Normal Distribution Calculator

In short The probability that a normal variable falls in a range is the area under its bell curve over that range, and every such area is found by converting the endpoints to z-scores and reading the standard normal distribution. With a mean of 100 and a standard deviation of 15, the chance of a value below 115 is 84.13%, above 130 is 2.28%, and between 115 and 130 is 13.59%. This tool does all four cases — below, above, between and outside — and shows the z-scores it used, so the working is visible rather than hidden behind a number. Three properties of the normal distribution are worth carrying around. Its mean and median are identical, so if your data has a mean well above its median, it is not normal and nothing here applies cleanly. Its middle 50% spans 1.349 standard deviations, which is a more useful landmark than the famous three. And it has no upper or lower limit: the curve assigns a tiny probability to every value, however extreme, which is exactly why it is the wrong model for anything with a hard floor at zero. That last point causes more real errors than any arithmetic. Heights and measurement errors are close to normal. Incomes, waiting times, insurance claims and file sizes are not, and using the normal distribution on them understates the tail badly.