Spearman`s Rho

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Spearman Rho Correlation
Introduction
 Spearman's rank correlation coefficient or Spearman's rho is named
after Charles Spearman
 Used Greek letter ρ (rho) or as rs (non- parametric measure of
statistical dependence between two variables)
 Assesses how well the relationship between two variables can be
described using a monotonic function
 Monotonic is a function (or monotone function) in mathematic that
preserves the given order.
 If there are no repeated data values, a perfect Spearman correlation of
+1 or −1 occurs when each of the variables is a perfect monotone
function of the other
1
Spearman Rho Correlation
 A correlation coefficient is a numerical measure or index of the amount
of association between two sets of scores. It ranges in size from a
maximum of +1.00 through 0.00 to -1.00
 The ‘+’ sign indicates a positive correlation (the scores on one variable
increase as the scores on the other variable increase)
 The ‘-’ sign indicates a negative correlation (the scores on one variable
increase, the scores on the other variable decrease)
2
Spearman Rho Correlation
Calculation
 Often thought of as being the Pearson correlation coefficient between
the ranked (relationship between two item) variables
 The n raw scores Xi, Yi are converted to ranks xi, yi, and the
differences di = xi − yi between the ranks of each observation on the
two variables are calculated
 If there are no tied ranks, then ρ is given by this formula:
3
Spearman Rho Correlation
Calculation
 Often thought of as being the Pearson correlation coefficient between
the ranked (relationship between two item) variables
 The n raw scores Xi, Yi are converted to ranks xi, yi, and the
differences di = xi − yi between the ranks of each observation on the
two variables are calculated
 If there are no tied ranks, then ρ is given by this formula:
4
Spearman Rho Correlation
Interpretation
 The sign of the Spearman correlation indicates the direction of
association between X (the independent variable) and Y (the
dependent variable)
 If Y tends to increase when X increases, the Spearman
correlation coefficient is positive
 If Y tends to decrease when X increases, the Spearman
correlation coefficient is negative
 A Spearman correlation of zero indicates that there is no
tendency for Y to either increase or decrease when X increases
5
Spearman Rho Correlation
Interpretation cont…/
 Alternative name for the Spearman rank correlation is the "grade
correlation” the "rank" of an observation is replaced by the "grade"
 When X and Y are perfectly monotonically related, the Spearman
correlation coefficient becomes 1

A perfect monotone increasing relationship implies that for any two
pairs of data values Xi, Yi and Xj, Yj, that Xi − Xj and Yi − Yj always
have the same sign
6
Spearman Rho Correlation
Example # 1


Calculate the correlation between
the IQ of a person with the number
of hours spent in the class per week
Find the value of the term d²i:
1. Sort the data by the first column
(Xi). Create a new column xi and
assign it the ranked values
1,2,3,...n.
2. Sort the data by the second column
(Yi). Create a fourth column yi and
similarly assign it the ranked values
1,2,3,...n.
3. Create a fifth column di to hold the
differences between the two rank
columns (xi and yi).
IQ, Xi
Hours of class per
week, Yi
106
7
86
0
100
27
101
50
99
28
103
29
97
20
113
12
112
6
110
17
7
Spearman Rho Correlation
Example # 1 cont…/
4. Create one final column to hold the value of column di squared.
IQ
(Xi )
Hours of class per week
(Yi)
rank xi
rank yi
di
d²i
86
0
1
1
0
0
97
20
2
6
-4
16
99
28
3
8
-5
25
100
27
4
7
-3
9
101
50
5
10
-5
25
103
29
6
9
-3
9
106
7
7
3
4
16
110
17
8
5
3
9
112
6
9
2
7
49
113
12
10
4
6
36
8
Spearman Rho Correlation
Example # 1- Result
 With d²i found, we can add them to find  d²i = 194
 The value of n is 10, so;
ρ=
1- 6 x 194
10(10² - 1)
ρ=
−0.18
 The low value shows that the correlation between IQ and hours spent
in the class is very low
9
Spearman Rho Correlation
Example # 2:
5 college students have the following rankings in Mathematics and
Science subject. Is there an association between the rankings in
Mathematics and Science subject.
Student
Ashley
David
Owen
Steven
Frank
Mathematic
class rank
1
2
3
4
5
Science
class rank
5
3
1
4
2
10
Spearman Rho Correlation
Example # 2 cont…/
 Compute Spearman Rho
n= number of paired ranks
d= difference between the paired ranks
(when two or more observations of one variable are the same, ranks
are assigned by averaging positions occupied in their rank order)
11
Spearman Rho Correlation
Example # 2 cont…/
Maths Rank
Science Rank
X-Y
(X-Y) 2
X
Y
D
d2
1
5
-4
16
2
3
-1
1
3
1
2
4
4
4
0
0
5
2
3
9
30
12
Spearman Rho Correlation
Example # 2- Result
Result: ρ = -0.5
There is a moderate negative correlation between
the Math and Science subject rankings of students
Students who rank high as compared to other
students in their Math subject generally have lower
Science subject ranks and those with low Math
rankings have higher Science subject rankings than
those with high Math rankings.
(The formula for Pearson r and Spearman rho are
equivalent when there are no tied)
13
Research Article
 Title:
“Motivation and Attitude in Learning English among UiTM
Students in the Northern Region of Malaysia”
 Purpose:
To describe the relationship between the students’ motivation and
attitude to their English Language performance
14
Article Cont…/
Method:
 Used a correlational research design
 Independent variables:
Motivation, attitude, and personal characteristics variables, as
measured by a self-report questionnaire
 Dependent variable:
English Language performance, measured by the UiTM
Preparatory English (BEL100) examination result
15
Article Cont…/
Method:
 Sampling Design
The subjects were 139 students from the Perlis Campus, 248 from the
Kedah Campus and 233 from the Pulau Pinang Campus.
 The selection criterion used in attaining the samples was to choose
those students who had just received their BEL100 examination result
regardless of their status whether as the first timer or repeater for that
particular paper.
16
Article Cont…/
Method:
 Questionnaire
Research instrument used was questionnaire that comprised questions
on personal characteristics, motivation and attitudes.
The instrument was adopted and adapted from Gardner andLambert
(1972) so that it is more appropriate, intelligible and meaningful for the
sample concerned.
The reliability test of the instrument produced a Cronbach Alfa of 0.757,
which was satisfactory and acceptable.
17
Article Cont…/
Method:
 Data Analysis
The data collected were computed and analyzed using the SPSS 12.
Each student’s score on the questionnaire was matched to his or her
BEL100 examination grade.
The statistical procedures used in this study were the descriptive
statistics – mean & standard deviation scores, frequency & percentage,
t-test, Spearman Rho Rank-Order Correlation Coefficient & ANOVA.
18
Article Cont…/
Result- Correlation between motivation in learning English &
English language performance
 Spearman Rho rank-order correlation coefficient test
Very weak relationship between Intrinsic Motivation & English
language performance, which is -.020.
 One-way ANOVA test
Intrinsic Motivation: Critical value of F at alpha = .05 is 2.70. The
obtained F value is 1.63, which is less than the critical value.
Extrinsic Motivation: Computed value for the correlation test
which is .043 and the obtained value of F for the one-way
ANOVA, 2.39.
This justifies there is no significant differences between Overall
Motivation (Intrinsic Motivation & Extrinsic Motivation) & English
language performance.
19
Article Cont…/
Result- Attitude in learning English & English language performance
 Spearman Rho rank-order correlation coefficient test
Exists a significant correlation (alpha = .01) between the attitude in
learning English & English language performance, which is -.152.
 One-way ANOVA test
F value from the one-way ANOVA test is 6.66, which is greater than the
critical value.
 Mean scores
Respondents received A - M = 3.06, B- M = 2.99, C - M = 2.93, D - M =
2.80), it can be concluded that the respondents who obtained an A
(high achievers) have better attitude in learning English compared to
the low achievers.
This justifies that there is a significant difference between the attitude in
learning English & English language performance.
20
Spearman Rho Correlation
References

http://en.wikipedia.org/wiki/Spearman's_rank_correlation_coefficient

http://davidmlane.com/hyperstat/A62436.html

http://www.wellesley.edu/Psychology/Psych205/Spearman.html

www.ccsenet.org/journal.html

www.statisticallysignificantconsulting.com

http://www.wikihow.com/Calculate- Spearman's-Rank-Correlation-Coefficient

http://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient
21
Thank you
Prepared by:
Rubiyatul Huda
Norsafrina
Salina
22
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