6.6: Introduction to Correlations
- Page ID
- 269735
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- Interpret correlations to determine how the variables vary.
The Correlation Coefficient
As a refresher, Pearson’s correlation coefficient, which is traditionally denoted by r, shows the relationship between two variables, and is a measure that varies from −1.00 to 1.00. When r=−1.00 it means that we have a perfect negative relationship, and when r=1.00 it means we have a perfect positive relationship. When r=0, there’s no relationship at all. If you look at Figure \(\PageIndex{1}\), you can see several plots showing what different correlations look like.
As you can see, strong correlations (shown on the bottom, r-values close to -1.00 or 1.00) are straight lines of dots. The closer to zero, the less linear the dots are, until you get to a correlation of zero (r=0.00) and there's just a random splatter of dots (top scatterplots). You can also see that positive correlations "point" up and to the right; as one variable increases, the other also increases. Negative correlations "point" down and to the right; as one variable increases, the other variable decreases.
Interpreting a Correlation
Naturally, in real life you don’t see many correlations of 1.00. So how should you interpret a correlation of, say r = .40? The honest answer is that it really depends on what you want to use the data for, and on how strong the correlations in your field tend to be. A friend of mine in engineering once argued that any correlation less than .95 is completely useless (I think he was exaggerating, even for engineering). On the other hand there are real cases – even in psychology – where you should really expect correlations that strong. For instance, one of the benchmark data sets used to test theories of how people judge similarities is so clean that any theory that can’t achieve a correlation of at least .90 really isn’t deemed to be successful. However, when looking for (say) elementary correlates of intelligence (e.g., inspection time, response time), if you get a correlation above .30 you’re doing very very well. In short, the interpretation of a correlation depends a lot on the context. That said, the rough guide in Table \(\PageIndex{1}\) is pretty typical.
| Correlation | Strength | Direction |
|---|---|---|
| -1.0 to -0.9 | Very strong | Negative |
| -0.9 to -0.7 | Strong | Negative |
| -0.7 to -0.4 | Moderate | Negative |
| -0.4 to -0.2 | Weak | Negative |
| -0.2 to 0 | Negligible | Negative |
| 0 to 0.2 | Negligible | Positive |
| 0.2 to 0.4 | Weak | Positive |
| 0.4 to 0.7 | Moderate | Positive |
| 0.7 to 0.9 | Strong | Positive |
| 0.9 to 1.0 | Very strong | Positive |
However, something that can never be stressed enough is that you should always look at the scatterplot before attaching any interpretation to the data. As was said in prior sections, a correlation might not mean what you think it means. The classic illustration of this is “Anscombe’s Quartet” (Anscombe, 1973), which is a collection of four data sets. Each data set has two variables, an X and a Y. For all four data sets the mean value for X is 9 (\( \bar{X}_{x-axis} = 9.00 \)) and the mean for Y is 7.5 (\( \bar{X}_{y-axis} = 7.50 \)). The, standard deviations for all X variables are almost identical, as are those for the the Y variables. And in each case the correlation between X and Y is r=0.816. You’d think that these four data sets would look pretty similar to one another. They do not. If we draw scatterplots of X against Y for all four variables, as shown in Figure \(\PageIndex{2}\) we see that all four of these are spectacularly different to each other.
The lesson here, which so very many people seem to forget in real life is “always graph your raw data”.
Now that you know a little more about what correlations can show, let's talk about what Pearson's correlation cannot show.
Reference
Anscombe, F. J. (1973). Graphs in statistical analysis, The American Statistician, 27(1), 17-21).