Family of $t$ Tests

$t$-tests is a family of Statistical tests that use $t$-statistics

  • critical values come from the $t$-distribution - used for calculating $p$-values

$t$ Tests

The following tests are $t$-tests:


Assumptions for $t$ tests are similar to the assumptions of the $z$-tests

  • Observations are independent (if less than 10% of population is sampled, then we can make sure it's satisfied)
  • Sample size is sufficiently large so C.L.T. holds
  • Moderate skew, few outliers (not too extreme)

$t$-tests vs $z$-tests

Sample Size

  • the sample size can be smaller than for $z$-tests
  • so it can be smaller than 30 - after 30 we can safely use $z$-tests with almost the same outcomes


  • the tails are thicker than for $N(0,1)$ and observations are more likely to fall within 2$\sigma$ from the mean
  • this is exactly the correction we need to account for poorly estimated Standard Error when the sample size is not big

Alternatives to $t$-Tests