# Pearson R and range of data

I have the following two lists:

x = [7,8, 9, 9, 9, 9, 10, 10, 10]

y = [9, 9, 10, 9, 8, 6, 10, 9, 8]

the Pearson r value is equal to 0 in this case.

However, the thing is, these numbers are from two parameters where we ask participants to select values for each parameter from a range of 0 to 10, and it just happens that all the participants rated high values.

If I add (0,0) as an extra point as follows to make it aware of the breadth of the range:

x = [7,8, 9, 9, 9, 9, 10, 10, 10, 0]

y = [9, 9, 10, 9, 8, 6, 10, 9, 8, 0]

the r value changes to 0.87.

I do know however that I cannot add (0,0) so how do I let it know that the range is from 0 to 10?

Adding more context to the question: The correlation was calculated for a test-retest reliability. I have the same set of users test a tool twice, and each time they would answer the same question. That's why their questions are close. So for example, person 1 answered 7/10 the first time and 9/10 the second. The 2nd person answered 8/9 first time and 9/10 the second time. Those seem to be highly correlated answers to me, they are consistingly high. What I don't understand is why the Pearson R is 0 here. I feel I'm doing something wrong.

• Why do you want to do that? // If you want to add $(0,0)$, you’re assuming that someone rating $0$ on the first would rate $0$ on the second. Given the lack of correlation, you don’t know if that point should be $(0,0)$, $(0,10)$, or $(10,0)$ (I also could see $(10,10)$ as being a related point). What happens to your correlation if you add all four of those points to your data?
– Dave
Feb 3, 2022 at 11:53

Why do you want to "let it know that the range is from 0 to 10"? This of course depends on why you calculated the Pearson $$r$$ as you did. If you are interested in the association between $$x$$ and $$y$$, I would recommend you also look at the corresponding confidence interval for $$r$$ (for example in R with the cor.test method). For the data you provided this confidence interval is $$[-.66, .66]$$. That is a lot of uncertainty!
• I suggest you inspect the data visually (e.g. a scatter plot of x vs. y). Then you should see why the correlation is $0$. As I said in the answer above I would not recommend to look at the point estimate of $r$ alone, but always consider the confidence interval (which is very big here), so you shouldn't take $0$ at face value. Feb 3, 2022 at 12:16