THE CORRELATION COEFFICIENT - cankaya.edu.tr

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THE CORRELATION

COEFFICIENT

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More Statistical Notation

Correlational analysis requires scores from two

variables. X stands for the scores on one

variable and Y stands for the scores on the

other variable. Usually, each pair of XY scores is

from the same participant.

3

Correlation Coefficient

• A correlation coefficient is the descriptive

statistic that, in a single number, summarizes

and describes the important characteristics in

a relationship

• It does so by simultaneously examining all

pairs of X and Y scores

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Understanding Correlational Research

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Drawing Conclusions

• The term correlation is synonymous with

relationship

• However, the fact that there is a relationship

between two variables does not mean that

changes in one variable cause the changes in

the other variable

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Plotting Correlational Data

• A scatterplot is a graph that shows the

location of each data point formed by a pair

of X-Y scores

• When a relationship exists, as the X scores

increase, the vertical height of the data points

changes, indicating that the Y scores are

changing

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Strong Relationship

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No Relationship

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Types of Relationships

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Linear Relationships

• In a linear relationship as the X scores

increase, the Y scores tend to change in only

one direction

– In a positive linear relationship, as the scores

on the X variable increase, the scores on the Y

variable also tend to increase

– In a negative linear relationship, as the scores

on the X variable increase, the scores on the Y

variable tend to decrease

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• When values on two variables tend to go in the

same direction, we call this a direct relationship.

• The correlation between children’s ages and heights

is a direct relationship.

• That is, older children tend to be taller than younger

children.

• This is a direct relationship because children with

higher ages tend to have higher heights.

Direct Relationship

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• When values on two variables tend to go in opposite

directions, we call this an inverse relationship.

• The correlation between students’ number of

absences and level of achievement is an inverse

relationship.

• That is, students who are absent more often tend to

have lower achievement.

• This is an inverse relationship because children with

higher numbers of absences tend to have lower

achievement scores.

Inverse Relationship

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A Scatterplot of a Positive Linear

Relationship

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A Scatterplot of a Negative Linear

Relationship

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Nonlinear Relationships

In a nonlinear, or curvilinear, relationship, as

the X scores change, the Y scores do not tend

to only increase or only decrease: At some point,

the Y scores change their direction of change.

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A Scatterplot of a Nonlinear

Relationship

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Question...

• Students with higher math scores tend to obtain

higher reading scores. Likewise, students with lower

math scores tend to obtain lower reading scores. Is

the relationship between math and reading scores

direct or inverse?

Direct Inverse

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Question...

• Students with high math scores tend to have higher

reading scores. Likewise, students with low math scores

tend to have lower reading scores. Is the correlation

coefficient between students’ math and reading scores

negative or positive?

– +

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Strength of the Relationship

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Strength

• The strength of a relationship is the extent to

which one value of Y is consistently paired with one

and only one value of X

• The larger the absolute value of the correlation

coefficient, the stronger the relationship

• The sign of the correlation coefficient indicates the

direction of a linear relationship

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Correlation Coefficients

• Correlation coefficients may range between -

1 and +1. The closer to

1 (-1 or +1) the coefficient is, the stronger

the relationship; the closer to 0 the

coefficient is, the weaker the relationship.

• As the variability in the Y scores at each X

becomes larger, the relationship becomes

weaker

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Computing Correlational Coefficients

23 ])()([])()([

))(()(

2222 YYNXXN

YXXYNr

Pearson Correlation Coefficient

• The Pearson correlation coefficient

describes the linear relationship between two

interval variables, two ratio variables, or one

interval and one ratio variable. The formula

for the Pearson

r is

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Spearman Rank-Order

Correlation Coefficient

)1(

)(61

2

2

NN

Drs

• The Spearman rank-order correlation

coefficient describes the linear relationship

between two variables measured using ranked

scores. The formula is

where N is the number of pairs of ranks and D is

the difference between the two ranks in each pair.

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X Y

1 8

2 6

3 6

4 5

5 1

6 3

Example 1

• For the following data set

of interval/ratio scores,

calculate the Pearson

correlation coefficient.

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])()([])()([

))(()(

2222 YYNXXN

YXXYNr

Example 1


• First, we must determine each X2, Y2, and XY

value. Then, we must calculate the sum of X,

X2, Y, Y2, and XY.

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X X2 Y Y2 XY

1 1 8 64 8

2 4 6 36 12

3 9 6 36 18

4 16 5 25 20

5 25 1 1 5

6 36 3 9 18

X = 21 X 2 = 91 Y = 29 Y 2 = 171 XY = 81

Example 1


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88.0374.139

123

]185[]105[

609486

])29()171(6[])21()91(6[

)29)(21()81(6

])()([])()([

))(()(

22

2222

YYNXXN

YXXYNr

Example 1


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X Y

1 5

2 2

3 6

4 4

5 3

6 1

Example 2

• For the following data set

of ordinal scores, calculate

the Spearman

rank-order correlation

coefficient.

30

)1(

)(61

2

2

NN

Drs

X Y D

1 5 -4

2 2 0

3 6 -3

4 4 0

5 3 2

Example 2

Spearman Correlation Coefficient

• First, we must calculate the difference

between the ranks for each pair.

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X Y D D2

1 5 -4 16

2 2 0 0

3 6 -3 9

4 4 0 0

5 3 2 4

D2 = 29

Example 2

Spearman Correlation Coefficient • Next, each D value is

squared.

• Finally, the sum of

the D2 values is

computed.

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45.045.11

120

1741

)125(5

)29(61

)1(

)(61

2

2

NN

Drs

Example 2

Spearman Correlation Coefficient

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• The illustration on the right

involves scores on a reading

readiness test, and scores

later obtained by these same

students on a reading

achievement test.

Reading

ReadinessScores

Reading

AchievementScores

Todd 10 19

Andrea 16 25

Kristen 19 23

Luis 22 31

Scott 28 27

Example 3

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• In summary, our six

values used to calculate

the correlation

coefficient are…

X2 X XY Y Y2

100 10 190 19 361

256 16 400 25 625

361 19 437 23 529

484 22 682 31 961

784 28 756 27 729

1985 95 2465 125 3205

Example 3

2465

3205

1985

125

95

5

2

2

XY

Y

X

Y

X

n

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The correlation between these students reading readiness

scores and later reading achievement scores is 0.75

75.0600

450

000,360

450

400900

450

625,15025,1690259925

875,11325,12

125320559519855

1259524655

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r

r

r

r

rXReadingReadinessScores

YReading

AchievementScores

Todd 10 19

Andrea 16 25

Kristen 19 23

Luis 22 31

Scott 28 27

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• Find the correlation

between spelling and

vocabulary

XY

Y

X

Y

X

n

2

2

XSpelling

YVocabulary

Sandra 8 10

Neil 5 6

Laura 4 7

Jerome 1 3

Example 4

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• You should get these

values:

141

194

106

26

18

4

2

2

XY

Y

X

Y

X

n

X2 X XY Y Y2

64 8 80 10 100

25 5 30 6 36

16 4 28 7 49

1 1 3 3 9

106 18 141 26 194

Example 4

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Example 4

2222 YYnXXn

YXXYnr

96.0100

96

000,10

96

100100

96

676776324424

468564

261944181064

26181414

22

r

r

r

r

rX

Spelling

YVocabulary

Sandra 8 10

Neil 5 6

Laura 4 7

Jerome 1 3

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What is a high correlation?

• Correlations above 0.80 generally are

regarded as high correlations

• Correlations between 0.50 and 0.80 are

typically considered a medium or moderate

correlations

• Correlations below 0.50 typically are regarded

as low correlations

• These ranges are not exact. Various people

associate somewhat different ranges with

high, moderate, and lower correlations.

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Restriction of Range

Restriction of range arises when the range

between the lowest and highest scores on one

or both variables is limited. This will reduce the

accuracy of the correlation coefficient,

producing a coefficient that is smaller than it

would be if the range were not restricted.

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Test 1(X)

Test 2(Y)

96 95

92 86

90 9888 91

86 93

85 8583 96

82 87

81 9080 92

78 94

77 88

76 9174 84

73 90

71 8169 83

67 8963 88

Heterogeneous students...

• Here are the scores of 19

students who obtained a

wide range of scores on two

tests

• When calculated, the

correlation between scores

on the two tests is 0.49

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Test 1(X)

Test 2(Y)

96 95

92 86

90 9888 91

86 93

85 8583 96

82 87

81 9080 92

78 94

77 88

76 9174 84

73 90

71 8169 83

67 8963 88

Homogeneous students... • Here are the scores of 19

students who obtained a wide

range of scores on two tests

• When calculated, the

correlation between scores

on the two tests is 0.49

• However, if we use only

students who obtained a

smaller range of scores on

Test 1, the correlation

between scores on the two

tests becomes 0.14

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Test 1(X)

Test 2(Y)

96 95

92 86

90 9888 91

86 93

85 8583 96

82 87

81 9080 92

78 94

77 88

76 9174 84

73 90

71 8169 83

67 8963 88

When students are more similar,

correlations are lower

• In this example, the

correlation became lower

because scores on the first

test became more similar.

The number of students also

did decrease, but the

number of students actually

does not decrease the

correlation coefficient

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Question...

• The heights and weights of 100 students

enrolled in 1st through 8th grades were

obtained. The correlation between heights

and weights was found to be 0.70. What

would the correlation had been if it were

based on 100 students enrolled in the 5th

grade?

Higher than 0.70

Perhaps Close to 0.70

Lower than 0.70

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Scatterplots -1

X Y

1 2

1 2

1 2

3 5

3 5

3 5

5 8

5 8

5 8

• Calculate the Pearson


and draw a scatterplot for

the data

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Scatterplots -1

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6

• r= +1.00

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Scatterplots -2 • Calculate the Pearson



the data

X Y

1 8

1 8

1 8

3 5

3 5

3 5

5 2

5 2

5 2

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6

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Scatterplots -2

• r= -1.00

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6

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the data

X Y

1 1

1 2

1 2

3 4

3 5

3 5

5 7

5 8

5 8

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Scatterplots -3

• r= + .98

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6

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the data

X Y

1 9

1 6

1 3

3 8

3 6

3 3

5 7

5 5

5 1

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Scatterplots -4

• r= -.28

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6

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the data

X Y

1 3

1 5

1 7

3 3

3 5

3 7

5 3

5 5

5 7

54

Scatterplots -5

• r= 0

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6

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