# 14.7.1: Table of Critical Values of r

- Page ID
- 22153

When using this table, we're following the general pattern of rejecting the null hypothesis when the critical value is larger than the calculated value. The only difference from what we've been doing is the rejecting or retaining the null hypothesis is not about whether group means are similar or not.

**(Critical \(<\) Calculated) \(=\) Reject null \(=\) There is a linear relationship. \(= p<.05 \)**

**(Critical \(>\) Calculated) \(=\) Retain null \(=\) There is not a linear relationship. \(= p>.05\)**

**Table of Critical Values of r**

Table \(\PageIndex{1}\) is a simplified and accessible version of the table in Real Statistics Using Excel by Dr. Charles Zaiontz. Table \(\PageIndex{1}\) shows the critical scores of Pearson's r for different probabilities (p-values) that represent how likely it would be to get a calculated correlation this extreme if the two variables were unrelated in the population, by the Degrees of Freedom (df), to represent the size of the sample). For Pearson's r, the Degrees of Freedom are N-2.

Degrees of Freedom (df) | p = 0.1 | p = 0.05 | p = 0.01 |
---|---|---|---|

1 | 0.988 | 0.997 | 1.000 |

2 | 0.900 | 0.950 | 0.990 |

3 | 0.805 | 0.878 | 0.959 |

4 | 0.729 | 0.811 | 0.917 |

5 | 0.669 | 0.754 | 0.875 |

6 | 0.621 | 0.707 | 0.834 |

7 | 0.582 | 0.666 | 0.798 |

8 | 0.549 | 0.632 | 0.765 |

9 | 0.521 | 0.602 | 0.735 |

10 | 0.497 | 0.576 | 0.708 |

11 | 0.476 | 0.553 | 0.684 |

12 | 0.458 | 0.532 | 0.661 |

13 | 0.441 | 0.514 | 0.641 |

14 | 0.426 | 0.497 | 0.623 |

15 | 0.412 | 0.482 | 0.606 |

16 | 0.400 | 0.468 | 0.590 |

17 | 0.389 | 0.456 | 0.575 |

18 | 0.378 | 0.444 | 0.561 |

19 | 0.369 | 0.433 | 0.549 |

20 | 0.360 | 0.423 | 0.537 |

21 | 0.352 | 0.413 | 0.526 |

22 | 0.344 | 0.404 | 0.515 |

23 | 0.337 | 0.396 | 0.505 |

24 | 0.330 | 0.388 | 0.496 |

25 | 0.323 | 0.381 | 0.487 |

26 | 0.317 | 0.374 | 0.479 |

27 | 0.311 | 0.367 | 0.471 |

28 | 0.306 | 0.361 | 0.463 |

29 | 0.301 | 0.355 | 0.456 |

30 | 0.296 | 0.349 | 0.449 |

35 | 0.275 | 0.325 | 0.418 |

40 | 0.257 | 0.304 | 0.393 |

45 | 0.243 | 0.288 | 0.372 |

50 | 0.231 | 0.273 | 0.354 |

60 | 0.211 | 0.250 | 0.325 |

70 | 0.195 | 0.232 | 0.302 |

80 | 0.183 | 0.217 | 0.283 |

90 | 0.173 | 0.205 | 0.267 |

100 | 0.164 | 0.195 | 0.254 |

150 | 0.134 | 0.159 | 0.208 |

200 | 0.116 | 0.138 | 0.181 |

250 | 0.104 | 0.124 | 0.162 |

300 | 0.095 | 0.113 | 0.148 |

400 | 0.082 | 0.098 | 0.128 |

500 | 0.073 | 0.088 | 0.115 |

700 | 0.062 | 0.074 | 0.097 |

1000 | 0.052 | 0.062 | 0.081 |

5000 | 0.023 | 0.028 | 0.036 |

Because tables are limited by size, not all critical values are listed. For example, if you had 100 participants, your Degrees of Freedom would be 98 (df=N-2=100-2=98=100). However, the table provides df=90 or df=100. 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
to your sample's Degrees of Freedom. For our example of r (98), that would mean that we would use the Degrees of Freedom of 100 because 98 is closer to 100 than to 90. That would mean that the critical r-value for r(98) would be 0.194604 for a p-value of 0.05.*closest* - Another option is to always we round down. For our example of N=100), we use the Degrees of Freedom of 90 because it is the next lowest df listed. That would mean that the critical r-value for r(98) would be 0.204968 for a p-value of 0.05. This option avoids inflating Type I Error (false positives).

Ask your professor which option you should use!

Whichever option you choose, your statistical sentence should include the actual degrees of freedom , regardless of which number is listed in the table; the table is used to decide if the null hypothesis should be rejected or retained.

## Contributors and Attributions

- Real Statistics Using Excel by Dr. Charles Zaiontz