# When can we use the z-score? (Since it's impossible to know the population SD)

Sometimes I see papers that report the z-score. I am confused because the z-score requires us to know the population standard deviation, which is impossible. What are some reasons people still report the z-score?? Can you give an example please? (The only thing I can think of is when the sample size is large enough so the t-distribution converges to a normal distribution. Also, if the sample is large enough, the sample SD will approximately converge to the population SD. However, even in that situation, shouldn't the researcher report the t-score just to be more sure and conservative?)

Thank you!

• If the purpose is descriptive, everything is fine, because the relevant standard deviation is the empirical SD of your data.
– whuber
Commented Feb 13, 2023 at 21:56

Remember that the T-statistic also requires your data to be normal, which is usually not true either. We end up using Z scores in large samples because of the central limit theorem, one version of which tells us that for arbitrary random variables $$\{X_i\}_{i=1}^n$$ with finite variance and the same unknown distribution $$F$$, the test statistic becomes normally distributed i.e. $$\sqrt{n}(\bar{X}_n - \mu_0)/s \stackrel{d}{\to} \mathcal{N}(0,1).$$ By abusing asymptotics, we basically end up using critical values like 1.96 (which is a z-critical value at the 5% level!) to reject the null that $$\mathbb{E}[X] = \mu_0$$.