MONT 106N, Identifying Patterns Discussion on Computing Correlation Coefficients

October 26, 2009

Background

The process for computing the correlation coefficient of two data sets x_{i} and y_{i}, i = 1, ..., n is as follows:

• First, compute ave_{x}, ave_{y}, and SD_{x}, SD_{y}.

• Then convert the x_{i} and y_{i} to standard units:

z_{i} = \frac{x_{i} − ave_{x}}{SD_{x}} , w_{i} = \frac{y_{i} − ave_{y}}{SD_{y}} .

• Finally,

r = ave of products z_{i}w_{i}

= \frac{1}{n}(z_{1}w_{1} + z_{2}w_{2} + ⋯ + z_{n}w_{n}).

We also saw that the computations could be arranged in a convenient tabular format to facilitate the various steps.

Practice Problem

The heights (in inches) and weights (in pounds) of n = 10 offensive backfield high school football players were measured as follows:

Player Height Weight
1  73  185
2  71  175
3  75  200
4  72  210
5  72  190
6  75  195
7  67  150
8  69  170
9  71  180
10  69  175

A) Construct a scatter plot for this data (use the graph paper provided). What can you say from the general shape of your scatter plot?

B) Compute ave_{x}, ave_{y}, SD_{x}, and SD_{y}. C) Convert all the x_{i} and y_{i} to the appropriate standard units. D) Compute the correlation coefficient r. What does the value say about the association between

height and weight for these football players. (Of course, by extension, one would like to say the same thing about all high school football offensive backfield players!)

E) Compute the equation of the SD-line and add it to your scatter plot.