Why are t-scores necessary?
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T-scores are used to account for the absence of population standard deviation or a small sample size.
PRO TIP: They're only used with sample means, not sample proportions!
When we don't have the population standard deviation (σ), we have to utilize our "best guess" at the population standard deviation with our sample standard deviation (s).
We have to provide room for error in case the sample standard deviation (s) is not an accurate representation of the population standard deviation (σ).
If our sample size is too small, then we can't assume that the sample distribution is a normal distribution (Central Limit Theorem). The size of the sample must be at least 30 for us to make the assumption that it's normal.
| Situation | t-score vs. z-score? | Why? |
|---|---|---|
| If you're using the sample standard deviation (s)... | t-score | Room for error in "s" being different than "σ". |
| You've got the population standard deviation (σ) but your sample size (n) is not greater than 30... | t-score | Central Limit Theorem supports our sampling distribution resembling the z-distribution once n >= 30. |
| You've got the population standard deviation (σ) and your sample is greater than 30... | z-score | We have enough evidence that our sampling distribution resembles the z-distribution. |
Strive to use z-scores... but when you can't (due to missing population standard deviation or small sample size), use t-scores!