What the normal distribution describes
The normal distribution is the familiar symmetric bell curve. It is fixed by two numbers: the mean μ, which sets the centre, and the standard deviation σ, which sets the spread. Many measurements that add up lots of small independent effects, such as manufacturing tolerances, measurement errors or standardised test scores, are close to normal, which is why it is the default model in statistics courses and quality control.
Four things the calculator can find
- P(X ≤ x): the probability of a value at or below x, the left tail.
- P(X ≥ x): the probability of a value at or above x, the right tail.
- P(a ≤ X ≤ b): the probability of landing between two values.
- Percentile → x: the inverse problem. Enter a percentage and get the value with that share of the distribution to its left, for example the cut-off for the top 2.5%.
The formulas
Every question starts by converting the value to a z-score, the number of standard deviations from the mean: z = (x − μ) ÷ σ. The left-tail probability is Φ(z), where Φ is the cumulative distribution function of the standard normal distribution. The right tail is 1 − Φ(z), which equals Φ(−z) by symmetry; the calculator uses the second form so that tiny right-tail probabilities keep their accuracy. A probability between two values is Φ(zb) − Φ(za). The inverse runs the other way: z = Φ−1(p) and then x = μ + σ·z. Φ has no closed form, so it is evaluated with a published double-precision approximation and checked against an independent reference to better than one part in ten million.
Worked example 1: a probability
Suppose a score is normally distributed with mean 100 and standard deviation 15. What share of scores are at or below 129.4? The z-score is (129.4 − 100) ÷ 15 = 1.9600, and Φ(1.96) = 97.5002%. So about 2.4998% of scores are above 129.4. Between 70.6 and 129.4, which is 1.96 standard deviations either side of the mean, lies 95.0004% of the distribution. This is the origin of the familiar rule that about 95% of a normal population lies within roughly two standard deviations of the mean.
Worked example 2: an inverse percentile
Using the same distribution, which score marks the 97.5th percentile? The inverse gives z = 1.959964, and x = 100 + 15 × 1.959964 = 129.3995. It is slightly below 129.4 because 1.96 is itself a rounded value of the exact z-score.
Area, not height
The curve on the chart shows probability density. Its height at a point is not a probability; only areas under it are. For a continuous distribution the probability of hitting one exact value is zero, so P(X ≤ x) and P(X < x) are the same number, and so are the between-probabilities with or without the end points.
When the normal model is the wrong model
The results are only as good as the normal assumption. Financial returns are a common case where it fails: large losses and gains happen more often than a bell curve predicts, and returns are often skewed. A normal model therefore tends to understate the probability of extreme moves. Our guide to risk measures beyond variance explains skewness, kurtosis and CVaR, and the article on Monte Carlo simulation shows how fat-tailed distributions are used instead. For how mean and volatility enter portfolio theory in the first place, see portfolio concepts. Use this calculator to understand what a normal model implies, not as a forecast of market returns or a risk limit.
FAQ
What is the difference between a percentile and a probability?
A probability answers "how likely is a value below 129.4?" (97.5002%). A percentile answers the reverse question, "which value has 97.5% of the distribution below it?" (129.3995). The first three modes give probabilities; the last gives a value.
What does the standard deviation do?
It sets the scale. Doubling σ doubles the distance from the mean to any given percentile and halves every z-score for a fixed value. σ must be greater than zero; with zero spread there is no distribution to calculate.
Why can't I enter 0% or 100% as a percentile?
The normal distribution has no smallest or largest value, so the 0th and 100th percentiles are at minus and plus infinity. The calculator supports percentiles from 0.0000001% to 99.9999999% and explains when an entry is outside that range rather than returning a made-up number.
How small can a tail probability be?
Very small. Eight standard deviations above the mean the right tail is about 6.2 × 10−16. The calculator shows such values in scientific notation instead of rounding them to 0%, because a tiny probability is not the same as impossible.
Can I use this for stock returns or trading risk?
Only as an illustration. Real return distributions have fatter tails than the normal model, so it understates the chance of large losses. For trading questions use tools built for them, and treat any normal-model probability as a lower bound on extreme-move risk rather than a prediction.
Formulas follow the standard definitions in the NIST/SEMATECH e-Handbook of Statistical Methods, normal distribution. Educational tool; it does not test whether your data are normal.