Z score calculator
How many standard deviations from the mean, and the percentile that implies — along with a straight answer about when that percentile means nothing at all. Nothing you type here leaves your browser — it is worked out on your own device, so there is nothing to send and nothing to log.
What a z score is
How many standard deviations a value sits from the mean. That is all, and it is pure arithmetic: subtract the mean, divide by the standard deviation. It works on any set of numbers.
The percentile is a different matter. Turning z into “you are in the 97.7th percentile” requires the data to actually follow a normal distribution, and a great deal of data does not.
Where the normal assumption holds and where it does not
Heights, measurement errors and test scores designed to be normal are all reasonably normal. Incomes, waiting times, house prices and file sizes are not — they are skewed, often heavily, and a z score of 2 on income data does not mean 97.7th percentile.
The z score itself is still a valid description of distance from the mean. It is the percentile that stops being true.
And the very small numbers are computed properly
Past about four standard deviations, the probability in the tail gets small enough that the usual way of computing it breaks.
Most implementations work out the tail as one minus the area on the other side. At z = 8 the area on the other side is 0.99999999999999938, and subtracting that from 1 in ordinary floating-point arithmetic leaves about two significant digits — the answer comes out about 7% wrong and looks entirely plausible. This page computes the tail directly and never subtracts.