Correlation Coefficients

Correlation Coefficients Definition

Let’s start with understanding what a correlation is first. Have you ever noticed that two things seem to be related? It can be as simple as the hotter it is outside, the more water you drink. You’ve noticed that when the temperature rises, your water consumption also increases. In this instance, you’re noting that these two factors are correlated.

In the example above, the two variables would be temperature and water consumption. You know these two variables are related, but you need to remember an essential part about correlations – correlation does not equal causation.

Now that we understand a correlation, what is a correlation coefficient?

So, you could look at temperature and water consumption and know they’re correlated, but a little more goes into understanding correlation coefficients.

A person drinking water on a hot day, freepik.com

Correlation Coefficient Interpretation

We now know what a correlation coefficient is, but how does it work?

Positive vs Negative Correlation

Let’s first break down positive and negative correlations. When two variables increase or decrease, that would be considered a positive correlation. A negative correlation isn’t actually when both variables decrease, but when the variables move in opposite directions – one increases and one decreases. This knowledge is vital for understanding the values of the correlation coefficient.

Correlation Coefficient Values

The correlation coefficient ranges on a scale from -1.00 to 1.00. -1.00 shows the strongest possible negative correlation, and 1.00 shows the strongest possible positive correlation. As you might guess, a correlation coefficient value of 0 indicates no correlation.

Correlation coefficients that are less than -0.80 or greater than 0.80 are significant. A correlation with a correlation coefficient of, for example, 0.21 does show a correlation, but it is not strong.

Correlation Coefficients Formula

Below is the formula for finding the correlation coefficient. It looks like a lot, but don’t be scared! Let’s break it down, so it’s more digestible.

$\mathrm{r}=\frac{\mathrm{n}\underset{}{(\sum }\mathrm{xy})-(\sum _{}\mathrm{x}\left)\right(\sum _{}\mathrm{y})}{\sqrt{[\mathrm{n}\sum _{}{\mathrm{x}}^2-{\left(\sum _{}\mathrm{x}\right)}^2][\mathrm{n}\sum _{}{\mathrm{y}}^2-{\left(\sum _{}\mathrm{y}\right)}^2]}}$

Above is the formula for finding the correlation coefficient. It looks like a lot, but don’t be scared! Let’s break it down so it’s more digestible.

• As stated earlier, the value of r represents the correlation coefficient. It’s what we’re trying to find.
• The value of n stands for the number of data points in the set (AKA, how many participants did you have?)
• The ∑ stands for “the summation of.” What that means is that all the values of each category are added together. So if you had ∑x and your x values were 80, 20, and 100, ∑x = 200.

The numerator would have the number of participants in the set multiplied by the summation of the x times y values. So, you’d multiply a participant’s x value by their y value, do this for every participant, then add them all together (and multiply by the total number of participants). Then, all the x-values (all x-values added together) are multiplied by the summation of all the y-values. This second value is subtracted from the first value to get your numerator.

The denominator has a little more going on. The number of participants is multiplied by the summation of all the x-values squared. So, you’d have to square each x-value, add them all up, and then multiply by the number of participants. Then, you would square the total x-values (add up the x-values and then square that number. The first value then subtracts this second value.

Correlation Coefficient Calculations, flaticon.com

The next part of the denominator is the same thing you just did, but replace the x-values with y-values. This second final number is multiplied by the final number from all the x-values. Finally, the square root is taken from this value you just got from multiplying.

Last but not least, the numerator value is divided by the denominator value to get your correlation coefficient!

Correlation Coefficients Example

An extremely common example of a correlation is between height and weight. In general, someone who is taller is going to be heavier than someone who is shorter. These two variables, height & weight, would be positively correlated since they either both increase or decrease. Let’s pretend you ran a study to see if these are correlated.

Your study consisted of ten data points from ten people.

1. 61 inches, 140 pounds

2. 75 inches, 213 pounds

3. 64 inches, 134 pounds

4. 70 inches, 175 pounds

5. 59 inches, 103 pounds

6. 66 inches, 144 pounds

7. 71 inches, 220 pounds

8. 69 inches, 150 pounds

9. 78 inches, 248 pounds

10. 62 inches, 120 pounds

You then either plug the data into SPSS or find the correlation coefficient by hand. Let’s gather values we know.

n = 10 (how many data points in the study?)

∑xy = 113676 (what are the x and y values multiplied and then all added together? For example, (61*140) + (75*213) + (64*134) + …)

∑x = 675 (add all the x values together)

∑y = 1647 (add all the y values together)

∑x² = 45909 (square all the x values then add them together)

∑y² = 291699 (square all the y values then add them together)

$\mathrm{r}=\frac{\mathrm{n}\underset{}{(\sum }\mathrm{xy})-(\sum _{}\mathrm{x}\left)\right(\sum _{}\mathrm{y})}{\sqrt{[\mathrm{n}\sum _{}{\mathrm{x}}^2-{\left(\sum _{}\mathrm{x}\right)}^2][\mathrm{n}\sum _{}{\mathrm{y}}^2-{\left(\sum _{}\mathrm{y}\right)}^2]}}$

Start with the numerator and plug in your values.

10(113676) - (675)(1647)

= 1136760 - 1111725

= 25035

Then the denominator.

(10*45909 - (675)²) (10*291699 - (1647)²)

= (459090 - 455625) (2916990 - 2712609)

= 3465*204381

= 708180165

Don’t forget to square root it!

= 2661.654684

Finally, divide the numerator by the denominator!

25035 / 26611.654684

= 0.950899

~ 0.95

As you correctly assumed, the height and weight of the data in this experiment are strongly correlated!

Correlation Coefficient Significance

A correlation coefficient is an essential tool for researchers in determining the strength of their correlational studies. Correlational research is an integral part of the field of psychology and the correlation coefficient serves as the benchmark for what a strong correlation looks like. Without it, there would be no parameters for what makes a strong correlation and what makes a weak or nonexistent one.