Z-Test

What Is a Z-Test?

A z-test is a hypothesis test that asks whether an observed result is far enough from an expected value that it is unlikely to be due to chance alone. It relies on the standard normal distribution, the bell-shaped curve with a mean of zero and a standard deviation of one, to judge how extreme the result is. The test produces a z-score, which measures how many standard error units the observed result lies from the hypothesized value.

How It Works

The test starts with a null hypothesis, usually a statement that there is no difference, such as an average return of zero. The z-score is then compared against the standard normal distribution to find a p-value, the probability of seeing a result at least as extreme if the null hypothesis were true. A small p-value suggests the observed difference is unlikely to be random noise.

For a single sample mean with a known population standard deviation, the formula is:

z = (sample mean - hypothesized mean) / (population standard deviation / √n)

Here n is the sample size. For comparing two proportions, a similar calculation uses a pooled standard error in the denominator.

Limitations and Why It Matters

Z-tests assume the sample is large enough for the results to be approximately normal, and that the population standard deviation is known or well estimated. When the standard deviation must be estimated from a small sample, a t-test is usually more appropriate. Investors encounter the logic of z-tests when reading research that claims a strategy beat the market, since the question is whether the result could plausibly be chance. Findings are often reported alongside a confidence interval for the same reason, showing a range of plausible values rather than a single verdict.