Correlation in Statistics

CORRELATION
Avjinder Singh Kaler and Kristi Mai

 Linear correlation coefficient, r, is a number that measures how well
paired sample data fit a straight-line pattern when graphed.
 Using paired sample data (sometimes called bivariate data), we find
the value of r (usually using technology), then we use that value to
conclude that there is (or is not) a linear correlation between the two
variables.
 In this section we will consider only linear relationships, which means
that when graphed, the points approximate a straight-line pattern.
 We will discuss methods of hypothesis testing for correlation.

Correlation – a correlation exists between two variables when the
values of one variable are somehow associated with the values of the
other variable.
• Can be positive, negative, non-existent, or non-linear
• A linear correlation exists between two variables when there is a
correlation and the plotted points of paired data result in a pattern that
can be approximated by a straight line.

We can often see a relationship between two variables by
constructing a scatterplot.
Scatter plots of paired data

1. The sample of paired data is a Simple Random Sample of
quantitative data
2. The pairs of data ( 𝑥,𝑦) have a bivariate normal distribution, meaning
the following:
• Visual examination of the scatter plot(s) confirms that the sample points
follow an approximately straight line(s)
• Because results can be strongly affected by the presence of outliers, any
outliers should be removed if they are known to be errors (Note: Use caution
when removing data points)
Note: These are the same as the Requirements for Simple Linear
Regression.

• Linear Correlation Coefficient (𝑟) – measures the strength of the linear
correlation between the paired quantitative 𝑥 and 𝑦 values in a sample
• Also known as the Pearson Product Moment Correlation Coefficient in honor of
Karl Pearson
• This is a Sample Statistic of the correlation that is linear between 𝑥 and 𝑦
• If this value is squared, the value is the Coefficient of Determination ( 𝑟2)
• Notation:
• 𝑟 : linear correlation coefficient for sample data
• 𝜌 : linear correlation coefficient for a population of paired data
• Formula for calculating 𝑟: 𝑟 =
𝑛 Σ𝑥𝑦 − Σ𝑥 ∗(Σ𝑦)
𝑛(Σ𝑥2)− Σ𝑥 2∗ 𝑛(Σ𝑦2)− Σ𝑦 2

1. – 1 ≤ r ≤ 1
2. If all values of either variable are converted to a different scale, the
value of r does not change.
3. The value of r is not affected by the choice of x and y.
4. r measures strength of a linear relationship.
5. r is very sensitive to outliers
• A single outlier can dramatically affect the value of r

• The value of 𝑟2 is the proportion of the variation in 𝑦 that is explained by
the linear relationship that exists between 𝑥 and 𝑦
• Thus, 𝑟2 is, also, the amount of variation in 𝑦 that is explained by the
regression line itself
• We may use 𝑟2 to describe the predictive power of the regression
equation

• To conclude that correlation implies causality
• Using data based on averages
• This type of data causes an inflated correlation coefficient
• To conclude that if there is no linear correlation, there is no correlation
at all

Hypotheses:
𝐻0: 𝜌 = 0
𝐻1: 𝜌 ≠ 0
(𝑛𝑜 𝑙𝑖𝑛𝑒𝑎𝑟 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛)
(𝑙𝑖𝑛𝑒𝑎𝑟 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑒𝑥𝑖𝑠𝑡𝑠)
These hypotheses can be equivalently tested with the following
hypotheses:
𝐻0: 𝛽1 = 0
𝐻1: 𝛽1 ≠ 0
Note: This equivalence will be important for the interpretation of the
technological output.

• Use Critical Value from Table A-6 (this is a simpler approach) and think
of the Linear Correlation Coefficient (𝑟) as a ‘test statistic’
• OR, use the following t-score test statistic with 𝑑𝑓 = 𝑛 − 2
𝑡 =
𝑟
1 − 𝑟2
𝑛 − 2
• This 𝑡 test statistic can be viewed in most technological output
corresponding to the test of significance for the slope in the regression
line (i.e. the second set of equivalent hypotheses listed above)

• If using critical values from Table A-6:
𝐼𝑓 𝑟 > 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒, 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻0
𝐼𝑓 𝑟 ≤ 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒, 𝑓𝑎𝑖𝑙 𝑡𝑜 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻0
This test is obviously a two-tailed hypothesis test based on the alternative
hypothesis. Visualize this by plotting the possible values for 𝑟 on a number line
with a labeled critical region.
• If using the t-score test statistic:
• Use statistical software to calculate the correct p-value that corresponds
with the test statistic. Then, base the conclusion on comparison between
the p-value and 𝛼

One-tailed tests can occur with a claim of a positive linear correlation or a
claim of a negative linear correlation. In such cases, the hypotheses will be as
shown here.
For these one-tailed tests, the P-value method can be used as well.

• Construct a scatter plot and verify that the pattern of the points is
approximately a straight line pattern without outliers
• Assess the linear correlation between two variables of interest and
create a regression equation
• Consider any effects of a pattern over time
• Perform a Residual Analysis:
• Construct a residual plot and verify that there is no pattern (other than a
straight line pattern) and also verify that the residual plot does not become
thicker or thinner
• Use a histogram, normal quantile plot, or Shapiro Wilk test of normality to
confirm that the values of the residuals have a distribution that is
approximately normal

• Measurement Error – could be described as ‘explainable’ outliers
• Nonlinear Associations – ignoring possible nonlinear relationships
• Extrapolation – predicting far beyond the scope of our available
data

The paired shoe / height data from five males are listed below.
Using StatCrunch, find the value of the correlation coefficient r.

Requirement Check:
The data are a simple random sample of quantitative data, the plotted
points appear to roughly approximate a straight-line pattern, and there
are no outliers.

A few technologies are displayed below, used to calculate the
value of r.

We found previously for the shoe and height example that r = 0.591.
With r = 0.591, we get r2 = 0.349.
We conclude that about 34.9% of the variation in height can be
explained by the linear relationship between lengths of shoe prints and
heights.

Conduct a formal hypothesis test of the claim that there is a linear
correlation between the two variables.
Use a 0.05 significance level.

We test the claim:
𝐻0: 𝛽1 = 0
𝐻1: 𝛽1 ≠ 0

We calculate the test statistic:
Table A-3 shows this test statistic yields a p-value that is greater
than 0.20.
2 2
0.591
1.269
1 1 0.591
2 5 2
r
t
r
n
  
 
 

StatCrunch provides a
P-value of 0.2937.
Because the p-value of 0.2937 is greater than the significance level of 0.05,
we fail to reject the null hypothesis.
We conclude there is not sufficient evidence to support the claim that there
is a linear correlation between shoe print length and heights of males.

With the test statistic, r = 0.591.
The critical values of r = ± 0.878 are found in Table A-6 with n = 5 and α = 0.05.
We fail to reject the null and conclude there is not sufficient evidence to
support the claim that there is a linear correlation between shoe print length
and heights of males.

Use the 5 pairs of shoe print lengths and heights to predict the height of
a person with a shoe print length of 29 cm.
The regression line does not fit the points well. The correlation is r = 0.591,
which suggests there is not a linear correlation (the p-value was 0.2937).
From StatCrunch,
The best predicted height is simply the mean of the sample heights:
177.3 cmy 

Use the 40 pairs of shoe print
lengths from Data Set 2 in
Appendix B to predict the height
of a person with a shoe print
length of 29 cm.
Now, the regression line does fit
the points well, and the
correlation of r = 0.813 suggests
that there is a linear correlation
since the p-value is < 0.0001.

The regression equation and scatterplot are shown below:

The given shoe length of 29 cm is not beyond the scope of the
available data, so substitute in 29 cm into the regression model:
A person with a shoe length of 29 cm is predicted to be 174.3 cm tall.
Using StatCrunch,
 
ˆ 80.9 3.22
80.9 3.22 29
174.3 cm
y x 
 


What if we have two or more explanatory variables?
Do we have a method for this? YES!! Of course, we do.
We may want to predict a sea turtle’s lifespan by more variables than
simply length of shell! I want to use variables that account for the diet,
exercise, mental health, and captivity status of the turtle! What about
variables that also account for the water quality in the turtle’s
surrounding environment? I want that too!!!
To do this, we can use something known as Multiple Linear Regression!

Correlation in Statistics

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