Normal Distribution Calculator

Every normal curve is the same shape, just shifted and stretched. So the calculator first turns your value into a z-score: how many standard deviations it sits from the mean.

Here $x$ is your cut-off, $\mu$ (mu) is the mean and $\sigma$ (sigma) is the standard deviation. The area under the standard curve to the left of $z$ is written $\Phi(z)$. That area is the probability of a value below $x$.

The whole area under the curve is 1, so the other cases follow by subtraction. The check in the working runs this backwards: it turns your area into a value again and confirms you get your cut-off back.

There is no simple formula for $\Phi$. The tool evaluates it with the complementary error function, which is accurate to about 15 significant digits, and matches the printed tables in Abramowitz and Stegun. To work backwards from a percentage, it inverts the curve to find $z$, then converts back with $x = \mu + z\sigma$.

Who fits a doorway range?

Say adult heights in a group are normal with a mean of 175 cm and a standard deviation of 7 cm. What share is between 170 and 185 cm? The z-scores are −0.714 and 1.429. The areas to their left are 0.2375 and 0.9234. Subtract them and you get 0.6859, or about 68.6% of people.

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A score above 130

IQ-style scores are scaled to a mean of 100 and a standard deviation of 15. A score of 130 is exactly z = 2. The area to its right is 0.0228, so about 2.3% of people score higher. That is roughly 1 in 44.

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Bolts outside tolerance

A machine cuts bolts with a mean length of 50 mm and a standard deviation of 0.2 mm. The spec allows 49.6 to 50.4 mm, which is 2 SD either side. About 4.55% of bolts fall outside. If the spread came down to 0.15 mm, the what-if table shows that share falling to 0.77%.

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The cut-off for the top 10%

A test has a mean of 500 and a standard deviation of 100. To beat 90% of test takers, you need the 90th percentile. Inverting the curve gives z = 1.2816, so the score is 500 + 1.2816 × 100 = 628.2.

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A probability here is a share of everything the curve describes. A result of 0.0228 for scores above 130 means about 2.3% of people. The tool also turns it into a "1 in N" figure, which is often easier to explain.

The z-score tells you how unusual a value is, whatever the units. About 68.3% of values sit within 1 SD of the mean, 95.4% within 2 SD and 99.7% within 3 SD (NIST e-Handbook). So a value beyond 2 SD is unusual and one beyond 3 SD is rare. The rule card gives these ranges in your own units.

For the percentile and middle modes, the answer is a value on your scale. A middle 95% of 70.6 to 129.4 for a mean of 100 and SD of 15 means 2.5% of values fall below 70.6 and 2.5% above 129.4.

Tail areas below 0.000001 are shown in scientific notation instead of being rounded to 0. The area above x is computed from the right tail itself rather than as 1 minus the area below, so it stays accurate even 10 standard deviations out. Real data are rarely that normal so far out anyway, so treat any tail beyond 4 SD as a rough guide.

• Entering the variance instead of the standard deviation. If a problem gives N(100, 225), the 225 is σ², so σ = 15.
• Reading the table the wrong way round. Φ(z) is the area to the left. For "more than" questions you need 1 − Φ(z), which is the Above x mode.
• Asking for an exact value. The chance of exactly 120 is zero on a continuous curve. If your data are whole numbers, use a range such as 119.5 to 120.5, which gives 0.0109.
• Using a percentage as a decimal, or the reverse. Type 90 for the 90th percentile here. Other tools may ask for 0.9.
• Assuming everything is normal. Incomes, waiting times and house prices have long right tails. A normal curve will understate their extremes, so check a histogram first.