# t-distribution table

If a **hypothesis** is to be tested with the t-test, the **t value** from the
calculated test must be compared with the critical t value. The
**critical t-value** can be read from the table below for a selected significance
level alpha. Usually the significance level alpha is 0.05. If the calculated chi-squared
value is smaller than the critical value, the null hypothesis can not be rejected.

## Probability of error

## Critical t-Value

## Table t-value

Area two tailed | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

df | 0 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 0.95 | 0.98 | 0.99 | 0.998 | 0.999 |

1 | 0 | 1 | 1.376 | 1.963 | 3.078 | 6.314 | 12.71 | 31.82 | 63.66 | 318.31 | 636.62 |

2 | 0 | 0.816 | 1.061 | 1.386 | 1.886 | 2.92 | 4.303 | 6.965 | 9.925 | 22.327 | 31.599 |

3 | 0 | 0.765 | 0.978 | 1.25 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 | 10.215 | 12.924 |

4 | 0 | 0.741 | 0.941 | 1.19 | 1.533 | 2.132 | 2.776 | 3.747 | 4.604 | 7.173 | 8.61 |

5 | 0 | 0.727 | 0.92 | 1.156 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 5.893 | 6.869 |

6 | 0 | 0.718 | 0.906 | 1.134 | 1.44 | 1.943 | 2.447 | 3.143 | 3.707 | 5.208 | 5.959 |

7 | 0 | 0.711 | 0.896 | 1.119 | 1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 4.785 | 5.408 |

8 | 0 | 0.706 | 0.889 | 1.108 | 1.397 | 1.86 | 2.306 | 2.896 | 3.355 | 4.501 | 5.041 |

9 | 0 | 0.703 | 0.883 | 1.1 | 1.383 | 1.833 | 2.262 | 2.821 | 3.25 | 4.297 | 4.781 |

10 | 0 | 0.7 | 0.879 | 1.093 | 1.372 | 1.812 | 2.228 | 2.764 | 3.169 | 4.144 | 4.587 |

11 | 0 | 0.697 | 0.876 | 1.088 | 1.363 | 1.796 | 2.201 | 2.718 | 3.106 | 4.025 | 4.437 |

12 | 0 | 0.695 | 0.873 | 1.083 | 1.356 | 1.782 | 2.179 | 2.681 | 3.055 | 3.93 | 4.318 |

13 | 0 | 0.694 | 0.87 | 1.079 | 1.35 | 1.771 | 2.16 | 2.65 | 3.012 | 3.852 | 4.221 |

14 | 0 | 0.692 | 0.868 | 1.076 | 1.345 | 1.761 | 2.145 | 2.624 | 2.977 | 3.787 | 4.14 |

15 | 0 | 0.691 | 0.866 | 1.074 | 1.341 | 1.753 | 2.131 | 2.602 | 2.947 | 3.733 | 4.073 |

16 | 0 | 0.69 | 0.865 | 1.071 | 1.337 | 1.746 | 2.12 | 2.583 | 2.921 | 3.686 | 4.015 |

17 | 0 | 0.689 | 0.863 | 1.069 | 1.333 | 1.74 | 2.11 | 2.567 | 2.898 | 3.646 | 3.965 |

18 | 0 | 0.688 | 0.862 | 1.067 | 1.33 | 1.734 | 2.101 | 2.552 | 2.878 | 3.61 | 3.922 |

19 | 0 | 0.688 | 0.861 | 1.066 | 1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.579 | 3.883 |

20 | 0 | 0.687 | 0.86 | 1.064 | 1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.552 | 3.85 |

21 | 0 | 0.686 | 0.859 | 1.063 | 1.323 | 1.721 | 2.08 | 2.518 | 2.831 | 3.527 | 3.819 |

22 | 0 | 0.686 | 0.858 | 1.061 | 1.321 | 1.717 | 2.074 | 2.508 | 2.819 | 3.505 | 3.792 |

23 | 0 | 0.685 | 0.858 | 1.06 | 1.319 | 1.714 | 2.069 | 2.5 | 2.807 | 3.485 | 3.768 |

24 | 0 | 0.685 | 0.857 | 1.059 | 1.318 | 1.711 | 2.064 | 2.492 | 2.797 | 3.467 | 3.745 |

25 | 0 | 0.684 | 0.856 | 1.058 | 1.316 | 1.708 | 2.06 | 2.485 | 2.787 | 3.45 | 3.725 |

26 | 0 | 0.684 | 0.856 | 1.058 | 1.315 | 1.706 | 2.056 | 2.479 | 2.779 | 3.435 | 3.707 |

27 | 0 | 0.684 | 0.855 | 1.057 | 1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.421 | 3.69 |

28 | 0 | 0.683 | 0.855 | 1.056 | 1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.408 | 3.674 |

29 | 0 | 0.683 | 0.854 | 1.055 | 1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.396 | 3.659 |

30 | 0 | 0.683 | 0.854 | 1.055 | 1.31 | 1.697 | 2.042 | 2.457 | 2.75 | 3.385 | 3.646 |

40 | 0 | 0.681 | 0.851 | 1.05 | 1.303 | 1.684 | 2.021 | 2.423 | 2.704 | 3.307 | 3.551 |

60 | 0 | 0.679 | 0.848 | 1.045 | 1.296 | 1.671 | 2 | 2.39 | 2.66 | 3.232 | 3.46 |

80 | 0 | 0.678 | 0.846 | 1.043 | 1.292 | 1.664 | 1.99 | 2.374 | 2.639 | 3.195 | 3.416 |

100 | 0 | 0.677 | 0.845 | 1.042 | 1.29 | 1.66 | 1.984 | 2.364 | 2.626 | 3.174 | 3.39 |

1000 | 0 | 0.675 | 0.842 | 1.037 | 1.282 | 1.646 | 1.962 | 2.33 | 2.581 | 3.098 | 3.3 |

## Calculate t-value

The t-distribution results from a combination of a random variable X with chi-squared distribution and a random variable Y with standard normal distribution to

where Y and X are independent and n is the number of degrees of freedom

**t-distribution table: ** The desired critical t-values can be read from the above
table of the t-distribution. In the case of a directional hypothesis, the area is read
off at the 1-alpha point; in the case of a non-directional hypothesis, the area is read
off at 1-alpha / 2.

This means that the critical t-value can be read from the
** t-test table ** above and can thus determine whether the null hypothesis is
rejected or not rejected. This means that a hypothesis test can be calculated with the
t-test table without statistical software.

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