2σ²)); P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ)Source: de Moivre (1733); Gauss, Theoria Motus (1809); Abramowitz & Stegun, Handbook of Mathematical Functions (1964), formula 7.1.26.
Worked example
Scores are normal with mean 100 and standard deviation 15. What share falls between 90 and 120?
- z for 90 = (90 − 100) ÷ 15 = −0.6667; Φ = 0.2525.
- z for 120 = (120 − 100) ÷ 15 = 1.3333; Φ = 0.9088.
- P(90 < X < 120) = 0.9088 − 0.2525 = 0.6563.
- Central 95% range = 100 ± 1.96 × 15 = 70.6 to 129.4.
About 65.6% of scores fall between 90 and 120, and 95% of the whole distribution lies between roughly 71 and 129.
Frequently asked questions
How do I know my data is normal?+
Plot a histogram and a Q-Q plot first. Formal tests such as Shapiro–Wilk reject almost any large real data set, so judgment about the shape matters more than the p-value.
What is the empirical rule?+
Roughly 68%, 95% and 99.7% of a normal distribution lie within one, two and three standard deviations of the mean.
Why is the probability of an exact value zero?+
For a continuous distribution, probability is area, and a single point has no width. Only intervals carry non-zero probability.
Is the standard normal a different distribution?+
It is the same family with mean 0 and standard deviation 1. Any normal variable becomes standard normal once you convert it to a z-score.
Last reviewed September 1, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.