Normal Distribution Calculator
Enter the mean, standard deviation and a value x to find its z-score and probabilities.
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Your results
Normal Distribution Calculator by TheCalcTools
How the Normal Distribution Calculator works
Enter the mean and standard deviation of the distribution and the value x.
The z-score counts how many standard deviations x is from the mean.
The cumulative probability comes from the standard normal distribution.
The formula
Worked example
Using the default values shown in the calculator:
| Mean (μ) | 100 |
|---|---|
| Standard deviation (σ) | 15 |
| Value (x) | 130 |
- z = (130 − 100) ÷ 15 = 2.00.
- P(X < 130) = 0.9772, a percentile of 97.7%.
- P(X > 130) = 0.0228.
Result:
| z-score | 2.0000 |
|---|---|
| P(X < x) | 0.9772 |
| P(X > x) | 0.0228 |
| P(−|z| < Z < |z|) | 0.9545 |
| Percentile | 97.7% |
Frequently asked questions
What is the 68-95-99.7 rule?
About 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
Is IQ normally distributed?
IQ scores are designed to have a mean of 100 and a standard deviation of 15 (or 16) on many tests, so the defaults show an example.
Sources and further reading
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This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.