•
If
r
equals −1 or 1 then all points fall directly on a straight line in a scatter plot.
•
Values of
r
close to −1 or 1 indicate a strong linear relationship between
X
and
Y
.
•
The closer
r
is to 0, the weaker the
linear
relationship between
X
and
Y
.
•
If
X
and
Y
are interchanged, the value of
r
does not change.
•
r
and the slope of the line in a scatter plot will have the same sign (positive or negative).
EXAMPLE:
(Adapted from Question 13, page 675 in
Business Statistics
, 2
nd
Canadian Ed. (2014). Sharpe, N.R., DeVeaux,
R.D., Velleman, P.F., Wright, D. Pearson Toronto)
Data on the number of sales associates working, and the
number of sales (in $1000s), were recorded for 10 randomly selected small book stores. The objective
of the study was to determine if there was a linear relationship between the number of sales associates
on the floor and the amount of business done in sales. A partial table of the data is presented below.
Number of
Salespeople
Sales (in
$1000s)
2
10
3
11
7
13
...
...
20
26
Since we have two quantitative variables, we can use a scatter plot to get a quick visual representation:
Here, the summary statistics are provided (since they are tedious to calculate by hand!):
SS
XX
= 286
.
4,
SS
YY
= 256
.
4,
SS
XY
= 261
.
6
We can now calculate the correlation as:
24

3.8
Linear Transformations
Linear transformations occur when our data is transformed from one set of units/measurements to
another. Common examples of linear transformations include:
•
Currency exchanges (i.e. Canadian to US dollars)
•
Temperature conversions (i.e. degrees Celsius to degrees Fahrenheit)
•
Weight changes (i.e. pounds to kilograms)
•
Calculating a Z-score!
We can transform our data by adding a constant to each observation, and/or multiplying each
observation by a constant.
Adding a Constant
Consider the following data set: 2, 4, 6, 8, 10.
The mean, median, variance, and standard deviation of this data set is:
Consider what happens when we add a constant (say, 3) to each observation, such that our new data
set is: 5, 7, 9, 11, 13. The mean, median, variance, and standard deviation of the new data set is:
In general, adding a constant to every observation in the data set will change the Measures of Centre,
but not the Measures of Variability.
Multiplying by a Constant
Consider our first data set again: 2, 4, 6, 8, 10
Now, consider what happens when we multiply each observation in the data set by a constant (say, 2),
such that our new data set is: 4, 8, 12, 16, 20.
The mean, median, variance, and standard deviation of the new data set is:
25