Subtract the mean, divide by the spread
z = (x minus the mean) divided by the standard deviation. The subtraction moves the centre of the distribution to zero and the division rescales it so one unit of z is one standard deviation. Whatever the original data measured — millimetres, dollars, exam marks, tensile strength — the result is unitless, which is the entire point. A z of 1.5 means the same thing in every dataset that produced it: this value sits one and a half standard deviations above centre.
That is what makes otherwise incomparable numbers comparable. A 78 on one exam and a 62 on another cannot be ranked until you know each mean and each spread; standardize both and the comparison is immediate. Same for a machinist deciding whether a part is further out of tolerance on diameter or on length, when the two dimensions have completely different tolerances.
The percentile conversion is where the assumption lives
Computing z requires nothing but a mean and a standard deviation, and it is valid for any dataset. Converting z into a percentile requires the shape of the distribution, and this page assumes a normal one. In a normal distribution z = 1 sits at the 84.1st percentile; in a right-skewed distribution — incomes, house prices, response times, most things with a floor at zero and no ceiling — it does not. Skew pushes the mean above the median, so a z of 0 no longer marks the halfway point, and the tail percentiles can be off by a lot.
If you have the raw data rather than just summary statistics, do not go through z at all. Rank the value directly against the list with the percentile calculator, which makes no distributional assumption. Use the normal conversion when summary statistics are all you have, and treat the percentile as an estimate rather than a fact.
Rescalings you will meet
Negative numbers and decimals are awkward to report, so most test scales are z with the axes moved. All of them carry identical information.
| Scale | Formula | Mean | SD | Where you see it |
|---|---|---|---|---|
| z-score | z | 0 | 1 | Statistics, process control |
| T-score | 50 + 10z | 50 | 10 | Psychological and fitness testing |
| IQ, Wechsler | 100 + 15z | 100 | 15 | Cognitive test reporting |
| Stanine | round(5 + 2z) | 5 | 2 | Educational testing, 1 to 9 |
An IQ of 130, a z of 2 and a T-score of 70 are three names for one position. Knowing that stops the scales seeming mysterious: 130 is not measuring an amount of anything, it is a rank expressed in units of 15.
The 68-95-99.7 rule
Under a normal distribution about 68 percent of values fall within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. It is the fastest sanity check available. If someone reports a mean of 50 with a standard deviation of 5 and then a value of 90, that is a z of 8, and under a normal distribution a z of 8 does not happen. Either the distribution is not normal or something is wrong with the data.
Manufacturing quality control reads the same rule from the other end. A process whose tolerance band sits six standard deviations from the mean produces roughly two defects per billion opportunities in the pure arithmetic, which is where the phrase six sigma comes from. Real implementations allow for the process mean drifting and quote a far higher defect rate, a useful reminder that these tail probabilities assume the distribution is both normal and stable, and real processes are frequently neither.
Which standard deviation to use
If you know the population parameters — a standardized test with published norms, a process with a long control history — use those. If you are working from a sample, use the sample standard deviation with the n minus 1 divisor, and remember that with a small sample the standard deviation is itself an estimate carrying meaningful uncertainty. A z computed from ten observations is a rough indication; the same z from ten thousand is a measurement. The standard deviation calculator returns both divisors side by side and labels which is which, so you can see how much the choice moves the answer for your data before it feeds into anything here.
Questions people ask
Is a negative z-score bad?
It means below average, which is only bad if higher is better. For a lap time, a defect count, a delivery delay or a golf score, below average is the good direction and a negative z is the result you want. The sign carries direction and nothing else. This matters when you standardize several measures and combine them, because a composite that adds z-scores without flipping the sign on the lower-is-better components will rank everything backwards on those components.
Can I average z-scores from different tests?
You can, and it is a standard way to build a composite index, but it comes with conditions. The tests should measure things you genuinely want weighted equally, since averaging z-scores gives each component equal weight by construction. Components that correlate heavily with each other effectively count more than once. And the average of z-scores is not itself a z-score on a known distribution, so converting the composite back to a percentile requires knowing the distribution of the composite, which is rarely normal just because the parts were.
Why does my z-score not match the one my software reports?
Almost always the standard deviation divisor. Dividing the sum of squared deviations by n gives the population standard deviation and by n minus 1 gives the sample one, and with small datasets the two differ noticeably — for eight values the gap is about 7 percent, which flows straight through to z. Spreadsheet functions split along the same line: STDEV.P against STDEV.S in Excel. Check which one produced the number you were given before assuming anything is broken.
How accurate is the percentile figure here?
The normal cumulative distribution has no closed form, so it is computed by approximation. This page uses the Zelen and Severo rational approximation, whose absolute error stays under 7.5 times ten to the minus eighth across the whole range — several decimal places tighter than anything you would report. The practical limits on accuracy are elsewhere: whether your data is really normal, and how well the mean and standard deviation you entered are estimated. Deep in the tails, past about z = 5, the normality assumption is doing all the work and the arithmetic none.