Z-Test for a Mean (Known σ)
Computes the z-test for a mean when the population standard deviation is known. It's the simplest case of hypothesis testing about a mean: the statistic measures how many standard errors separate the observed mean from the hypothesized value. When σ is known (or the sample is large), the normal distribution is used instead of the t. The tool returns the z statistic and the two-sided p-value. Enter the sample mean, the hypothesized mean, the population standard deviation and the sample size.
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Z-Test for a Mean (Known σ)
Computes the z-test for a mean when the population standard deviation is known. It's the simplest case of hypothesis testing about a mean: the statistic measures how many standard errors separate the observed mean from the hypothesized value. When σ is known (or the sample is large), the normal distribution is used instead of the t. The tool returns the z statistic and the two-sided p-value. Enter the sample mean, the hypothesized mean, the population standard deviation and the sample size.
The simplest hypothesis test about a mean
Before the t distribution entered the scene, hypothesis testing about a mean lived in a simpler world, in which the population standard deviation was treated as known. That case still shows up plenty: in quality control with well-characterized processes, or whenever the sample is large enough for the deviation estimate to be reliable. The z-test is the tool for that scenario.
The logic is to measure distance in standardized units. The z statistic counts how many standard errors separate the observed mean from the hypothesized mean. Because the population deviation is taken as known, the reference distribution is the normal, without the heavy-tail correction that the t applies. A z large in magnitude means the sample mean is too far from what was expected to be a coincidence.
Enter the sample mean, the hypothesized mean, the population standard deviation and the sample size. The tool returns the z statistic and the two-sided p-value. When the population deviation is unknown and estimated from the data themselves, the correct move is to swap this test for the t-test, especially in small samples.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.