df | One-tail probability (P) | Two-tail probability (P) | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.10 | 0.05 | 0.025 | 0.01 | 0.005 | 0.001 | 0.0005 | 0.20 | 0.10 | 0.05 | 0.02 | 0.01 | 0.005 | 0.002 | 0.0005 | |

1 | 3.078 | 6.314 | 12.706 | 31.821 | 63.657 | 318.309 | 636.619 | 1.376 | 3.078 | 6.314 | 12.706 | 31.821 | 63.657 | 318.309 | 636.619 |

2 | 1.886 | 2.920 | 4.303 | 6.965 | 9.925 | 22.327 | 31.599 | 1.061 | 1.886 | 2.920 | 4.303 | 6.965 | 9.925 | 22.327 | 31.599 |

3 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 | 10.214 | 12.924 | 0.978 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 | 10.214 | 12.924 |

4 | 1.533 | 2.132 | 2.776 | 3.747 | 4.604 | 7.173 | 8.610 | 0.941 | 1.533 | 2.132 | 2.776 | 3.747 | 4.604 | 7.173 | 8.610 |

5 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 5.893 | 6.869 | 0.920 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 5.893 | 6.869 |

6 | 1.440 | 1.943 | 2.447 | 3.143 | 3.707 | 5.208 | 5.959 | 0.906 | 1.440 | 1.943 | 2.447 | 3.143 | 3.707 | 5.208 | 5.959 |

7 | 1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 4.785 | 5.408 | 0.896 | 1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 4.785 | 5.408 |

8 | 1.397 | 1.860 | 2.306 | 2.896 | 3.355 | 4.501 | 5.041 | 0.889 | 1.397 | 1.860 | 2.306 | 2.896 | 3.355 | 4.501 | 5.041 |

9 | 1.383 | 1.833 | 2.262 | 2.821 | 3.250 | 4.297 | 4.781 | 0.883 | 1.383 | 1.833 | 2.262 | 2.821 | 3.250 | 4.297 | 4.781 |

10 | 1.372 | 1.812 | 2.228 | 2.764 | 3.169 | 4.144 | 4.587 | 0.879 | 1.372 | 1.812 | 2.228 | 2.764 | 3.169 | 4.144 | 4.587 |

11 | 1.363 | 1.796 | 2.201 | 2.718 | 3.106 | 4.025 | 4.437 | 0.876 | 1.363 | 1.796 | 2.201 | 2.718 | 3.106 | 4.025 | 4.437 |

12 | 1.356 | 1.782 | 2.179 | 2.681 | 3.055 | 3.930 | 4.318 | 0.873 | 1.356 | 1.782 | 2.179 | 2.681 | 3.055 | 3.930 | 4.318 |

13 | 1.350 | 1.771 | 2.160 | 2.650 | 3.012 | 3.852 | 4.221 | 0.870 | 1.350 | 1.771 | 2.160 | 2.650 | 3.012 | 3.852 | 4.221 |

14 | 1.345 | 1.761 | 2.145 | 2.624 | 2.977 | 3.787 | 4.140 | 0.868 | 1.345 | 1.761 | 2.145 | 2.624 | 2.977 | 3.787 | 4.140 |

15 | 1.341 | 1.753 | 2.131 | 2.602 | 2.947 | 3.733 | 4.073 | 0.866 | 1.341 | 1.753 | 2.131 | 2.602 | 2.947 | 3.733 | 4.073 |

16 | 1.337 | 1.746 | 2.120 | 2.583 | 2.921 | 3.686 | 4.015 | 0.865 | 1.337 | 1.746 | 2.120 | 2.583 | 2.921 | 3.686 | 4.015 |

17 | 1.333 | 1.740 | 2.110 | 2.567 | 2.898 | 3.646 | 3.965 | 0.863 | 1.333 | 1.740 | 2.110 | 2.567 | 2.898 | 3.646 | 3.965 |

18 | 1.330 | 1.734 | 2.101 | 2.552 | 2.878 | 3.610 | 3.922 | 0.862 | 1.330 | 1.734 | 2.101 | 2.552 | 2.878 | 3.610 | 3.922 |

19 | 1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.579 | 3.883 | 0.861 | 1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.579 | 3.883 |

20 | 1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.552 | 3.850 | 0.860 | 1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.552 | 3.850 |

21 | 1.323 | 1.721 | 2.080 | 2.518 | 2.831 | 3.527 | 3.819 | 0.859 | 1.323 | 1.721 | 2.080 | 2.518 | 2.831 | 3.527 | 3.819 |

22 | 1.321 | 1.717 | 2.074 | 2.508 | 2.819 | 3.505 | 3.792 | 0.858 | 1.321 | 1.717 | 2.074 | 2.508 | 2.819 | 3.505 | 3.792 |

23 | 1.319 | 1.714 | 2.069 | 2.500 | 2.807 | 3.485 | 3.767 | 0.858 | 1.319 | 1.714 | 2.069 | 2.500 | 2.807 | 3.485 | 3.767 |

24 | 1.318 | 1.711 | 2.064 | 2.492 | 2.797 | 3.467 | 3.743 | 0.857 | 1.318 | 1.711 | 2.064 | 2.492 | 2.797 | 3.467 | 3.743 |

25 | 1.316 | 1.708 | 2.060 | 2.485 | 2.787 | 3.450 | 3.722 | 0.856 | 1.316 | 1.708 | 2.060 | 2.485 | 2.787 | 3.450 | 3.722 |

26 | 1.315 | 1.706 | 2.056 | 2.479 | 2.779 | 3.435 | 3.700 | 0.856 | 1.315 | 1.706 | 2.056 | 2.479 | 2.779 | 3.435 | 3.700 |

27 | 1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.421 | 3.682 | 0.856 | 1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.421 | 3.682 |

28 | 1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.408 | 3.665 | 0.855 | 1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.408 | 3.665 |

29 | 1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.396 | 3.649 | 0.855 | 1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.396 | 3.649 |

30 | 1.310 | 1.697 | 2.042 | 2.457 | 2.750 | 3.385 | 3.646 | 0.855 | 1.310 | 1.697 | 2.042 | 2.457 | 2.750 | 3.385 | 3.646 |

60 | 1.296 | 1.671 | 2.000 | 2.390 | 2.660 | 3.145 | 3.460 | 0.889 | 1.296 | 1.671 | 2.000 | 2.390 | 2.660 | 3.145 | 3.460 |

120 | 1.289 | 1.658 | 1.980 | 2.358 | 2.617 | 3.055 | 3.365 | 0.884 | 1.289 | 1.658 | 1.980 | 2.358 | 2.617 | 3.055 | 3.365 |

1000 | 1.282 | 1.646 | 1.962 | 2.330 | 2.581 | 2.935 | 3.136 | 0.879 | 1.282 | 1.646 | 1.962 | 2.330 | 2.581 | 2.935 | 3.136 |

∞ | 1.282 | 1.645 | 1.960 | 2.326 | 2.576 | 2.937 | 3.090 | 0.879 | 1.282 | 1.645 | 1.960 | 2.326 | 2.576 | 2.937 | 3.090 |

## How to Use the t-Distribution Critical Values Table

The t-distribution critical values table is used in hypothesis testing when the sample size is small and the population standard deviation is unknown. It helps you find the critical value to compare your t-statistic against to determine whether to reject the null hypothesis.

### Steps to Use the Table

**Step 1:**Determine the**degrees of freedom (df)**for your test. Typically, the degrees of freedom is the sample size minus one (*n - 1*).**Step 2:**Identify whether your test is**one-tailed**or**two-tailed**. One-tailed tests assess the direction of the effect (greater or less than), while two-tailed tests check for any significant difference in either direction.**Step 3:**Choose your desired**significance level (P)**. Common levels of significance are 0.10, 0.05, and 0.01 for one-tailed tests, and 0.20, 0.10, 0.05, and 0.01 for two-tailed tests.**Step 4:**Locate the appropriate row for your degrees of freedom and find the intersection with the column representing your significance level. The resulting value is your critical value.

### Example: Two-Tailed Test with df = 10 and P = 0.05

For a two-tailed test with 10 degrees of freedom and a significance level of P = 0.05, look for the row labeled "10" and the column labeled "0.05" in the two-tailed section. The critical value is approximately 2.228.

### Interpreting the Critical Value

The critical value represents the threshold beyond which your t-statistic must lie for you to reject the null hypothesis. If your computed t-statistic exceeds this critical value, you have enough evidence to reject the null hypothesis at the chosen significance level.

### Further Reading

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.