Advanced Higher

STATISTICS

Linear Regression

To see if there is a relationship between two variables, we draw a scatter-graph. It is then possible to draw a ‘best-fit’ line through the

points on the graph. To draw the straight line, we try to have as many points as possible on either side of the line.

This is pretty difficult so we calculate a line of regression instead: this is called linear regression.

Advanced Higher

STATISTICS

The formula for drawing the linear regression

The equation for a straight line is Y = a + bX

·‘a’ is known as the ‘intercept’. oIt is given by the formula: a = ӯ - bẊ

·‘b’ is the ‘regression coefficient’

·It is given by the formula: b = ∑(Ẋ-X)(y-ӯ) ∑(X-Ẋ)2

Advanced Higher

STATISTICS

First: draw yourself a scatter-graph to show the relationship between two

variablesSite Discharge M3/sec

Suspended load g/m3

1 0.45 10.8

2 0.42 9.7

3 0.51 11.2

4 0.55 11.3

5 0.68 12.5

6 0.75 12.8

7 0.89 13.0

8 0.76 12.7

9 0.96 13.0

10 1.26 17.4

0.2 0.4 0.6 0.8 1 1.2 1.40

2

4

6

8

10

12

14

16

18

20

Series1

Increasing river discharge

Incr

easi

ng s

uspe

nded

load

Scatter graph showing the relationship between discharge and suspended on the River Farg

Advanced Higher

STATISTICS

Second: complete Pearsons Product to get the values you need for ‘b’ b = ∑(Ẋ-X)(y-ӯ)

∑(X-Ẋ)2Site Discharge M3/sec

Suspended load g/m3

1 0.45 10.8

2 0.42 9.7

3 0.51 11.2

4 0.55 11.3

5 0.68 12.5

6 0.75 12.8

7 0.89 13.0

8 0.76 12.7

9 0.96 13.0

10 1.26 17.4

Advanced Higher

STATISTICS

Second: complete Pearsons Product to get the values you need for ‘b’

b = ∑(Ẋ-X)(y-ӯ) ∑(X-Ẋ)2

b = 4.65 0.62 b = 7.5

Advanced Higher

STATISTICS

Third: calculate ‘a’ using the formula

a = 12.44 – (7.5 x 0.723) a = 7.02

a = ӯ - bẊ

a = ӯ - bẊ

Advanced Higher

STATISTICS

Fourth: figure out some ‘x’ & ‘y’ values

• To draw the straight line you have to calculate the values of ‘y’ for values of ‘x’ that you choose.• For example, if x = 0, then y = a + (b x 0)• For example, if x = 20, then y = a + (b x 20)

• To draw the straight line you have to calculate the values of ‘y’ for values of ‘x’ that you choose.• if x = 0, then y = 7.02 + (7.5 x 0) = 7.02• if x = 0.6, then y = 7.02 + (7.5 x 0.6) = 11.52• if x = 1.2, then y = 7.02 + (7.5 x 1.2) = 16.02

b = 7.5a = 7.02

if x = 0, y = 7.02if x = 0.6, y = 11.52if x = 1.2, y = 16.02

Advanced Higher

STATISTICS

Fifth: plot your straight line!

if x = 0, y = 7.02if x = 0.6, y = 11.52if x = 1.2, y = 16.02