## Critical values for correlation coefficient

In the realm of statistics, the correlation coefficient is a powerful tool for assessing relationships between variables. It quantifies the strength and direction of these relationships, but how can you determine if a correlation is statistically significant or simply the result of chance? This is where critical values for the correlation coefficient come into play. In this article, we’ll dive into the significance of critical values and how they help us interpret the correlations found in data.

**What is the Correlation Coefficient?**

The correlation coefficient, denoted as “r,” is a numerical measure that quantifies the degree to which two variables are linearly related. It can range from -1 to 1, with -1 indicating a perfect negative linear relationship, 1 indicating a perfect positive linear relationship, and 0 indicating no linear relationship.

**The Importance of Critical Values**

When we calculate a correlation coefficient, we obtain a sample value (r) that estimates the population correlation (ρ). However, in statistics, we often need to determine whether this sample correlation is significantly different from zero, suggesting a genuine relationship between variables rather than a random occurrence.

**Critical Values Explained**

Critical values are thresholds that help us decide whether a correlation coefficient is statistically significant. These values are associated with a chosen level of significance, typically denoted as “α.” The most commonly used levels of significance are 0.05 (5%) and 0.01 (1%).

**Hypothesis Testing**

To determine if a correlation is statistically significant, we use hypothesis testing. The null hypothesis (H0) states that the population correlation is equal to zero (no relationship), while the alternative hypothesis (H1) asserts that the population correlation is not zero (a relationship exists).

- If the absolute value of the calculated sample correlation (|r|) is greater than the critical value, we reject the null hypothesis. This suggests that there is a statistically significant correlation between the variables.
- If |r| is less than the critical value, we fail to reject the null hypothesis, indicating that the correlation is not statistically significant.

## Pearson’s Correlation Table

df = n – 2 n = # of pairs of data |
Level of significance for two-tailed test | |||

.10 | .05 | .02 | .01 | |

1 | .988 | .997 | .9995 | .9999 |

2 | .900 | .950 | .980 | .990 |

3 | .805 | .878 | .934 | .959 |

4 | .729 | .811 | .882 | .917 |

5 | .669 | .754 | .833 | .874 |

6 | .622 | .707 | .789 | .834 |

7 | .582 | .666 | .750 | .798 |

8 | .549 | .632 | .716 | .765 |

9 | .521 | .602 | .685 | .735 |

10 | .497 | .576 | .658 | .708 |

11 | .476 | .553 | .634 | .684 |

12 | .458 | .532 | .612 | .661 |

13 | .441 | .514 | .592 | .641 |

14 | .426 | .497 | .574 | .628 |

15 | .412 | .482 | .558 | .606 |

16 | .400 | .468 | .542 | .590 |

17 | .389 | .456 | .528 | .575 |

18 | .378 | .444 | .516 | .561 |

19 | .369 | .433 | .503 | .549 |

20 | .360 | .423 | .492 | .537 |

21 | .352 | .413 | .482 | .526 |

22 | .344 | .404 | .472 | .515 |

23 | .337 | .396 | .462 | .505 |

24 | .330 | .388 | .453 | .495 |

25 | .323 | .381 | .445 | .487 |

26 | .317 | .374 | .437 | .479 |

27 | .311 | .367 | .430 | .471 |

28 | .306 | .361 | .423 | .463 |

29 | .301 | .355 | .416 | .456 |

30 | .296 | .349 | .409 | .449 |

35 | .275 | .325 | .381 | .418 |

40 | .257 | .304 | .358 | .393 |

45 | .243 | .288 | .338 | .372 |

50 | .231 | .273 | .322 | .354 |

60 | .211 | .250 | .295 | .325 |

70 | .195 | .232 | .274 | .302 |

80 | .183 | .217 | .256 | .284 |

90 | .173 | .205 | .242 | .267 |

100 | .164 | .195 | .230 | .254 |