Bowman’s Website

March 21, 2010

Statistics Notes — Correlation

Filed under: Statistics, Statistics Notes — Tags: — bowman @ 12:56 pm

In statistics, correlation (often measured as a correlation coefficient) indicates the strength and direction of a linear relationship between two random variables.

The data can be represented by the ordered pairs (x, y) where x is the independent, or explanatory, variable and y is the dependent, or response, variable.

A scatter plot can be used to determine whether a linear (straight line) correlation exists between two variables. In a scatter plot, the ordered pairs (x, y) are graphed as points in a coordinate plane.

The following scatter plots show several types of correlations.

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Interpreting correlation using a scatter plot can be subjective. A more precise way to measure the type and strength of a linear correlation between two variables is to calculate the correlation coefficient.

The correlation coefficient is a measure of the strength and the direction of a linear relationship between two variables. The symbol r represents the sample correlation coefficient. The formula for r is

where n is the number of pairs of data.

The population correlation coefficient is represented by ρ (the lowercase Greek letter rho).

The range of the correlation coefficient is –1 to 1.

If x and y have a strong positive linear correlation, r is close to 1.

If x and y have a strong negative linear correlation, r is close to –1.

If there is no linear correlation or a weak linear correlation, r is close to 0.

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Example. Calculate the correlation coefficient for the advertising expenditures and company sales data. What can you conclude?

Advertising expenses (1000s of $), x

Company sales (1000s of $), y

2.4

225

1.6

184

2.0

220

2.6

240

1.4

180

1.6

184

2.0

186

2.2

215

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Let’s look at the scatter plot.

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Guidelines.

1. Find the sum of the x values.

2. Find the sum of the y values

3. Multiply each x value by its corresponding y value and find the sum.

4. Square each x value and find the sum.

5. Square each y value and find the sum.

6. Use these five sums to calculate the correlation coefficient.

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Advertising expenses (1000s of $), x

Company sales (1000s of $), y

xy

x2

y2

2.4

225

540

5.76

50625

1.6

184

294.4

2.56

33856

2.0

220

440

4

48400

2.6

240

624

6.76

57600

1.4

180

252

1.96

32400

1.6

184

294.4

2.56

33856

2.0

186

372

4

34596

2.2

215

473

4.84

46225

∑x = 15.8

∑y = 1634

∑xy = 3289.8

∑x2 = 32.44

∑y2 = 337,558

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Using these sums and n = 8, the correlation coefficient is

Because r is close to 1, there is a strong positive linear correlation. As the amount of spending on advertising increases, the company sales also increase.

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Find the sums.

These are your 5 sums. Use these and n = 8 to determine the correlation coefficient.

Because r is close to 1, there is a strong positive linear correlation. As the amount of spending on advertising increases, the company sales also increase.

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If your calculator is like mine, you needed to turn on a feature. I didn’t know until I saw that r was missing. r should be below the a and b.

Here’s what we need to do. Turn on DiagnosticOn. DiagnosticOn is within our CATALOG. Now we have the ability to find r using our lists.

Because r is close to 1, there is a strong positive linear correlation. As the amount of spending on advertising increases, the company sales also increase.

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Example. Calculate the correlation coefficient for the income level and donating percent data. What can you conclude?

Income level (in 1000s of $), x

Donating percent, y

42

9

48

10

50

8

59

5

65

6

72

3

.

Find the five sums and use the correlation coefficient formula.

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Using these sums and n = 6, the correlation coefficient is

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Use the Linear Regression feature to check. Nice work.

Because r is close to –1, there is a strong negative linear correlation. As the level of income rises, the percentage of donating decreases.

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