How could we attribute labels for correlation coefficients in order to facilitate reading the data specially for non-technical people or in qualitative analyses?

For example:

  • $\rho > 0.9$ - strongly correlated
  • $\rho > 0.7$ - moderately correlated
  • $\rho > 0.4$ - weakly correlated
  • etc.
  • $\rho < -0.9$ - strongly uncorrelated
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    $\begingroup$ I am not sure if such labelling would be universally acceptable. For instance, in the social sciences people might jump from joy when they get a correlation of 0.5. $\endgroup$ – JohnK Jan 14 '16 at 12:41
  • $\begingroup$ The whole point is that it is a quantitative scale.... @JohnK is essentially right: your best guidance is what is considered striking in your field. There are fields in which correlations of 0.9 signal utterly inadequate laboratory technique. $\endgroup$ – Nick Cox Jan 14 '16 at 12:50
  • $\begingroup$ You might find this cartoon helpful! xkcd.com/1478 $\endgroup$ – Adrian Esterman Jan 15 '16 at 1:53
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    $\begingroup$ @AdrianEsterman That's about p-values rather than correlations... and is trying to make a funny point about significance-hunting. $\endgroup$ – Glen_b Jan 15 '16 at 9:30

It's really hard to say. The problem is that what might be considered a strong correlation varies across disciplines.

For example: I ask two people to measure the height of a group of children and correlate them. If I find a correlation of 0.7 I'm going to consider that very weak, and think that something has gone wrong somewhere.

Similarly, if I give those same children two different tests of cognitive ability, and find a correlation of 0.7, that would seem to be about the ballpark I'd expect.

If I try to correlate that cognitive ability with some sort of measure of behavior, if I found a correlation of 0.7 I'll assume I've put the wrong variables in my model. If I get 0.3 I'll be pretty excited.

In the social sciences Jacob Cohen suggested a rule of thumb which has become codified into some sort of law (and is often misunderstood).

  • A correlation of 0.1 is small.
  • A correlation of 0.3 is medium.
  • A correlation of 0.5 is large.

But these are very general guidelines that are often (in my opinion) overinterpreted and/or misunderstood. (I've seen submitted articles where the authors have written [something like] "The correlation was 0.48 so it was medium, as it was below 0.5". The wikipedia page has more: https://en.wikipedia.org/wiki/Effect_size

Some interesting correlations are also presented on this page: http://web.sonoma.edu/users/s/smithh/psysurvey490/toc/effectsize.pdf For example, the correlation between ever having smoked and having lung cancer is around 0.1. A small correlation, but given the seriousness of cancer, you might not want to say that.

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In general, we can say that correlation can be positive, negative, or zero. Also, correlation may be strong, moderate, or weak. The range for correlation coefficient ‘R’ is from -1 to +1.

A value close (How much close, will be dependent on the problem) to +/- 1 will represent strong (positive/negative) correlation and a value close (How much close, will be dependent on the problem) to 0 will represent weak correlation. Any value in the middle will represent moderate correlation.

  • If the two variables deviate in the same direction, i.e., increase (or decrease) in one variable results in an increase (or decrease) in the other, then the variables are said to be directly or positively correlated.
  • If the two variables deviate in the opposite direction, i.e., increase (or decrease) in one variable results in an decrease (or increase) in the other, then the correlation is said to be diverse or negative.
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  • 2
    $\begingroup$ All correct and sensible, but ignores the OP's question of precisely what is (in your terms) strong, moderate, or weak. (Whether it's a good question is a large issue; I personally agree in wording being dependent on the problem, but that is also an empty answer.) $R$ is in my experience much, much rarer notation than $r$. $\endgroup$ – Nick Cox Jan 15 '16 at 9:09

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