Normal Distribution Calculator

Calculate z-scores, percentiles, and probabilities with a bell curve that shades the area below your value.

Normal Distribution Calculator tool

Loading...

Example

Input: IQ scores: mean 100, standard deviation 15, X value 130

Result: Z-score 2; P(X < x) = 0.97725 (97.7250%); P(X > x) = 0.02275 (2.2750%); percentile 97.72th

z = (130 - 100) / 15 = 2, and the standard normal CDF gives Phi(2) = 0.97725, so about 2.3% of scores are above 130.

Z-Score Reference Table

zP(Z < z)P(Z > z)Within plus or minus |z|
-30.13%99.87%99.73%
-22.28%97.72%95.45%
-115.87%84.13%68.27%
050.00%50.00%0.00%
184.13%15.87%68.27%
1.28290.01%9.99%80.02%
1.64595.00%5.00%90.00%
1.9697.50%2.50%95.00%
297.72%2.28%95.45%
2.32699.00%1.00%98.00%
2.57699.50%0.50%99.00%
399.87%0.13%99.73%

Computed with this calculator (mean 0, SD 1), rounded to 2 decimals. 1.645, 1.96 and 2.576 are the 90%, 95% and 99% two-sided values.

  1. Enter the mean and standard deviation that define your normal distribution.
  2. Input a raw value in the X Value field to convert it to a z-score and find its corresponding percentile.
  3. Read the probabilities of a value falling below x, above x, and within plus or minus |z| standard deviations of the mean.
  4. Observe the shaded region under the bell curve, which shows the probability of a value falling below x.
  5. For the probability between two values, enter each value in turn and subtract the two below-x probabilities.

Z-Scores and the Bell Curve

The normal distribution is the bell-shaped curve that describes many measurements, from heights to test scores. Enter its mean and standard deviation plus a value x. The calculator turns x into a z-score and finds the share of the curve below it, above it, and within the same distance of the mean. For IQ scores with a mean of 100 and a standard deviation of 15, a score of 130 has a z-score of 2. That puts it at about the 97.7th percentile.

The z-score says how many standard deviations x is from the mean: z = (x - mean) / SD. Positive values are above the mean and negative values below it. Z-scores let you compare results on different scales. Scoring 80 on a test with a mean of 70 and an SD of 5 gives z = 2. Scoring 85 on a test with a mean of 75 and an SD of 10 gives z = 1. So the first result is more unusual, even though the raw score is lower.

The results cover the common questions. P(X < x) is the share below x, which is the same as the percentile. P(X > x) is the share above. The central result is the share within plus or minus |z| of the mean. That is where the 68-95-99.7 rule comes from, as the table on this page shows. For the chance between two values, run each one and subtract the two below-x results. The bell curve shades the area below x and redraws as you type.

The answers are only as good as the bell-curve assumption. Heights and measurement errors fit it well. Incomes, waiting times and counts often do not. The tool uses a standard approximation that is accurate to about seven digits. It shows very small tail areas in scientific notation instead of rounding them to zero. To get the mean and SD from raw data, start with the Standard Deviation Calculator.

Formula / How It Works

z = (x - mu) / sigma | P(X < x) = Phi(z) | Phi(z) = (1/2)(1 + erf(z / sqrt(2)))

Sources

Frequently Asked Questions

A z-score tells you how many standard deviations a value is from the mean. A z-score of 0 is exactly average, 1 is one standard deviation above, and -1.5 is one and a half below. It lets you compare values measured on different scales, such as scores from two different tests.
Find P(X < x) for each value, then subtract the smaller from the larger. With a mean of 100 and SD of 15, P(X < 115) is 84.13% and P(X < 85) is 15.87%, so 68.27% of values fall between 85 and 115. Enter each value in turn to get the two figures.
For a normal distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. It gives quick estimates without a table. The exact figures from this calculator are 68.27%, 95.45% and 99.73%.
For a two-sided 95% range, z is about 1.96. That leaves 2.5% in each tail, which is why P(X < x) at z = 1.96 shows 97.50%. The matching values are 1.645 for 90% and 2.576 for 99%. These are the same z-scores used to size surveys and set error margins.
When it clusters around a middle value and tails off evenly on both sides, as heights and measurement errors tend to. Data with a hard floor or a long tail, such as incomes or wait times, usually is not normal. Plot a histogram before you rely on normal probabilities.

Related Tools

Standard Deviation Calculator

Calculate mean, variance, and standard deviation for both population and sample data sets instantly.

Statistics and Data

Probability Calculator

Calculate basic, conditional, and Bayesian probability with odds, an independence check, and a Bayes breakdown.

Statistics and Data

Sample Size Calculator

Determine the required sample size for surveys given confidence level, margin of error, and population.

Statistics and Data

Mean Median Mode Calculator

Find the mean, median, mode, and range of any data set with a frequency table and distribution display.

Statistics and Data

GPA Calculator

Calculate your semester and cumulative GPA on a 4.0 or 5.0 scale with support for weighted and unweighted grades.

Education and Academic

Combination and Permutation Calculator

Calculate nCr and nPr with step-by-step factorial breakdowns. Understand when order matters.

Statistics and Data