📐 Z-Test Calculator
Enter your sample mean, known population σ and size to run a z-test, with the z statistic, exact p-value and whether the result is significant.
z statistic
1.667
Standard error
3
p-value
0.0956
Not significant at α = 0.05
p = 0.0956 ≥ 0.05, so you fail to reject the null hypothesis, not enough evidence of a real difference at this level.
The z-test assumes the population standard deviation (σ) is known and uses the standard normal distribution. Use it when σ is known or the sample is large; when σ is estimated from the sample, prefer the t-test instead. p-values use an exact normal CDF. 🔒 Computed in your browser.
How the z-test calculator works
The z-test compares a sample mean to a hypothesized value (one-sample) or two sample means to each other (two-sample) when the population standard deviation σ is known. The tool computes the standard error, the z statistic (how many standard errors the observed difference is from zero), and the p-value from the exact standard-normal distribution for a one- or two-tailed test, then compares it to your chosen significance level α.
The z-test is the right test when σ is genuinely known or the sample is large enough that the sample SD is a reliable stand-in, otherwise the t-test, which accounts for the extra uncertainty of estimating σ, is the correct choice. The two tests converge as the sample grows. A non-significant result is not proof of “no difference”; it only means the evidence is insufficient at your α. Everything runs locally in your browser.
Frequently asked questions
When should I use a z-test instead of a t-test?
Use a z-test when the population standard deviation (σ) is known, or the sample is large (often n ≥ 30) so the sample SD reliably estimates σ. Use a t-test when σ is unknown and estimated from a smaller sample. It accounts for that extra uncertainty.
What does the z statistic mean?
It’s how many standard errors the observed difference lies from the value expected under the null hypothesis. A larger absolute z means the result is further from “no effect,” which corresponds to a smaller p-value.
What is a one-tailed vs two-tailed test?
A two-tailed test checks for any difference (≠); a one-tailed test checks for a difference in a specific direction (> or <). One-tailed tests have more power in that direction but must be chosen before seeing the data.
Does a significant result prove my hypothesis?
No. It means the data are unlikely under the null hypothesis at your chosen α, so you reject the null. It doesn’t prove the alternative or measure the effect’s size; pair it with an effect-size measure like Cohen’s d.
Is my data uploaded?
No, the z statistic and p-value are computed entirely in your browser and the tool works offline.
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