Normal Distribution Calculator
(Below, Above, Between, Outside and Inverse)
Enter the mean and standard deviation, then a value or two. The calculator returns the probability and z-scores, or a value for a given probability.
What do you want to find?
Use 0 and 1 to work with z-scores
Normal Probability
💡 Formula Guide
This normal distribution calculator finds the probability that a normally distributed value falls below, above, between or outside given values, for any mean and standard deviation. It also works backwards from a probability to the value that cuts it off.
From a Value to a Probability
Test scores with a mean of 75 and a standard deviation of 10: the values 70 and 90 have z-scores of −0.5 and 1.5. Φ(1.5) = 0.933193 and Φ(−0.5) = 0.308538, so 62.47% of scores fall between 70 and 90.
| Question | Result |
|---|---|
| P(X < 90) | 0.933193 |
| P(X > 90) | 0.066807 |
| P(70 < X < 90) | 0.624655 |
| P(X < 70 or X > 90) | 0.375345 |
Φ is the area under the standard normal curve to the left of z. For a continuous distribution P(X < x) and P(X ≤ x) are the same.
From a Probability to a Value
The score that 95% of students fall below is 75 + 1.644854 × 10 = 91.45. The middle 95% of scores lie between 75 − 1.959964 × 10 = 55.40 and 94.60. Choose left, right or middle in the inverse mode to match how the probability is stated.
The 68–95–99.7 Rule
| Within the mean ± | Share of values |
|---|---|
| 1 standard deviation | 68.27% |
| 2 standard deviations | 95.45% |
| 3 standard deviations | 99.73% |
These shares hold for any normal distribution, which makes z-scores a quick check. A value 2 standard deviations above the mean is above about 97.7% of values. When counts come from many independent yes-or-no trials, the Binomial Probability Calculator gives the exact answer that the normal curve approximates.
How to Use the Normal Distribution Calculator
- Choose what to find: Pick below, above, between, outside or inverse.
- Enter the mean and standard deviation: Use 0 and 1 to work directly with z-scores.
- Enter the value or values: Type x, or x₁ and x₂ for a range.
- For the inverse: Type a probability and choose whether it is the area to the left, the right or in the middle.
Frequently Asked Questions
How do I find the probability between two values?
Convert both values to z-scores with z = (x − μ) ÷ σ, look up Φ for each, and subtract. For a mean of 75 and a standard deviation of 10, P(70 < X < 90) = Φ(1.5) − Φ(−0.5) = 0.9332 − 0.3085 = 0.6247.
How do I calculate a z-score?
Subtract the mean from the value and divide by the standard deviation. A score of 90 with a mean of 75 and a standard deviation of 10 has z = 1.5.
What is the probability above a value?
It is 1 minus the probability below it. P(X > 90) = 1 − 0.9332 = 0.0668 for the example above.
How do I find the value for a given percentile?
Use the inverse mode. The 95th percentile of a normal distribution with mean 75 and standard deviation 10 is 75 + 1.6449 × 10 = 91.45.
What does the 68–95–99.7 rule say?
About 68.27% of values lie within one standard deviation of the mean, 95.45% within two and 99.73% within three.
Is this the same as normalcdf and invNorm?
Yes. The between mode matches normalcdf(lower, upper, μ, σ) and the inverse mode with a left-tail probability matches invNorm(p, μ, σ) on a TI calculator.
Formulas and worked examples checked in October 2026. Probabilities use a numerical approximation of the normal CDF accurate to about seven significant figures.