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Correlation
Indicates the relationship between two dependent variables (x and y)
Symbol: r (Pearson correlation coefficient)
-1< r < 1
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Positive Correlation
As value of variable X increases, value of variable Y increases
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Strong Positive Corr.
r = .80
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Strong positive correlation:
- low variability of the data
- one variable acts as a good predictor of the second variable.
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Weaker Positive Corr.
r = .50
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Weaker correlations:
- More variability in the data
- Less predictability
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Negative correlation
As value of variable X increases, value of variable Y decreases.
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Strong negative Corr.
r = -.80
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Social Activities
Grades
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# of bananas eaten
StatisticsAbility
No Correlation r = 0
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Variables measured on different scales
e.g. Height and weight
Correlation formula converts the scores into z-score to make them comparable
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Limit of correlation is that itidentifies a relationship, but is NOT identifying cause
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r
n XY X Y
n X X n Y Y
2 2 2 2
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r
n XY X Y
n X X n Y Y
2 2 2 2
Multiply each value of X by its corresponding Y.
Add the products.
Multiply sum by n
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r
n XY X Y
n X X n Y Y
2 2 2 2
Add all the X values. Add all the Y values.
Multiply the two sums.
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r
n XY X Y
n X X n Y Y
2 2 2 2
Variation of the formula for SS for X
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r
n XY X Y
n X X n Y Y
2
22
2
Variation of the formula for SS of Y
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Quiz One Quiz Four
(X) (Y)
28 30.5
28 29.5
30 39
29 36.5
21 30.5
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Quiz One Quiz Four
(X) (Y)
28 30.5
28 29.5
30 39
29 36.5
21 30.5
X = 136 Y = 166
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Quiz One Quiz Four
(X) X2 (Y)
28 784 30.5
28 784 29.5
30 900 39
29 841 36.5
21 441 30.5
X2 = 3750
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Quiz One Quiz Four
(X) (Y) Y2
28 30.5 930.25
28 29.5 870.25
30 39 1521
29 36.5 1332.25
21 30.5 930.25
Y2 = 5584
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Quiz One Quiz Four
(X) (Y) (X)(Y)
28 30.5 854
28 29.5 826
30 39 1170
29 36.5 1058.5
21 30.5 640.5
XY = 4549
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X = 136 Y = 166
X2 = 3750 Y2 = 5584
XY = 4549
n = 5
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r
5 4549 136 166
5 3750 136 5 5584 1662 2
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22745 22576
18750 18496 27920 27556
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169
254 364
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169
92456
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169
304 07.
r .55
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Critical Value:
df = n -- 2
r crit. = .878
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df Critical r
3 .878
5 .754
10 .576
20 .423
50 .273
100 .195
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How much variability of X and Y is jointly shared?
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How much of the variability of X can be accounted for by variability in Y?
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r2 = the strength or the amount of shared variability
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r r2
.9 .81 or 81%
.8 .64 or 64%
.7 .49 or 49%
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df Critical r
3 .878
5 .754
10 .576
20 .423
50 .273
100 .195
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y bx a
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b
n xy x y
n x x
2 2
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y bx a
y bx a
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y bx a
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# of absences test grade 8 35 0 48 3 43 2 41 5 39 6 36 X = 24 Y = 242
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# of absences test grade 8 64 35 1225 0 0 48 2304 3 9 43 1849 2 4 41 1681 5 25 39 1521 6 36 36 1296 X2= 138 Y2 = 9876
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# of absences test grade 8 35 280 0 48 0 3 43 129 2 41 82 5 39 195 6 36 216 xy = 902
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r
n XY X Y
n X X n Y Y
2
22
2
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r
6 902 24 242
6 138 24 6 9876 2422 2
( )
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396
417 59
95
.
.r
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b
n xy x y
n x x
2
2
b
5412 5808
6 138 242
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57.1
252
396
b
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y bx a
y bx a
46
24
X
X
33.406
242
y
y
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a )4)(57.1(33.40
40 33 6 28
46 61
. .
.
a
a
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y bx a
y
y
y
157 5 46 61
7 85 46 61
38 76
. .
. .
.