Normal Distribution Calculator
Give a mean and standard deviation, pick a range, and this normal distribution calculator returns its probability and shades it on the bell curve. Switch to inverse to get the value at any probability.
P(85 < X < 130)
- P(85 < X < 130) Φ(2.00) - Φ(-1.00) = 0.97725 - 0.15866: the area left of 130 minus the area left of 85.
- 0.8186
- As a percent
- 81.86%
- z for 85 (85 - 100) / 15. Every normal distribution is the standard one stretched and shifted, so the area is read at this z.
- -1.00
- z for 130 (130 - 100) / 15. Every normal distribution is the standard one stretched and shifted, so the area is read at this z.
- 2.00
- P(X < 85)
- 0.1587
- P(X > 130)
- 0.0228
- Outside the range The complement: the two add to 1.
- 0.1814
| Percentile | Value | z |
|---|---|---|
| 1st | 65.10 | -2.3263 |
| 2.5th | 70.60 | -1.96 |
| 5th | 75.33 | -1.6449 |
| 10th | 80.78 | -1.2816 |
| 25th | 89.88 | -0.6745 |
| 50th | 100.00 | 0.00 |
| 75th | 110.12 | 0.6745 |
| 90th | 119.22 | 1.2816 |
| 95th | 124.67 | 1.6449 |
| 97.5th | 129.40 | 1.96 |
| 99th | 134.90 | 2.3263 |
How the Normal Distribution Calculator works
A normal distribution is the bell curve: symmetric about its mean, with a spread set by its standard deviation. Two numbers describe it completely. About 68.27% of it lies within one standard deviation of the mean, 95.45% within two and 99.73% within three.
Every probability on the curve is an area under it, and every normal curve is the standard one (mean 0, SD 1) shifted and stretched. So the method is two steps. First standardize each value with z = (x - μ) / σ. Then read the area at those z values from the standard normal.
Take scores on a test scaled to a mean of 100 and a standard deviation of 15, and ask what share falls between 85 and 130. The z values are (85 - 100) / 15 = -1.00 and (130 - 100) / 15 = 2.00. The area left of z = 2.00 is 0.97725 and the area left of z = -1.00 is 0.15866, so the share between is 0.97725 - 0.15866 = 0.81859, or 81.86%. P(X < 85) is 0.1587 and P(X > 130) is 0.0228.
The calculator draws the bell curve for your mean and SD and shades the region it measured, with the axis marked at whole standard deviations.
Run it backward and it works as an inverse normal distribution calculator: give a probability and it returns the value. The value with 95% of the curve below it is μ + 1.6449σ, which for the test scores is 124.67. The middle 95% runs from μ - 1.96σ to μ + 1.96σ, about 70.6 to 129.4. The table under the result lists the value at eleven common percentiles.
For a continuous distribution the probability of any single exact value is 0, so P(X < a) and P(X ≤ a) are the same number. That changes when a normal curve stands in for a count, such as a binomial with many trials, where adding or taking away half a unit at each end (the continuity correction) brings the approximation closer.
All of this assumes the data really are close to normal. Measurement errors and averages of many independent values usually are. Incomes, house prices and waiting times are skewed, and a normal curve will understate how often their extreme values turn up.
Normal Distribution Calculator FAQ
How do you find a probability on a normal distribution?
Standardize the value with z = (x - μ) / σ, then take the area under the standard normal curve on the side you want. For a range, subtract the area left of the lower end from the area left of the upper end. With mean 100 and SD 15, P(85 < X < 130) is 0.8186.
What does an inverse normal distribution calculator do?
It runs the curve backward: give a probability and it returns the value. It finds the z with that much of the standard normal below it, then converts with x = μ + zσ. The value with 95% below it is μ + 1.6449σ.
Is P(X < a) the same as P(X ≤ a)?
Yes, for a normal distribution. It is continuous, so the probability of landing on exactly a is 0, and including or excluding the endpoint changes nothing.
What is the 68-95-99.7 rule?
In a normal distribution, 68.27% of values lie within one standard deviation of the mean, 95.45% within two and 99.73% within three. It is a property of the bell curve, so it only describes data that are roughly normal.
What is the difference between a normal and a standard normal distribution?
The standard normal has mean 0 and standard deviation 1. Any normal distribution converts to it by subtracting the mean and dividing by the standard deviation, which is the z score. The z score calculator and its z-table work on the standard one directly.