Linear Project

Scatterplots, Linear Regression, and Correlation

When we have a set of data, often we would like to develop a model that fits the data.

First we graph the data points (x, y) to get a scatterplot. Take the data, determine an appropriate scale

on the horizontal axis and the vertical axis, and plot the points, carefully labeling the scale and axes.

Summer Olympics:

Men's 400 Meter Dash

Winning Times

Year (x)

Time(y)

(seconds)

1948 46.20

1952 45.90

1956 46.70

1960 44.90

1964 45.10

1968 43.80

1972 44.66

1976 44.26

1980 44.60

1984 44.27

1988 43.87

1992 43.50

1996 43.49

2000 43.84

2004 44.00

2008 43.75

Burger Fat (x)

(grams)

Calories (y)

Wendy's Single 20 420

BK Whopper Jr. 24 420

McDonald's Big Mac 28 530 Wendy's Big Bacon

Classic 30 580

Hardee's The Works 30 530 McDonald's Arch

Deluxe 34 610 BK King Double

Cheeseburger 39 640 Jack in the Box

Jumbo Jack 40 650

BK Big King 43 660

BK King Whopper 46 730 Data from 1997

If the scatterplot shows a relatively linear trend, we try to fit a linear model, to find a line of best fit.

We could pick two arbitrary data points and find the line through them, but that would not necessarily

provide a good linear model representative of all the data points.

A mathematical procedure that finds a line of "best fit" is called linear regression. This procedure is also

called the method of least squares, as it minimizes the sum of the squares of the deviations of the points

from the line. In MATH 107, we use software to find the regression line. (We can use Microsoft Excel, or

Open Office, or a hand-held calculator or an online calculator --- more on this in the Technology Tips

topic.)

Linear regression software also typically reports parameters denoted by r or r 2 .

The real number r is called the correlation coefficient and provides a measure of the strength of the

linear relationship.

r is a real number between −1 and 1.

r = 1 indicates perfect positive correlation --- the regression line has positive slope and all of the data

points are on the line.

r = −1 indicates perfect negative correlation --- the regression line has negative slope and all of the

data points are on the line

The closer |r| is to 1, the stronger the linear correlation. If r = 0, there is no correlation at all. The

following examples provide a sense of what an r value indicates.

Source: The Basic Practice of Statistics, David S. Moore, page 108.

Notice that a positive r value is associated with an increasing trend and a negative r value is associated

with a decreasing trend. The strongest linear models have r values close to 1 or close to −1.

The nonnegative real number r 2 is called the coefficient of determination and is the square of the

correlation coefficient r.

Since 0 ≤ |r| ≤ 1, multiplying through by |r|, we have 0 ≤ |r| 2 ≤ |r| and we know that −1 ≤ r ≤ 1.

So, 0 ≤ r 2 ≤ 1. The closer r

2 is to 1, the stronger the indication of a linear relationship.

Some software packages (such as Excel) report r 2 , and so to get r, take the square root of r

2 and

determine the sign of r by observing the trend (+ for increasing, − for decreasing).