Z-Score Calculator

The z-score, also called the standard score, z-value or normal score, is a dimensionless quantity giving the signed number of standard deviations by which a value sits above or below the mean. Values above the mean have positive z-scores; values below it have negative ones. A z-score of 0 is the mean exactly.

Calculate it by subtracting the population mean from the raw score, then dividing that difference by the population standard deviation:

where x is the raw score, μ is the population mean and σ is the population standard deviation. For a sample the formula is the same shape, with the sample mean x̄ and sample standard deviation s in place of the population values.

That substitution is not quite free, and most explanations skip it. Once σ is estimated from the sample rather than known, the statistic follows a t-distribution rather than a normal one. With 30 or more observations the difference is small enough to ignore. Below that, the t-distribution has heavier tails, and using a z-table will make your result look more significant than it is.

Z-scores are used to run z-tests, build prediction intervals, drive process control, compare scores measured on different scales, and more.

What standardising actually does

Subtracting the mean shifts the distribution so it is centred on zero. Dividing by the standard deviation rescales it so one unit equals one standard deviation. The result always has a mean of 0 and a standard deviation of 1, whatever the original units were, which is what makes scores from different scales comparable.

Here is the part that trips people up: standardising changes location and scale, and nothing else. It does not make a skewed distribution normal. If your data has a long right tail, the z-scores have exactly the same long right tail, and reading probabilities off a normal table will be wrong. A z-score becomes a probability only once you know the underlying distribution is approximately normal.

Comparing across scales is where it earns its keep. A 1350 on the SAT and a 30 on the ACT are meaningless side by side until both become z-scores against their own distributions. The same logic applies to a child's height on a growth chart, a factory measurement against its process limits, and a feature column being standardised before it goes into a machine learning model.

Reading probabilities from a z-score

For a normal distribution the z-score maps directly to an area under the curve, which is the probability of landing below that point. The empirical rule is the shorthand version: about 68.27% of values fall within one standard deviation of the mean, 95.45% within two and 99.73% within three.

Those figures come straight out of the table below. A z of 2 gives an area of 0.47725 from the mean, doubling to 0.9545 for the range from -2 to +2, and leaving 0.0455 split between the two tails.

Direction matters when you use this for testing. A two-tailed test at the 5% level puts 2.5% in each tail and has critical values of ±1.96. A one-tailed test at the same level puts the whole 5% at one end, with a critical value of 1.645. The 1% two-tailed value is 2.576. Mixing the two up is one of the most common errors in undergraduate statistics, and it always makes a result look stronger than it is.

Z-table

A z-table, also called a standard normal table or unit normal table, lists the probability that a statistic falls below, above or between points on the standard normal distribution. On the graph of that distribution, z = 0 is the centre of the curve, a positive z lies to the right of the mean and a negative z lies to the left. Several different layouts exist.

The values in the table below are the area between z = 0 and the given z-score. Read the whole number and first decimal place down the left column, and the second decimal along the top: z = 1.06 sits at row 1.0, column 0.06, giving 0.35543. Add 0.5 to get the area to the left of that point, which is 0.85543.

Because the curve is symmetric, negative z-scores need no separate table. The area between 0 and -1.06 is the same 0.35543, and the area to the left of -1.06 is 0.5 minus that, which is 0.14457.

Z table from mean (0 to Z)

Where z-scores get used

Outlier detection is the most common informal use. A rule of thumb flags any observation with a z beyond ±3, since under a normal distribution roughly 1 value in 370 sits that far out. The catch is that the mean and standard deviation are themselves distorted by the outlier you are hunting, so a single extreme value can hide itself by inflating σ. The modified z-score avoids this by using the median and the median absolute deviation instead, as 0.6745(x − median)/MAD, with anything past 3.5 treated as an outlier.

Manufacturing uses z-scores as the language of process capability. Specification limits are set a number of standard deviations from the target, and the Six Sigma programme is named for putting them six away. Quality charts plot measurements in standard deviation units for exactly this reason: it makes any process comparable to any other regardless of what is being made.

Finance uses the term for two unrelated things, which is worth untangling. Standardising returns is the ordinary meaning. The Altman Z-score, published by Edward Altman in 1968, is something else entirely: a weighted formula of five accounting ratios that predicts corporate bankruptcy, where a score below 1.8 signals distress. It is not a standard score, and the two should never be confused.

Notes on the three calculators

The first converts a raw score into a z-score and shows the three probabilities that follow from it, with the matching area shaded on the curve. The second works in both directions: type a z-score to get its probabilities, or type any one of the five probabilities to recover the z-score, which is what looking backwards through a z-table amounts to. The third gives the area between two bounds, which is the calculation behind any confidence interval.

All the probabilities here come from a double precision algorithm rather than a rounded printed table, so they stay accurate far into the tails where a table runs out of digits. Areas are shown to five decimal places. The normal distribution has been studied since Abraham de Moivre approximated the binomial with it in 1733, with Laplace and Gauss developing it further, and the standard normal table exists because for two centuries there was no other practical way to get these numbers.