What a z-score actually tells you
A z-score answers one question: how far does a value sit from the mean of its group, measured in standard deviations rather than in the group's original units? A z-score of 2 means a value sits exactly two standard deviations above the mean; a z-score of −1.5 means one and a half standard deviations below it. The sign always shows the direction, above or below, and the size shows the distance.
x is the raw value being checked, μ (mu) is the mean of the group it came from, and σ (sigma) is that group's standard deviation. Subtracting the mean centres the value on zero, and dividing by the standard deviation rescales that distance into a common unit that works the same way whatever the original numbers looked like. A z-score of 0 always means "exactly average"; nothing else about the original data needs to be known to read that much.
Standardising across different scales
The reason z-scores exist at all is that raw numbers on different scales cannot be compared directly. Take two tests marked out of different totals: test A has a class mean of 65 and a standard deviation of 8, while test B has a mean of 28 and a standard deviation of 3. A student scores 77 on test A and 34 on test B. The raw numbers make test A look like the stronger result, since 77 is a bigger number than 34, but the two tests are not on the same scale, so that comparison is meaningless on its own.
Standardising both scores clears that up. Test A: z = (77−65)÷8 = 1.5. Test B: z = (34−28)÷3 = 2.0. Once both are converted to standard deviations from their own mean, test B is actually the stronger performance relative to its class, half a standard deviation further above average than test A, even though its raw mark was the smaller of the two. This is the whole point of a z-score: it puts values from entirely different scales, currencies, units or grading systems onto one shared, comparable footing.
Reading the percentile, and one-tailed versus two-tailed probabilities
Alongside the z-score itself, the first tab works out three related probabilities, each answering a slightly different question about a normal distribution with the mean and standard deviation entered.
P(Z < z), sometimes called the percentile, is the probability of landing at or below this value; a z-score of 2.00 gives a percentile of about 97.72%, meaning the value sits higher than roughly 97.72% of a normal distribution shaped like this one. P(Z > z) is the opposite tail, the probability of landing above the value, and the two always add up to exactly 1. Both of those are one-tailed probabilities, since each only looks in one direction from the value.
The two-tailed figure, P(|Z| > |z|), instead asks how likely a value is to sit at least this far from the mean in either direction, above or below. It is calculated as twice the smaller one-tailed probability, and it is the figure to reach for whenever "unusual" means unusual in either direction, such as a quality-control measurement that is a problem whether it runs too high or too low. A one-tailed test only makes sense when a result in one particular direction is the only one being checked for.
Working out the probability between two values
The second tab answers a different but related question: what proportion of a normal distribution falls between two values, rather than above or below just one? It works out a z-score for each bound, a and b, then subtracts the smaller cumulative probability from the larger one.
Φ (the Greek letter Phi) is the standard notation for the cumulative distribution function used throughout this page, the same calculation behind P(Z<z) on the other tab. Take a mean of 100 and a standard deviation of 15, and ask what share of the distribution falls between 85 and 115: 85 gives z = −1.00 and 115 gives z = 1.00, and Φ(1.00) − Φ(−1.00) works out to almost exactly 0.6827, or 68.27%. That is the familiar "68% within one standard deviation" rule of thumb, arrived at directly from the formula rather than assumed.
How the probability is worked out
The standard normal cumulative distribution function has no closed algebraic form; every calculator and statistics package works from an approximation rather than an exact formula. This page uses a widely published polynomial approximation, the one credited to Zelen and Severo and printed as formula 26.2.17 in Abramowitz and Stegun's Handbook of Mathematical Functions, with a stated maximum error of about 0.000000075 (7.5×10⁻⁸). That is far tighter than the four decimal places shown on this page, so nothing meaningful is lost to the approximation itself; any rounding visible in the results comes from the four-decimal-place display, not from the underlying calculation.
Extremely large z-scores, beyond about ±6, sit so far into the tail that the true probability, while never exactly zero, is too small to show at four decimal places; the calculator flags this rather than silently displaying 0.0000 or 1.0000 as if that were exact.
Why a z-score assumes roughly normal data
Turning a z-score into a percentile or a probability only works cleanly if the underlying data is at least roughly bell-shaped, the classic normal distribution. The z-score itself, the plain distance from the mean measured in standard deviations, can be calculated for any list of numbers regardless of its shape. The probability figures are a different matter: they describe how a genuinely normal distribution behaves, and a heavily skewed dataset, one with a long tail in one direction, or one with distinct clusters rather than a single peak, will not match those percentages nearly as closely, even though the arithmetic still runs without complaint.
Many real measurements, exam scores across a large cohort, adult heights, repeated measurement error, are close enough to normal for the percentages here to be a genuinely useful guide. Small samples, strongly skewed data such as income or waiting times, or anything with a hard floor or ceiling close to the values in question, are worth treating with more caution. The standard deviation calculator is the place to check the spread of a raw list before standardising it here, and includes its own check of how many of your values actually land within one and two standard deviations of the mean.
Questions people ask
What does a z-score of 0 mean?
It means the value is exactly equal to the mean, with no distance above or below it at all. A z-score only ever moves away from 0 once a value sits above or below the average of its group.
How do I read a negative z-score?
A negative z-score simply means the value sits below the mean rather than above it. The size of the number still measures the distance in standard deviations; a z-score of −2 is exactly as far from the mean as +2, just in the opposite direction.
What is the difference between a one-tailed and a two-tailed probability?
A one-tailed probability, P(Z<z) or P(Z>z), only looks in a single direction from the value. A two-tailed probability, P(|Z|>|z|), asks how likely a value is to sit at least this far from the mean in either direction, and is calculated as twice the smaller one-tailed figure. Use two-tailed whenever an unusually low result and an unusually high result would both count as noteworthy.
Why does the calculator need a standard deviation as well as a mean?
The mean alone only says where the centre of a distribution sits, not how spread out the values around it are. A z-score measures distance from the mean in units of the standard deviation, so without it there is no way to say whether a given raw difference from the mean counts as small and ordinary or large and unusual.
Does my data have to be normally distributed for a z-score to make sense?
The z-score itself can be calculated for any set of numbers with a mean and a standard deviation. The percentiles and probabilities on this page describe a normal distribution specifically, so they are most accurate when the underlying data is at least roughly bell-shaped; strongly skewed or clustered data will not match those percentages as closely, even though the z-score arithmetic still works.
What happens if I enter a standard deviation of zero or a negative number?
The calculator shows a calm message rather than a broken result. A standard deviation of exactly zero means every value in the group is identical, so dividing by it has no meaningful answer, and a negative standard deviation cannot describe a real distribution at all.
How accurate are the probabilities on this page?
They come from a published polynomial approximation to the standard normal cumulative distribution function, with a stated maximum error of about 0.000000075, far tighter than the four decimal places displayed. Beyond roughly ±6 standard deviations the true probability becomes too small to show at that precision, and the calculator notes this rather than displaying an exact-looking 0 or 1.
This calculator assumes the mean and standard deviation entered are accurate for the group being described, and that the underlying data is reasonably close to a normal distribution. It does not adjust for sampling error in an estimated mean or standard deviation.