Normal Distribution Calculator: P(X < x)
This normal distribution calculator finds probabilities under the bell curve for any mean and standard deviation. Choose the area to the left of a value, to the right of a value, between two values, or outside two values, and it gives the probability, the percent and the z-scores. The inverse mode works backward from a percentile or probability to the matching x value, for example the score that marks the top 10%. It defaults to the standard normal distribution (mean 0, SD 1), and it works just as well for test scores, heights or manufacturing tolerances.
Normal Distribution Calculator
Areas under the bell curve: P(X < x), P(X > x), between two values, or the value for a percentile.
How to use the Normal Distribution Calculator
- Enter the mean and standard deviation.
- Choose the type of probability or the inverse mode.
- Enter x, or a and b, or the percentile.
- Read the probability, percent and z-scores.
Normal distribution
z = (x - ฮผ) ÷ ฯ and P(X < x) = ฮฆ(z), the standard normal cumulative distribution. The inverse finds z with ฮฆ(z) = p, then x = ฮผ + zฯ.
Features
- Left, right, between and outside areas
- Inverse normal (percentile to x)
- Any mean and SD
- z-scores shown
- Accurate to about 7 decimal places
- Copy and print
Frequently asked questions
How do I find a normal probability?
Convert x to a z-score, z = (x - ฮผ) ÷ ฯ, then find the area under the standard normal curve. With mean 100 and SD 15, P(X < 130) is about 0.9772.
What is the 68-95-99.7 rule?
About 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD.
How do I find the value for a percentile?
Use inverse mode. The 90th percentile is ฮผ + 1.2816 ฯ, so with mean 100 and SD 15 it is about 119.2.
What is the standard normal distribution?
A normal distribution with mean 0 and SD 1. Any normal variable can be converted to it using z-scores.
How accurate are the results?
The cumulative probability uses a high-precision approximation accurate to about 1 × 10^-7, which is more than enough for homework and reports.