This document discusses correlation and the correlation coefficient (r). It begins by defining r as a measure of the direction and strength of a linear relationship between two variables. r ranges from -1 to 1, with values closer to these extremes indicating a stronger linear relationship. r is calculated using the means and standard deviations of both variables and does not distinguish which is the explanatory or dependent variable. While r describes the strength of linear relationships, it does not capture nonlinear relationships between variables. r can also be influenced by outliers in the data.

1. INTRODUCTION TO
STATISTICS & PROBABILITY
Chapter 2:
Looking at Data–Relationships (Part 2)
Dr. Nahid Sultana
1

2. 2
Chapter 2:
Looking at Data–Relationships
2.1: Scatterplots
2.2: Correlation
2.3: Least-Squares Regression
2.5: Data Analysis for Two-Way Tables

3. Objectives
 The correlation coefficient “r”
 r does not distinguish between x and y
 r has no units of measurement
 r ranges from -1 to +1
 Influential points
2.2: Correlation
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4. The correlation coefficient "r"
 The correlation coefficient is a measure of the direction and
strength of a linear relationship.
 It is calculated using the mean and the standard deviation of both
the x and y variables.
 Correlation can only be used to describe quantitative variables.
Categorical variables don’t have means and standard deviations.
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5. The correlation coefficient “r“ (Cont…)
Time to swim: = 35, sx = 0.7
Pulse rate: = 140, sy = 9.5
x
y
5
r =
1
n −1
xi − x
sx






i=1
n
∑
yi − y
sy






 Suppose that we have data
on variables x and y for n
individuals.
 The means and standard
deviations of the two variables
are and for the x-values,
and and for y-values.
 The correlation r between x
and y
x
y

6. “r” does not distinguish x & y
The correlation coefficient, r,
treats x and y symmetrically.
"Time to swim" is the explanatory variable here, and belongs on
the x axis. However, in either plot r is the same (r=-0.75).
r = -0.75 r = -0.75
r =
1
n −1
xi − x
sx






i=1
n
∑
yi − y
sy






6

7. Changing the units of variables does
not change the correlation coefficient
"r“.
"r" has no unit r = -0.75
r = -0.75
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standardized
value of x
(unit less)
standardized
value of y
(unit less)

8. "r" ranges from -1 to +1
Properties of Correlation
 r is always a no. between –1 and 1.
 r > 0 indicates a positive association.
r < 0 indicates a negative association.
 Values of r near 0 indicate a very
weak linear relationship.
 The strength of the linear relationship
increases as r moves away from 0
toward –1 or 1.
 The extreme values r = –1 and r = 1
occur only in the case of a perfect
linear relationship.
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9. 9
“r” increases as variation decreases
When variability in
one or both variables
decreases, the
correlation coefficient
gets stronger
( closer to +1 or -1).

10. Correlation only describes linear
relationships
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No matter how strong the association,
r does not describe curved relationships.

11. 11
Influential points
Correlations are calculated using
means and standard deviations,
and thus are NOT resistant to
outliers.
Just moving one point away from
the general trend here decreases
the correlation from -0.91 to -
0.75

12. 12
12
Influential points (Cont…)