Statistics For Dummies
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Sometimes, you may want to see how closely two variables relate to one another. In statistics, we call the correlation coefficient r, and it measures the strength and direction of a linear relationship between two variables on a scatterplot. The value of r is always between +1 and –1. To interpret its value, see which of the following values your correlation r is closest to:

  • Exactly –1. A perfect downhill (negative) linear relationship

  • –0.70. A strong downhill (negative) linear relationship

  • –0.50. A moderate downhill (negative) relationship

  • –0.30. A weak downhill (negative) linear relationship

  • 0. No linear relationship

  • +0.30. A weak uphill (positive) linear relationship

  • +0.50. A moderate uphill (positive) relationship

  • +0.70. A strong uphill (positive) linear relationship

  • Exactly +1. A perfect uphill (positive) linear relationship

If the scatterplot doesn’t show that there’s at least somewhat of a linear relationship, the correlation doesn’t mean much. Why measure the amount of linear relationship if there isn’t much of one?

However, you can think of this idea of no linear relationship in two ways: 1) If no relationship at all exists, calculating the correlation doesn’t make sense because correlation only applies to linear relationships, and 2) If a strong relationship exists but it’s not linear, the correlation may be misleading, because in some cases a strong curved relationship exists. That’s why it’s critical to check out the scatterplot first.

Scatterplots with correlations of a) +1.00; b) –0.50; c) +0.85; and d) +0.15.

Scatterplots with correlations of a) +1.00; b) –0.50; c) +0.85; and d) +0.15
The above figure shows examples of what various correlations look like, in terms of the strength and direction of the relationship. Figure (a) shows a correlation of nearly +1, Figure (b) shows a correlation of –0.50, Figure (c) shows a correlation of +0.85, and Figure (d) shows a correlation of +0.15.

Comparing Figures (a) and (c), you see Figure (a) is nearly a perfect uphill straight line, and Figure (c) shows a very strong uphill linear pattern (but not as strong as Figure (a)). Figure (b) is going downhill, but the points are somewhat scattered in a wider band, showing a linear relationship is present, but not as strong as in Figures (a) and (c). Figure (d) doesn’t show much of anything happening (and it shouldn’t, since its correlation is very close to 0).

Many folks make the mistake of thinking that a correlation of –1 is a bad thing, indicating no relationship. Just the opposite is true! A correlation of –1 means the data are lined up in a perfect straight line, the strongest negative linear relationship you can get. The “–” (minus) sign just happens to indicate a negative relationship, a downhill line.

How close is close enough to –1 or +1 to indicate a strong enough linear relationship? Most statisticians like to see correlations beyond at least +0.5 or –0.5 before getting too excited about them. Don’t expect a correlation to always be 0.99 however; remember, these are real data, and real data aren’t perfect.

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Deborah Rumsey has a PhD in Statistics from The Ohio State University (1993). Upon graduating, she joined the faculty in the Department of Statistics at Kansas State University, where she won the distinguished Presidential Teaching Award and earned tenure and promotion in 1998. In 2000, she returned to Ohio State and is now a Statistics Education Specialist/Auxiliary Faculty Member for the Department of Statistics. Dr. Rumsey has served on the American Statistical Association’s Statistics Education Executive Committee and is the Editor of the Teaching Bits section of the Journal of Statistics Education. She’s the author of the books Statistics For Dummies and Statistics Workbook For Dummies (Wiley). She also has published many papers and given many professional presentations on the subject of Statistics Education. Her particular research interests are curriculum materials development, teacher training and support, and immersive learning environments. Her passions, besides teaching, include her family, fishing, bird watching, driving a new Kubota tractor on the family “farm,” and Ohio State Buckeye football (not necessarily in that order).

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