Normal Distribution Calculator
Calculate probabilities and quantiles for any normal distribution. Choose a mode: find P(X < x), P(X > x), P(x1 < X < x2), the outside-range probability, or reverse-solve to find x given a probability. Enter the mean, standard deviation, and the value(s) you care about to get a full breakdown with step-by-step working.
Formula
Worked example
For x = 120, mean = 100, sigma = 15: z = (120 - 100)/15 = 1.3333. Phi(1.3333) = 90.88%, so 90.88% of values fall below 120 (left tail) and 9.12% fall above (right tail), placing x at the 90.9th percentile. Inverse: to find x at the 95th percentile, Phi_inv(0.95) = 1.6449, so x = 100 + 1.6449 * 15 = 124.67.
What the normal distribution describes
The normal distribution, the familiar bell curve, is the most important model in statistics because so many natural and measured quantities cluster symmetrically around an average. Heights, test scores, measurement errors and sample means all tend toward this shape. A normal distribution is fully defined by just two numbers: the mean, which sets where the peak sits, and the standard deviation, which sets how wide the bell is. Once you know those two parameters, the curve tells you exactly how likely any range of values is, which underpins confidence intervals, hypothesis tests and quality control.
From a raw value to a z-score
A z-score restates a raw value as the number of standard deviations it sits away from the mean, using z = (x - mu)/sigma. A z of 0 means the value equals the mean; a z of +2 means it is two standard deviations above it; a negative z means it is below the mean. Because the z-score strips out the original units, it lets you compare values drawn from completely different normal distributions on one common scale. Converting to z is the first step in nearly every probability lookup, because it maps any normal distribution onto the single standard normal curve.
Reading the cumulative probability and tail areas
The left-tail cumulative probability P(X < x), written Phi(z), is the area under the curve to the left of your value. It equals the proportion of the population that falls below x, and it is x's percentile rank. This calculator uses a high-accuracy approximation of the error function (accurate to about 7 decimal places), so you never need a printed z-table. The right-tail area P(X > x) is simply 1 minus the left tail. Because the curve is symmetric, the area below the mean is always 50%, so any value below the mean is below the 50th percentile.
Between-values and outside-range probabilities
To find the probability that a measurement lands inside a range, use the between-values mode. The calculator converts both bounds to z-scores, looks up their cumulative probabilities, and subtracts: P(x1 < X < x2) = Phi(z2) - Phi(z1). The outside-range mode returns the complementary probability, 1 minus the inside probability, which is the combined area in both tails. This two-tailed probability is the one used for two-sided hypothesis tests: a two-tailed p-value is the probability that a result at least as extreme as yours could occur by chance.
Inverse (reverse) normal: finding x from a probability
The inverse normal (quantile function) solves the problem in reverse: given a target cumulative probability p, what is the value x such that P(X < x) = p? This is how you find the 95th-percentile score for a test, the cutoff for the top 1% of incomes, or the critical value for a one-tailed hypothesis test. This calculator uses Peter Acklam's rational approximation of the inverse CDF, accurate to about 9 significant digits. Simply enter the target probability between 0 and 1, and the calculator reports both the z-score and the raw x-value for your distribution.
Key normal distribution percentiles and z-scores
| Percentile | z-score | Common use |
|---|---|---|
| 1st | -2.3263 | Extreme low outlier threshold |
| 5th | -1.6449 | One-tailed 5% critical value |
| 10th | -1.2816 | One-tailed 10% critical value |
| 16th | -1.0000 | One sigma below mean |
| 25th | -0.6745 | Lower quartile |
| 50th | 0.0000 | Median (equals mean) |
| 75th | +0.6745 | Upper quartile |
| 84th | +1.0000 | One sigma above mean |
| 90th | +1.2816 | One-tailed 10% critical value |
| 95th | +1.6449 | One-tailed 5% critical value |
| 97.5th | +1.9600 | Two-tailed 5% (alpha=0.05) |
| 99th | +2.3263 | Extreme high outlier threshold |
| 99.9th | +3.0902 | Three-sigma boundary |
Commonly needed z-scores with their cumulative left-tail probabilities.
Frequently asked questions
What is the difference between a z-score and a probability?
A z-score tells you how far a value sits from the mean in standard deviation units, while the probability P(X < x) tells you what fraction of the distribution falls below that value. You convert the raw value to a z-score first, then look up the area under the standard normal curve to the left of that z-score. The z-score is a position; the cumulative probability is the area to its left.
How do I find the probability that a value falls between two numbers?
Select the "Between two values" mode, enter your mean, standard deviation, lower bound (x1) and upper bound (x2). The calculator converts both bounds to z-scores, looks up Phi(z1) and Phi(z2), and returns Phi(z2) - Phi(z1) as the between-values probability. For example, with mean 100 and sigma 15, the probability of a value between 85 and 115 is Phi(1) - Phi(-1) = 84.13% - 15.87% = 68.27%, which is the empirical one-sigma rule.
What does the inverse (reverse) normal calculate?
The inverse normal answers: "what value of x gives a cumulative probability of p?" For example, if you want the 90th percentile of a distribution with mean 100 and sigma 15, enter 0.9 as the probability. The calculator applies the inverse CDF to get z = 1.2816, then computes x = 100 + 1.2816 * 15 = 119.22. This is used in grading, quality control, hypothesis testing critical values and percentile-based thresholds.
How accurate is the probability this calculator returns?
Cumulative probabilities use the Abramowitz and Stegun error-function approximation, accurate to about 1.5e-7 in absolute error, matching a printed z-table to four decimal places. The inverse CDF uses Peter Acklam's rational approximation, accurate to about 9 significant digits. Both are effectively exact for all practical statistical work.
Why does the standard deviation have to be greater than zero?
The standard deviation measures spread, and the z-score formula divides by it. A standard deviation of zero would mean every value equals the mean, leaving no distribution to speak of and making the division undefined. The calculator therefore requires sigma greater than zero and returns no result until you enter a valid positive spread.
What is the empirical (68-95-99.7) rule?
For any normal distribution, about 68.27% of values fall within one standard deviation of the mean (between mu - sigma and mu + sigma), about 95.45% fall within two standard deviations, and about 99.73% fall within three standard deviations. You can verify these in the between-values mode by setting x1 = mean - k*sigma and x2 = mean + k*sigma for k = 1, 2 or 3.