AP* Statistics Tutorial: Correlation and Linearity
Correlation coefficients measure the strength of
association between two variables. The most common correlation
coefficient, called the
Pearson product-moment correlation coefficient,
measures the strength of the
linear association between variables.
In this tutorial, when we speak simply of a correlation
coefficient, we are referring to the Pearson product-moment
correlation. Generally, the correlation coefficient of a
sample
is denoted by
r, and the correlation coefficient of a
population
is denoted by
ρ or R.
How to Interpret a Correlation Coefficient
The sign and the
absolute value
of a correlation coefficient
describe the direction and the magnitude of the relationship
between two variables.
- The value of a correlation coefficient ranges between -1 and
1.
- The greater the absolute value of a correlation coefficient,
the stronger the linear relationship.
- The strongest linear relationship is indicated by a correlation
coefficient of -1 or 1.
- The weakest linear relationship is indicated by a correlation
coefficient equal to 0.
- A positive correlation means that if one variable gets bigger,
the other variable tends to get bigger.
- A negative correlation means that if one variable gets bigger,
the other variable tends to get smaller.
Keep in mind that the Pearson product-moment correlation coefficient
only measures
linear relationships. Therefore, a correlation of 0 does not
mean zero relationship between two variables; rather, it means
zero linear relationship. (It is possible for two
variables to have zero linear relationship and a strong
curvilinear relationship at the same time.)
Scatterplots and Correlation Coefficients
The scatterplots
below show how different patterns of data produce different degrees of
correlation.
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 |
 |
Maximum positive correlation
(r = 1.0) |
Strong positive correlation
(r = 0.80) |
Zero correlation
(r = 0) |
 |
 |
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Minimum negative correlation
(r = -1.0) |
Moderate negative correlation
(r = -0.43) |
Strong correlation & outlier
(r = 0.71) |
Several points are evident from the scatterplots.
- When the
slope
of the line in the plot is negative, the correlation is
negative; and vice versa.
- The strongest correlations (r = 1.0 and r = -1.0 ) occur
when data points fall exactly on a straight line.
- The correlation becomes weaker as
the data points become more scattered.
- If the data points fall in a random pattern, the correlation
is equal to zero.
- Correlation is affected by
outliers.
Compare the first
scatterplot with the last scatterplot. The single outlier
in the last plot greatly reduces the correlation
(from 1.00 to 0.71).
How to Calculate a Correlation Coefficient
If you look in different statistics textbooks,
you are likely to find different-looking (but equivalent) formulas for
computing a correlation coefficient. In this section, we
present several formulas that you may encounter.
The most common formula for computing a product-moment correlation coefficient (r)
is given below.
Product-moment correlation
coefficient.
The correlation r between two variables is:
r = Σ (xy) / sqrt [ ( Σ x2 ) * ( Σ y2 ) ]
where Σ is the summation symbol,
x = x i - x,
x i is the x value for observation i,
x is the mean x value,
y = y i - y,
y i is the y value for observation i, and
y is the mean y value.
The formula below uses population means and population standard deviations
to compute a population correlation coefficient (ρ) from population data.
Population correlation
coefficient.
The correlation ρ between two variables is:
ρ = [ 1 / N ] * Σ { [ (Xi - μX) / σx ] * [ (Yi - μY) / σy ] }
where N is the number of
observations in the population, Σ is the summation symbol,
X i is the X value for observation i,
μ X is the population mean for variable X,
Y i is the Y value for observation i,
μ Y is the population mean for variable Y,
σ x is the population standard deviation of X, and
σ y is the population standard deviation of Y.
The formula below uses sample means and sample standard deviations
to compute a correlation coefficient (r) from sample data.
Sample correlation
coefficient.
The correlation r between two variables is:
r = [ 1 / (n - 1) ] * Σ { [ (xi - x) / sx ] * [ (yi - y) / sy ] }
where n is the number of
observations in the sample, Σ is the summation symbol,
x i is the x value for observation i,
x is the sample mean of x,
y i is the y value for observation i,
y is the sample mean of y,
s x is the sample standard deviation of x, and
s y is the sample standard deviation of y.
The interpretation of the sample correlation coefficient depends on how the
sample data is collected. With a
simple random sample,
the sample correlation coefficient is an unbiased estimate of the population
correlation coefficient.
Each of the latter two formulas can be derived from the first formula.
Use the second formula when you have data from the entire population.
Use the third formula when you only have sample data.
When in doubt, use the first formula. It is always correct.
Fortunately, you will rarely have to compute a correlation
coefficient by hand. Many software packages (e.g., Excel) and most
graphing calculators
have a correlation function that will do the job for you.
Note: Sometimes, it is not clear whether a software package or a
graphing calculator is computing a population correlation coefficient or a
sample correlation coefficient. For example, a casual user
might not realize that Microsoft uses a
population correlation coefficient (ρ) for the Pearson function
in its Excel software.
Test Your Understanding of This Lesson
Problem 1
A national consumer magazine reported the following correlations.
- The correlation between car weight and car reliability is -0.30.
- The correlation between car weight and annual maintenance cost
is 0.20.
Which of the following statements are true?
I. Heavier cars tend to be less reliable.
II. Heavier cars tend to cost more to maintain.
III. Car weight is related more strongly to reliability than to
maintenance cost.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III
Solution
The correct answer is (E). The correlation between car weight
and reliability is negative. This means that reliability tends to
decrease as car weight increases. The correlation between car
weight and maintenance cost is positive. This means that maintenance
costs tend to increase as car weight increases.
The strength of a relationship between two variables is indicated by the
absolute value
of the correlation coefficient. The correlation between car weight
and reliability has an absolute value of 0.30. The correlation
between car weight and maintenance cost has an absolute value of
0.20. Therefore, the relationship between car weight and
reliability is stronger than the relationship between car weight
and maintenance cost.
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