Pearson’s Correlation CoefficientWhen we describe the association between two

variables, we can use a scatterplot to help our description.

However, words like strong, moderate, and weak to determine the strength of the relationship can be very subjective. (Remember the saying. “Beauty is in the eye of the beholder”)

So statisticians have another tool, a numeric measure, to help us clarify and be somewhat consistent when we describe these relationships.

What it MeasuresIn the lesson on scatterplots, we indicated

that a tighter oval around the data points indicated a stronger relationship. In some sense this must mean that the closer the points are to each other, the stronger the relationship.

Pearson’s Correlation Coefficient helps us to numerically measure this “spread” of our data.

So how does it Measure this “spread”?We know that in describing a distribution

that we think about both the “center” of the distribution, and the “spread” of the distribution.

When looking at the relationship between two variables, we need to consider both the “center” and the “spread” of each, and how the combination of these two distributions interact.

Properties of r“r” is unitless, which allows us to change

scales or calculate the relationship between two variables that are not the same units

“r” measures the linear relationship between two quantitative variables.

-1 ≤ r ≤ 1The sign of “r” indicates the direction of

the relationshipThe closer “r” is to either +1 or -1, the

stronger the relationship.The closer “r” is to 0, the weaker the

relationship.

Numeric GuidelinesPhysical Sciences

“Hard Sciences” ≥ .80---Strong

.50 --.80—Moderate≤ .50—Weak

Social Sciences“Soft Sciences”≥ .50---Strong

.30 --.50—Moderate≤ .30—Weak

Remember that these numbers are just guidelines. Each set of data is different and the context for the data must be considered.

The Formula

yx syy

sxx

n 11

Notice that the formula is adding terms together (we’ll talk about what those terms are shortly) and then dividing that sum by 1 less than the number of data points we have. So, it appears that we are looking for “an average” of sorts.

The Formula (cont.)

yx syy

sxx

nr

11

Now the terms that we are adding together are the product of z-scores. Remember that a z-score is the number of standard deviations a piece of data is from the mean of the distribution.So each term is the product of the z-scores in each direction (x and y) for each point. So, how can we calculate this value?

Back to the Burgers Fat Calories

19 410

31 580

34 590

35 570

39 640

39 680

43 660C

alor

ies

400

450

500

550

600

650

700

Fat18 20 22 24 26 28 30 32 34 36 38 40 42 44

Burgers Scatter Plot

Calculating “r”

yx syy

sxx

n 11

•We can calculate “r” using this formula and the lists.L1X (amount of fat)L2Y (calories)L3Zx (x-xbar)/sx

L4Zy (y-ybar)/sy

L5L3*L4

•Once these lists are created, find the sum of L5 and then divide by n-1

Another formula for “r”

yx syy

sxx

n 11

yyxxssn yx

)1(

1

yyxxyyxxssn nnyx

...)1(

111

Starting with our original formulaNow, the standard deviation of our x-values and the y-values are constants once our data has been collected, so they will be the same for each term in the summation.This means that we can factor those out of the sum leaving:Now, expanding the summation gives us:

Another formula for “r” (cont.) yyxxyyxx

ssn nnyx

...)1(

111

yxyxyxyxyxyxyxyxssn nnnnyx

...)1(

11111

yxnyxxyyxssn iiiiyx

)1(

1

Now, using the distributive property to multiply the binomials in each term gives:

Then, collapsing the sums gives:

Another formula for “r” (cont.)

yxnyxnyxnyxssn iiyx

)1(

1

yxnyxxyyxssn iiiiyx

)1(

1

yx

iiii

yx ssnyxnyxyxnyx

ssn 1)1(1

Now, the ∑xi and the ∑yi can be written as nxbar and nybar

But two of the last three terms cancel each other out, so we are left with:

Evaluating the FormulaThis formula is

helpful to us because our calculator gives us each of the terms we see here

With our data in the lists, L1(fat) and L2(calories) in this case, we calculate the 2-var stats to find these values.

yx

ii

ssnyxnyxr

1

Calculating “r”Now, calculate the 2-

var stats for L1, L2STAT

CALC

This gives us all the values we need to calculate “r”

We can then describe numerically the relationship between amount of fat and calories in a burger.

Calculating “r”

)8146.89)(8042.7(17)590)(2857.34)(7(145640

r

yx

ii

ssnyxnyxr

1

5866.4205059.4040

r

r= .9606

Now, substituting the values for each of the variables we find that the correlation coefficient, r=.96, indicating a strong, linear correlation in which as the amount of fat in the burger increases, so does the calories

Outliers??What about the point

we determined was a long way away from the rest of our data----our possible “outlier”

Since this burger appears to be somewhat different than the rest of our data, it would be wise to report the correlation both with and without it.

Cal

orie

s

400

450

500

550

600

650

700

Fat18 20 22 24 26 28 30 32 34 36 38 40 42 44

Burgers Scatter Plot

•If we calculate the correlation coefficient without this piece of data, we find that it drops to .8367. This change indicates that this piece of data is unusual

And yet another way to find “r”In the next section, we will look at even

another way to find the value of Pearson’s correlation coefficient.

For now, either method used in this lesson is appropriate.

Additional ResourcesThe Practice of Statistics—YMM

Pg 128 – 136The Practice of Statistics—YMS

Pg 140-149The Basic Practice of Statistics—Moore

Pg 88-94