Z-Score Calculator

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Calculate Z-ScoreFind Value from Z-Score

Value (x)

Mean (μ or x̄)

Standard Deviation (σ or s)

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Standardize using a stated reference

To calculate z, enter a raw value x, reference mean μ and positive reference SD σ. The formula is z=(x−μ)/σ. The reverse mode calculates x=μ+zσ from a supplied z-score. Use values and reference summaries measured in the same units.

A z-score expresses distance from the supplied mean in SD units. Subtracting a mean and dividing by SD does not change a skewed distribution into a normal one. The displayed percentile is an approximate normal-model cumulative probability, not an empirical rank calculated from your data.

Reproduce the worked example

Select Load worked example in the raw-value mode. With x=85, μ=75 and σ=5:

$z=\frac{85-75}{5}=\mathbf2.$

The value is two SDs above the mean. Under a suitable normal model, P(Z≤2)≈0.9772, or the 97.72nd percentile, and P(Z>2)≈0.0228. Reverse mode with z=2 and the same reference recovers x=85.

These reference parameters are hypothetical. The same standardized distance can correspond to a different tail probability in a non-normal population.

What the output does not establish

The visual marker shows a location on a standardized scale; it is not a measured distribution. Values outside its displayed range are pinned at the edge. A z value alone does not prove an observation is an error or justify deleting it.

Do not choose a z-test merely because sample size exceeds 30. Inference about a mean with population SD estimated from the sample generally uses t. This calculator standardizes a value; it does not run a hypothesis test or estimate a proportion interval.

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Read z-scores, compare the normal reference, or check the normal model lesson.