Transcript 38 Correlation - WordPress.com

“Teach A Level Maths”
Vol. 2: A2 Core Modules
Product Moment Correlation
Coefficient
© Christine Crisp
Product Moment Correlation Coefficient
Statistics 1
AQA
EDEXCEL
OCR
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Product Moment Correlation Coefficient
We used the following data for finding the equation of a
regression line
x
y
1
5
2
3
3
1
The diagram of the data looked like this
Enter the data again into your calculator and this
time look under regression for a value labelled r.
r  1
Product Moment Correlation Coefficient
r  1
r is the product moment correlation coefficient.
The product moment correlation coefficient measures the
scatter of the data. Since these points all lie on the
regression line we have perfect ( negative ) correlation.
Product Moment Correlation Coefficient
NOTES
•
The value of r, the product moment correlation
coefficient, always satisfies 1  r  1.
•
If 0  r  1 we have positive correlation and the
regression line slopes from bottom left to top right.
•
If 1  r  0 we have negative correlation and
the regression line slopes from top left to bottom
right.
Product Moment Correlation Coefficient
•
If r = 1 or r = +1 we have perfect correlation and
all the points lie on the least squares regression line.
•
For values of r close to 1 or +1 the correlation is
strong and the points lie close to the line.
•
As values of r move towards zero, the correlation
becomes weak. The points scatter further and
further from the line.
Strong correlation does not necessarily mean there is
a causal relationship
For example, the data we met earlier giving the
populations of birds in woodland and farmland areas
showed very little scatter about the regression line and
hence strong correlation. However, it is most unlikely
that the lack of birds in the woods causes a lack of birds
on farms. Numbers in both cases are likely to be linked
to availability of food.
Product Moment Correlation Coefficient
Exercise
Draw sketches showing about 10 points and a regression
line for each of the following:
(a) Data with perfect positive correlation
(b) Data with strong negative correlation
(c) Data with weak positive correlation
(d) Data with weak negative correlation
( Work with a partner if you like and do 2 each. Help
each other. )
Product Moment Correlation Coefficient
Solutions:
Our diagrams are not going to look exactly alike. Try to
decide if they have the same important feature.
(a) Data with perfect positive correlation
r 1
(b) Data with strong negative correlation
( r   0  93 )
Product Moment Correlation Coefficient
Solutions:
(c) Data with weak positive correlation
( r  0  46 )
(d) Data with weak negative correlation
( r   0  44 )
Product Moment Correlation Coefficient
For the height and foot length data,
Foot
length
(cm)
Foot length and height
of UK children
Height (cm)
the equation of the y on x regression line shown is
y  0  14x  1  98
and the product moment correlation coefficient is r  0 75
This value shows strong positive correlation. Taller
children have bigger feet!
Product Moment Correlation Coefficient
Exercise
1. Find the equation of the least squares regression line
of y on x, and the value of the product moment
correlation coefficient for the following sets of data.
For each set, interpret the value choosing from the
following words: “strong”,”weak”,”positive”,”negative”.
(a)
x
y
10
4
15
9
12
8
9
5
16
14
7
4
21
9
9 11
11 14
(b)
x
1
2
3
4
5
6
7
8
y
21
24
12
17
12
11
7
3
Answer: (a)
r  0 42
Weak, positive
(b) r   0  91 Strong, negative
Product Moment Correlation Coefficient
2(a) Using the bean data that we met before,
find the product moment correlation coefficient.
Weight (g)
0·
7
1·
1·
2
2·
0·
9
2·
1·
4
2·
1·
2
2·
1·
1
2·
1·
0
2·
0·
9
1·
1·
0
2·
0·
8
1·
Source: O.N.Bishop7
2
0
3
4
2
0
9
1
6
Length (cm)
(b) What does the answer to (a) tell you?
( You need to answer by using the mathematical
words AND referring to the beans. )
Answer:
(a)
r  0 90
(b) There is a strong, positive correlation between weight
and length. This means that the heavier beans are
longer.
Product Moment Correlation Coefficient
If you are not given raw data and you need to find the
product moment correlation coefficient, you can use
your formula booklet with summary data.
The formula is
r
S xy
S xx S yy
where, as before
S xy   xy 
S xx   x 
2
 x 
2
n
and
 x  y 
n
S yy   y 
2
 y 
2
n
The formulae booklets also give r in a simplified form
but it’s not very simple!
Product Moment Correlation Coefficient
e.g.1 Find the value of the correlation coefficient for
10 pairs of observations relating 2 variables x and
y where:
x  29
y  42


2
x
  397
Solution:
S xy   xy 
S xx   x 2 
S yy   y 2 

y 2  6728
 xy 792
S xy
r
S xx S yy
 x  y 
n
 x 
2
( 29)( 42)
 670 2
 792
10
n
29 2
 397 
 312 9
10
n
42 2
 6551 6
 6728 
10
 y 2
Product Moment Correlation Coefficient
e.g.1 Find the value of the correlation coefficient for
10 pairs of observations relating 2 variables x and
y where:
x  29
y  42


2
x
  397
S xx  312 9
S xy  670 2
r

S xy
S xx S yy

 xy 792
y 2  6728
S yy  6551 6
670  2
312  9  6551 6
 0  47 (2 d . p.)
Product Moment Correlation Coefficient
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
Product Moment Correlation Coefficient
SUMMARY
•
The product moment correlation coefficient, r,
measures the scatter of data.
•
The value of r, always satisfies 1  r  1.
•
If 0  r  1 we have positive correlation and the
regression line slopes from bottom left to top right.
•
If 1  r  0 we have negative correlation and
the regression line slopes from top left to bottom
right.
Product Moment Correlation Coefficient
•
If r = 1 or r = +1 we have perfect correlation and
all the points lie on the least squares regression line.
•
For values of r close to 1 or +1 the correlation is
strong and the points lie close to the line.
•
As values of r move towards zero, the correlation
becomes weak. The points scatter further and
further from the line.
Strong correlation does not necessarily mean there is
a causal relationship
For example, the data we met earlier showed a high
correlation between the number of birds in woodland and
farmland areas, but it is most unlikely that the lack of
birds in the woods causes a lack of birds on farms.
Numbers in both cases are likely to be linked to
availability of food.
Product Moment Correlation Coefficient
For the height and foot length data,
Foot length and height of UK
children
the equation of the y on x regression line shown is
y  0  14x  1  98
and the product moment correlation coefficient is
r  0 75
This value shows strong positive correlation. Taller
children have bigger feet!