Z Test Calculator

Work out z test instantly with clear inputs, formula shown and shareable results.

About Z Test Calculator

The z-test applies when the population standard deviation is known, so the standard error σ/√n needs no estimation. The statistic is compared against the standard normal distribution.

Formula

One-sample z

z = (x̄ − μ₀) / (σ/√n)

Worked example

With the default values below, Z Test Calculator returns z statistic of 1.86666667. Change any input above and every figure here is recalculated in your browser.

Inputs used

Sample mean: 103.5
Hypothesised mean: 100
Known population standard deviationminimum 0: 15
Sample sizeminimum 1: 64

Results

z statistic: 1.86666667
Two-tailed p value: 0.06194803
Standard error: 1.87500000
Decision at 5%: Fail to reject the null hypothesis at the 5% level

Step by step

1Values used
2One-sample z
3z statistic= 1.86666667
4Two-tailed p value= 0.06194803
5Standard error= 1.87500000

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For x̄ = 103.5 against 100 with σ = 15 and n = 64?

The standard error is 1.875, so z = 1.8667 and the two-tailed p is about 0.0619 — not significant at 5%.

When is a z-test appropriate?

Rarely in practice, since σ is seldom known. It is also a good approximation to the t-test for large samples.