Normal distribution calculator
Enter the mean and SD, get the probability.
How do I find a probability in a normal distribution?
Turn the value into a z-score, z = (x − μ) ÷ σ, and look up the area under the bell curve. With a mean of 100 and SD of 15, a score above 130 is z = 2, which has a probability of 0.0228 (2.28%). The z-score says how many standard deviations the value lies above (positive) or below (negative) the mean (OpenStax, Introductory Statistics 2e, section 6.1).
What is the 68-95-99.7 rule?
About 68.27% of values fall within 1 standard deviation of the mean, 95.45% within 2 and 99.73% within 3. The table shows these ranges for your numbers.
How do I find the probability between two values?
Subtract the lower area from the upper one. With a mean of 100 and SD of 15, P(85 < X < 115) = 0.6827.
How do I find the value at a percentile?
Choose "Value at a percentile": the 90th percentile of a normal distribution with mean 100 and SD 15 is 119.22 (z = 1.2816).
Worked example: exam scores
OpenStax (section 6.2) models exam scores as normal with mean 63 and standard deviation 5. A score above 65 has z = (65 − 63) ÷ 5 = 0.4, so P(X > 65) = 0.3446: about 34% of students did better. The 90th percentile is 63 + 5 × 1.2816 = 69.4, and the 70th percentile is 65.6. Choose “P(X > x)” with x = 65, or “Value at a percentile” with 90 and 70.
What is the standard normal distribution?
The one with mean 0 and standard deviation 1 — the z-table. Enter 0 and 1 to use it directly.
Why does P(X = x) come out as zero?
Because the normal distribution is continuous: the probability sits in ranges, not at single points, so P(X < x) and P(X ≤ x) are the same number. To ask about a measurement recorded to the nearest whole unit, ask for a range, such as P(64.5 < X < 65.5) for a reported 65.
How do I find the probability outside a range?
Choose “P(X < a or X > b)”, which adds the two tails. With a mean of 100 and SD of 15, the chance of landing below 70 or above 130 is 0.0455 — twice the single-tail 0.0228 for z = 2, because the range is symmetric around the mean. This is the same arithmetic behind a two-sided p-value.
Which z-values come up most often?
Three cut-offs appear in most confidence intervals and tests: z = 1.645 leaves 5% in the upper tail (a 90% two-sided interval), z = 1.96 leaves 2.5% (95%), and z = 2.576 leaves 0.5% (99%). Enter the percentile (95, 97.5 or 99.5) in “Value at a percentile” with mean 0 and SD 1 to reproduce them.
Common mistakes with the normal distribution
First, typing the variance where the standard deviation belongs: if the variance is 25, σ is 5. Second, assuming data are normal because they are numbers. Income, waiting times and counts near zero are usually skewed, and the normal probabilities will be off, especially in the tails; check the shape with a histogram or the skewness and kurtosis calculator. Third, choosing the wrong tail: “P(X > x)” and “P(X < x)” always add to 1. Fourth, confusing a value with its percentile: the 90th percentile is a score, not “90 points”.
Sources: OpenStax, Introductory Statistics 2e, sections 6.1 (The Standard Normal Distribution) and 6.2 (Using the Normal Distribution). Related: z-score calculator, empirical rule calculator, skewness and kurtosis calculator, p-value calculator and standard deviation calculator.