Normal Distribution Calculator
Enter a mean and a standard deviation, choose a question, and the calculator gives the area under the bell curve with the matching z-scores and a shaded graph. Ask for the probability that a value is below, above, between or outside limits, or run it backward to find the value that has a given share of the distribution below it, such as the 95th percentile, together with the central interval holding that share. Everything runs in your browser.
The normal distribution is the bell curve described by a mean and a standard deviation. The probability of a range is the area under the curve over it, found from the z-score (how many standard deviations a value sits from the mean). About 68.27% of values fall within one standard deviation of the mean, 95.45% within two and 99.73% within three. The inverse question runs the other way: given a share, it finds the value with that share below it, such as the 95th percentile. Real data are only approximately normal, so treat results as a model. Everything is worked out in your browser.
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Student ToolsAbout the Normal Distribution Calculator
The normal distribution is the symmetric bell curve described by a mean and a standard deviation. The chance that a value falls in a range is the area under the curve over that range. Converting a value to a z-score, the number of standard deviations it lies from the mean, lets one table cover every normal distribution. The cumulative distribution is computed here from the complementary error function, accurate to about 15 digits and still precise far out in the tails, where the chance of a z-score above 6 is about 1 in 1 billion. The empirical rule gives the best known areas: 68.27% of values within one standard deviation of the mean, 95.45% within two and 99.73% within three. The inverse uses Acklam's rational approximation with one refinement step, which gives z = 1.959964 for the 97.5th percentile, the value behind 95% confidence intervals, and 1.644854 for the 95th percentile. On the IQ scale, with a mean of 100 and a standard deviation of 15, a score of 115 is one standard deviation up and has about 84.13% of scores below it. Few real measurements are exactly normal, so treat the answers as a model, and use the t distribution when a standard deviation is estimated from a small sample.
How do I find the probability on a normal distribution?▾
Convert the value to a z-score by subtracting the mean and dividing by the standard deviation, then look up the area below it. The calculator does this for you and handles above, between and outside ranges as well.
What is the 68-95-99.7 rule?▾
About 68.27 percent of values lie within one standard deviation of the mean, 95.45 percent within two and 99.73 percent within three. It is a quick check for whether a value is unusual.
What z-score gives 95 percent?▾
For a central 95 percent interval the z-score is 1.96, which cuts off 2.5 percent in each tail. For a one-sided 95 percent limit it is 1.645. Use the percentile tab to find any other value.
What is the difference between below and outside?▾
Below x is the area to the left of one value. Outside a and b is the area in both tails, below the lower limit and above the upper one, which equals one minus the area between them.
Is my data uploaded?▾
No. The calculation runs in your browser and nothing you enter is sent anywhere.