# Null Hypothesis Testing with Pearson's r

**When to Use**

This is straightforward. Do this when you have a sample value of Pearson’s r and you want to test the null hypothesis that the value of r in the population is zero. For example, imagine that each of 50 subjects judges the intelligence level of a stimulus person on a 1 to 7 scale and also rates their mood on a 0 to 10 scale. Imagine also that for this sample of 50 subjects, Pearson’s r is +.23. We want to test the null hypothesis that Pearson’s r in the population is actually 0 and that this sample correlation of .23 represents nothing more than sampling error.

**The Test Statistic**

The test statistic in this case is t. However, this statistic is not usually computed or reported. Instead, we can proceed straight to a table (see below).

**The p Value**

p is the probability of a value of Pearson’s r as extreme as the one you got if the null hypothesis were true. Again, you do not have to figure out the exact p value. Instead, you can just figure out whether p is lower than your α level (.05). The table at right lists these __critical r values__ for α levels of .10, .05, .02, and .01. Note that it also lists different critical values depending on the degrees of freedom, which is the number of cases minus two (df = N – 2). Again, notice from the table that as the sample size gets smaller, the r value needed to reject the null hypothesis gets larger.

**The Decision**

You decide to reject the null hypothesis if your sample r value is more extreme than the critical r value. This means that the probability of getting an r value at least as extreme as yours, if the null hypothesis were true, is less than 5% (or whatever α is). So you decide that the null hypothesis is *not* true. You fail to reject the null hypothesis if your sample r value is less extreme than the critical t value. This means that the probability of getting an r value at least as extreme as yours, if the null hypothesis were true, is greater than 5%. So you decide that the null hypothesis *could* be true.

** Example**

A group of 50 subjects rates the intelligence of a stimulus person and rates their moods. Pearson’s r for the sample is +.23. I can look in the table for 48 degrees of freedom. It is not there, but 50 is pretty close. For an α level of .05, the critical value is .27. So I fail to reject the null hypothesis; the correlation is not statistically significant. In other words, I cannot conclude that my sample correlation reflects anything other than sampling error.

** Expressing the Result in Writing**

Here is how such a result might be expressed in APA style: “The correlation between intelligence ratings and mood was not statistically significant, *r*(48) = .23, *p* > .05.

TABLE: CRITICAL VALUES OF *r*

df (n-2) |
4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

Crit. r |
.811 | .754 | .707 | .666 | .632 | .602 | .576 | .553 | .532 | .514 | .497 | .482 | .468 |

df (n-2) |
17 | 18 | 19 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | |

Crit. r |
.456 | .444 | .433 | .423 | .349 | .304 | .273 | .250 | .232 | .217 | .205 | .195 |

Click the link below for a complete table of critical values of r: http://www.gifted.uconn.edu/siegle/research/Correlation/corrchrt.htm