Z-Score Calculator

A z-score (standard score) says how many standard deviations a value sits above or below the mean. It is calculated as z = (x − μ) / σ. A z of 0 is exactly average, +1 is one standard deviation above, and −2 is two standard deviations below. Because it removes the original units, z lets you compare values from different scales, such as an exam score and a height.

Subtract the mean from the raw score, then divide by the standard deviation: z = (x − μ) / σ. For a score of 85 with mean 70 and standard deviation 10, z = (85 − 70) / 10 = 1.5. If you only have raw data, first find the mean and standard deviation of the data, which the From data mode does for you.

Find the cumulative area to the left of z under the standard normal curve, Φ(z), and multiply by 100. A z of 1.5 gives Φ(1.5) ≈ 0.9332, so the value is at about the 93rd percentile — higher than roughly 93% of values. This assumes the data are approximately normally distributed.

For a right-tailed test the p-value is P(Z > z) = 1 − Φ(z); for a left-tailed test it is Φ(z); for a two-tailed test it is 2 × P(Z > |z|). With z = 1.96 the two-tailed p-value is about 0.05. The calculator shows all three so you can pick the one that matches your hypothesis.

In a normal distribution about 68% of values have |z| < 1, about 95% have |z| < 2 and about 99.7% have |z| < 3. A z-score beyond ±2 is usually called unusual, and beyond ±3 it is often treated as a possible outlier. Whether a high or low z is 'good' depends on what is being measured.

Yes. A negative z-score means the value is below the mean. A z of −1.2 is 1.2 standard deviations below average and corresponds to about the 11.5th percentile. The sign shows direction and the size shows distance from the mean, so z = −1.2 and z = +1.2 are equally far from average.

Use the population standard deviation (divide by n) when your data include every member of the group you care about. Use the sample standard deviation (divide by n − 1) when the data are a sample used to estimate a larger population. The sample version is slightly larger, so z-scores computed with it are slightly smaller in size.

For a two-sided 95% confidence interval the critical value is z = ±1.96, leaving 2.5% in each tail. For 90% it is ±1.645 and for 99% it is ±2.576. A one-sided 95% bound uses z = 1.645. The Critical z mode computes these for any confidence level or significance level.

Subtract the smaller cumulative area from the larger: P(z₁ < Z < z₂) = Φ(z₂) − Φ(z₁). For example, the area between z = −1 and z = 1 is 0.8413 − 0.1587 = 0.6827. In Between mode you can enter two raw scores with a mean and standard deviation and the tool converts both for you.

Rearrange the formula: x = μ + z × σ. With mean 100, standard deviation 15 and z = 2, the raw score is 100 + 2 × 15 = 130. The z → score mode also accepts a percentile, converting it to z with the inverse normal function first.