Calculates both the probability density and the cumulative probability for a value within a normal (Gaussian) distribution.
How it works
Given a mean, standard deviation, and a value, the calculator applies the standard normal probability density and cumulative distribution formulas directly.
What this does not include
This does not include standardizing a single reading into a z-score on its own — for that simpler transformation, use this site’s z-score calculator instead.
How to use this calculator
- Enter the value, the distribution’s mean, and its standard deviation.
A worked example
Mean 100, standard deviation 15, x = 100 (exactly at the mean): density = 0.0266, cumulative probability = 0.5 — exactly half the distribution lies below the mean, as expected for a symmetric curve.
Same distribution, x = 115 (one standard deviation above the mean): cumulative probability = 0.8413, meaning about 84% of values fall at or below 115.
What the variables mean
| Variable | Meaning |
|---|---|
| x | The value being evaluated |
| Mean | The distribution’s center |
| Standard deviation | How spread out the distribution is |
Edge cases worth knowing
A standard deviation of zero has no defined distribution — every value would be identical, with no spread for a probability density to describe.
The cumulative probability always approaches, but never quite reaches, 0 or 1 at the extreme tails — a normal distribution’s tails extend infinitely in both directions, however negligibly thin.
Frequently asked questions
What’s the difference between the density and the cumulative probability?
The density describes how likely values near x are relative to other values; the cumulative probability is the total probability of observing any value at or below x.
Why does the cumulative probability equal 0.5 at the mean?
Because a normal distribution is perfectly symmetric around its mean, exactly half the probability lies below it and half above.
What method computes the cumulative probability?
A standard numerical approximation to the error function (Abramowitz and Stegun), the same widely used method behind most statistical software’s normal distribution calculations.
