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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.

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  • $\begingroup$ 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? $\endgroup$
    – Dave
    Feb 3, 2022 at 11:53

2 Answers 2

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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!

Your data are consistent with both large positive and large negative correlations, so my advice would be to obtain more data!

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  • $\begingroup$ The correlation was calculated for a test-retest reliability ( statology.org/test-retest-reliability ). So I have the same set of users answer the same question regarding an experience that they have twice. 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. $\endgroup$
    – BKS
    Feb 3, 2022 at 12:12
  • $\begingroup$ 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. $\endgroup$
    – Stefan
    Feb 3, 2022 at 12:16
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Reading your edit, it looks like people are rating higher the second time and lower the second time in about equal amounts with about equal magnitudes, giving you little insight into a relationship between the two.

You say that all values are fairly high, but that is a property of the marginal distributions, not the relationship between the marginal distributions. Correlation has to do with the relationship between the marginal distributions. You know to expect a high value the second time because the second rating is pretty much always high.

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