# 8.3.1: Table of Critical t-scores

- Page ID
- 22081

## Student's \(t\)** Distribution**

Table \(\PageIndex{1}\), which follows Figure \(\PageIndex{1}\), shows the critical t-score. If the absolute value of your calculated t-score is bigger (more extreme) than the critical t-score, then you reject the null hypothesis. In Figure \(\PageIndex{1}\), the critical t-score is represented by the line; if the absolute value of the calculated t-score is to the left (where the alpha sign is, \(α\) ), then the null hypothesis should be rejected.

Remember

**Critical < |Calculated| = Reject null = means are different = p<.05**

**Critical > |Calculated| = Retain null = means are similar = p>.05**

## Table of Critical Values for **Student's **\(t\)

Table \(\PageIndex{1}\) shows the critical t-scores for different probabilities (p-values) that represent how likely it would be to get a calculated t-scores this big if the sample was really from the population, by the Degrees of Freedom (df, to represent the size of the sample). More information Degrees of Freedom is below the table. If we think that the sample is not from the population, we would expect a larger t-score and want a small p-value.

Degrees of Freedom (df) | p = 0.10 | p = 0.05 | p = 0.025 | p = 0.01 |
---|---|---|---|---|

1 |
3.078 | 6.314 |
12.706 | 31.821 |

2 |
1.886 | 2.920 |
4.303 | 6.965 |

3 |
1.638 | 2.353 |
3.182 | 4.541 |

4 |
1.533 | 2.132 |
2.776 | 3.747 |

5 |
1.476 | 2.015 |
2.571 | 3.365 |

6 |
1.440 | 1.943 |
2.447 | 3.143 |

7 |
1.415 | 1.895 |
2.365 | 2.998 |

8 |
1.397 | 1.860 |
2.306 | 2.896 |

9 |
1.383 | 1.833 |
2.262 | 2.821 |

10 |
1.372 | 1.812 |
2.228 | 2.764 |

11 |
1.363 | 1.796 |
2.201 | 2.718 |

12 |
1.356 | 1.782 |
2.179 | 2.681 |

13 |
1.350 | 1.771 |
2.160 | 2.650 |

14 |
1.345 | 1.761 |
2.145 | 2.624 |

15 |
1.341 | 1.753 |
2.131 | 2.602 |

16 |
1.337 | 1.746 |
2.120 | 2.583 |

17 |
1.333 | 1.740 |
2.110 | 2.567 |

18 |
1.330 | 1.734 |
2.101 | 2.552 |

19 |
1.328 | 1.729 |
2.093 | 2.539 |

20 |
1.325 | 1.725 |
2.086 | 2.528 |

21 |
1.323 | 1.721 |
2.080 | 2.518 |

22 |
1.321 | 1.717 |
2.074 | 2.508 |

23 |
1.319 | 1.714 |
2.069 | 2.500 |

24 |
1.318 | 1.711 |
2.064 | 2.492 |

25 |
1.316 | 1.708 |
2.060 | 2.485 |

26 |
1.315 | 1.706* |
2.056 | 2.479 |

27 |
1.314 | 1.703 |
2.052 | 2.473 |

28 |
1.313 | 1.701 |
2.048 | 2.467 |

29 |
1.311 | 1.699 |
2.045 | 2.462 |

30 |
1.310 | 1.697 |
2.042 | 2.457 |

40 |
1.303 | 1.684 |
2.021 | 2.423 |

60 |
1.296 | 1.671 |
2.000 | 2.390 |

100 |
1.290 | 1.660 |
1.984 | 2.364 |

\(\infty\) |
1.282 | 1.645 |
1.960 | 2.326 |

### Degrees of Freedom

- One-Sample t-test: N-1
- Independent Sample t-test: N
_{1}+ N_{2}- 2 - Dependent Sample t-test: N-1 (in which N is the
*number of pairs*)

Because tables are limited by size, not all critical t-scores are listed. There are a couple of options when your Degrees of Freedom is not listed on the table.

- One option is to use the Degrees of Freedom that is
*closest*to your sample's Degrees of Freedom. For example, if your df = 49. you would use the df row for 40, (so a p=0.05 would have a critical value of 1.684). If your df=55, you would use the df row for 60. It's sorta silly, but, mathematically, any score is closer to df=100 than infinity (**\(\infty\)**), so if your sample is more than 100 scores, use df=100. - Another option is to always we round down. For our example of df=49, we would still use the df row for 40. If your df=55, you would still use the df row for 40. And if your sample is more than 100 scores, use df=100. This option avoids inflating Type I Error (false positives).

Ask your professor which option you should use!

## Contributors and Attributions

Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/30189442-699...b91b9de@18.114.