Z-score Calculator

Frequently Asked Questions

Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) / σ. For example, a value of 85 against a mean of 72 and a standard deviation of 6 gives z = (85 − 72) / 6 = 2.17. Enter the same three numbers in the Raw Value mode of this calculator and you also get the percentile and p-values alongside the z-score.

It tells you how many standard deviations your value sits from the mean, in a unit that works for any measurement. A z of 0 is exactly average; a z of +2 is two standard deviations above average, which is roughly the 98th percentile; a z of −1.5 is one and a half standard deviations below, near the 7th percentile. Because the unit is standard deviations, a z of 2 means the same rarity for an exam score, a blood test, or a manufacturing reading.

It depends on the direction you care about. On a test or any "higher is better" measure, z = +2 or above is excellent (top 2.3%). On a defect count or a risk measure, the same magnitude is alarming. Between −1 and +1 lies the ordinary middle 68% of any bell-shaped data. Values beyond ±3 occur in less than 0.3% of a normal distribution and usually justify a second look, whether celebratory or investigative.

z = (x − μ) / σ, where x is the value, μ is the population mean, and σ is the population standard deviation. When you only have a sample, use the sample mean and sample standard deviation as estimates of μ and σ. The result expresses the gap between your value and the mean in units of standard deviation.

Yes. A negative z-score simply means the value is below the mean. A z of −1.8 sits 1.8 standard deviations below average, around the 4th percentile. Negative z-scores are common in quality control, growth monitoring, and finance, where values below a reference matter as much as values above it. The sign carries the direction; the magnitude carries the rarity.

z = 1.645. The 95th percentile leaves 95% of the curve below it and 5% above, and that area split occurs 1.645 standard deviations above the mean. Enter 95 in the Percentile mode of this calculator to confirm, or interpolate between Φ(1.6) = 0.9452 and Φ(1.7) = 0.9554 in the z-score table.

It depends on the tail. For a one-tailed test, p = 0.05 corresponds to z = 1.645 (or −1.645 in the other direction). For a two-tailed test, p = 0.05 corresponds to z = ±1.96, because the 5% is split into two 2.5% tails. This calculator reports both p-values with every z-score.

Because it is the two-tailed 5% cutoff. Roughly 95% of a normal distribution lies between −1.96 and +1.96, so a z beyond those bounds is unusual enough to reject a null hypothesis at the conventional 0.05 significance level. That single convention makes ±1.96 the threshold behind 95% confidence intervals, two-tailed significance tests, and most published "statistically significant" claims.

The z-score is undefined. A standard deviation of zero means every value in the reference set is identical, so there is no spread to measure distance against – division by zero. This calculator flags the input as an error rather than returning a number. If you see this with real data, check for a duplicated single value or a data entry mistake.

Find your z-score's row and take the area from the column. With 0.1-step tables like the ones on this page, interpolate between neighbouring rows for a second decimal: for z = 1.96, average the gap between Φ(1.9) = 0.9713 and Φ(2.0) = 0.9772 to get 0.975. For negative z, either use the negative table directly or apply symmetry: Φ(−z) = 1 − Φ(z).

Both standardise a value against a mean and spread, but a t score uses the t distribution, which is wider than the normal when the sample is small. With 30 or more observations the two are nearly identical. Below that, the t cutoffs for significance grow larger than the z cutoffs, correctly demanding more evidence from less data. Use z when σ is known or the reference set is large; use t when σ is estimated from a small sample.

Use the Percentile mode of this calculator: enter the area as a percentage and it inverts the normal curve to find the matching z. The 90th percentile gives z = 1.282, the 95th gives 1.645, the 97.5th gives 1.960, and the 99th gives 2.326. For areas below 50% the z is negative: the 5th percentile is −1.645.

Yes. The z-score itself is pure arithmetic and valid for any distribution, but converting it to a percentile or p-value assumes the standard normal shape. If your data is skewed, the true fraction of values below yours can differ from the normal-based estimate. For roughly symmetric data the approximation is excellent; for heavily skewed data, report the observed data percentile instead.

The standard deviation is the unit in which z-scores are measured. The z formula divides by σ precisely so the answer reads "how many standard deviations from the mean". Our Standard Deviation Calculator computes σ from a raw list of values; this page then uses that σ to score any value you choose. The two tools together cover the full workflow from raw data to standardised position.

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