How it works
Every normal distribution is the same shape stretched and shifted, so any question about one is answered by converting to the standard normal — mean 0, standard deviation 1 — and reading off the area under that curve.
z = (x − μ) ÷ σ
The z-score says how many standard deviations a value sits from the mean. An IQ of 115 with μ = 100 and σ = 15 has z = 1: exactly one standard deviation above average, which puts it at roughly the 84th percentile.
Reading the result
- P(X < x) — the area to the left, which is the percentile.
- P(X > x) — the area to the right, which is 1 − P(X < x).
- P(a < X < b) — the area between the two values, found from the corresponding cumulative probabilities.
- Two-tailed — the chance of being that far from the mean in either direction, which is what most significance tests ask.
Why P(X = x) is always zero
The normal distribution is continuous, so probability is area under the curve — and a single point has no width. The probability of a value being exactly 115.000… is zero, however likely values near 115 are.
One practical consequence: “less than” and “at most” give identical answers here. That is unlike the binomial distribution, where those two differ by exactly P(X = k) and the distinction matters a great deal.
The empirical rule
| Within | Contains | Outside |
|---|---|---|
| ±1σ | 68.27% | about 1 in 3 |
| ±2σ | 95.45% | about 1 in 22 |
| ±3σ | 99.73% | about 1 in 370 |
| ±1.96σ | 95.0004% | about 1 in 20 |
| ±1.9599639845σ | exactly 95% | exactly 1 in 20 |
The last two rows are the ones that turn up in practice: 1.96, not 2, is the multiplier behind a 95% confidence interval, and the familiar “two standard deviations” is a rounding of it. Strictly, 1.96 is itself a rounding — it captures 95.0004%, and the figure that gives exactly 95% is 1.9599639845.
Going backwards: percentiles and critical values
The second tab answers the reverse question — given a probability, which value sits there? That is how percentiles and critical values are found: the 95th percentile of the standard normal is 1.645, and the value cutting off 2.5% in each tail is about 1.96.
A note on accuracy
Unlike most calculators on this site, this one cannot give an exact answer, and that is a property of the mathematics not a shortcut. The normal CDF has no elementary closed form: it must be approximated.
The common approach is a short series from Abramowitz and Stegun that is accurate to about 1.5×10 −7 — fine for a printed table, but less reliable in the tails. This calculator uses Hart’s algorithm, with tail-aware calculations designed to preserve numerical accuracy deep into the tails. Measured against known values, its relative error is at worst about 3×10 −11 in the tested tail region and below 10 −15 elsewhere. The inverse calculation starts with Acklam’s approximation and adds a Halley refinement step.
Working from raw data
This page takes the mean and standard deviation as given. If you have a data set instead and need them worked out — including the choice between the sample and population divisor — the z-score calculator does that and standardises every value.
Frequently asked questions
What is a z-score?
How many standard deviations a value is from the mean: z = (x − μ) ÷ σ. A z of 1 is one standard deviation above the mean, −2 is two below. Converting to z is what lets a single table answer questions about any normal distribution.
Why is the probability of an exact value zero?
Because the distribution is continuous and probability is area under the curve. A single point has no width, so it encloses no area. Only ranges have non-zero probability, which is why "less than" and "at most" mean the same thing for a normal distribution.
Where does 1.96 come from?
It is the z-score with 2.5% of the distribution beyond it in each tail, leaving exactly 95% in the middle. The familiar "two standard deviations" is a rounding of 1.96, and the exact figure is what confidence intervals actually use.
Can the standard deviation be zero?
Not for a normal distribution. With σ = 0 there is no spread — every value would equal the mean — and the formula divides by σ. The calculator asks for a positive value instead of returning an infinity.