Normal Distribution Probability Calculator
Probability below, above and within a value on a normal curve, with the z-score and the odds of exceeding it.
Results
The dashed line marks the value you entered.
What this tool does
Heights, measurement errors, exam scores and thousands of other quantities pile up around a mean in the same bell shape, and the normal distribution is the mathematics of that pile. Once you have the mean and the standard deviation, every question about proportions has a single answer — what fraction lies below a value, above it, or inside a symmetric band around the mean.
Formula
P(X < x) = Phi((x - average) / standard deviation)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mu | Mean (average) | — |
sd | Standard deviation (sample) | — |
xx | Value | — |
PB | Probability below | % |
PA | Probability above | % |
ZZ | Z-score | — |
PW | Probability within the same distance of the mean | % |
OI | One in every | — |
Worked example
- Mean (average)100
- Standard deviation (sample)15
- Value130
- Probability below97.7250 %
- Probability above2.2750 %
- Z-score2.0000
- Probability within the same distance of the mean95.4500 %
- One in every43.96
Limitations
- Standard deviation and variance are calculated for a sample (dividing by n − 1). For a full population, divide by n instead.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
Frequently asked questions
How accurate is the approximation used?
The cumulative normal has no closed form, so this uses the Abramowitz and Stegun rational approximation 26.2.17, which is accurate to better than 7.5 x 10^-8 in probability across the whole range. That is far tighter than any statistical table and more than enough for any practical use, including the far tails.