Hypothesis Testing > Testing a Single Mean

When you test a single mean, you’re comparing the mean value to some other hypothesized value. Which test you run depends on if you know the population standard deviation(σ) or not.

## Known population standard deviation

If you know the value for σ, then the population mean has a normal distribution: use a one sample z-test. The z-test uses a formula to find a z-score, which you compare against a critical value found in a z-table. The formula is:

Unknown population standard deviation

If you don’t know the population standard deviation, use the t-test. The t-score formula is almost identical to the z-score formula, except that σ (the population standard deviation) has been replaced by s (the sample standard deviation). The formula is:

.

The test is run the same way: use the formula to calculate your t-score and then compare it to a value found in a table (this time you’ll use the t-table).

**See**: One Sample T-Test example.

## Non Parametric Tests for Testing a Single Mean

Non parametric (“distribution free”) tests don’t assume your data comes from a certain distribution, like the normal distribution. So if you have data that isn’t normally distributed, you should use one of these alternatives:

- One sample Wilcoxon test (assumes your data comes from a symmetric distribution).
- One sample sign test (has no assumption about the shape of the distribution).

Both tests use medians instead of means. You don’t want to compare means for non-normally distributed data because the mean is very affected by outliers and skewness. As you don’t know the shape of the potential distribution, running a test for a mean would give you a very high probability of your results being wrong. The median on the other hand, is resistant to outliers and changes in skew.

## References

Hahs-Vaughn, D. (2020). An Introduction to Statistical Concepts 4th Edition. Routledge.