KENDALL’S COEFFICIENT
OF CONCORDANCE ()
PRETEST- Problem
Kendall’s Coefficient of Concordance ()
Jegie 177
Problem
Three judges rated eight essays with the following results. Calculate the coefficient of
concordance for these data.
Judge
Essay
1
2
3
1
2
3
4
5
6
7
8
8
6
4
1
3
2
5
7
7
5
6
2
3
1
4
8
8
6
5
1
2
3
4
7
Solve the problem by getting the following requirements:
1).
Specific Problem
2).
Null Hypothesis
3).
Alternative Hypothesis
4).
Statistical Method Used
5).
Level of Significance
6).
Region of Rejection
7).
Computation of the value of W
8).
Decision
9).
Conclusion
PRETEST- Problem
Kendall’s Coefficient of Concordance ()
I -
Jegie 178
Specific Problem
Is there an agreement among the three judges in rating the eight essays?
II -
Null Hypothesis
There is no agreement among the three judges in rating the eight judges.
III -
Alternative Hypothesis
There is an agreement among the three judges in rating the eight judges.
IV -
Level of Significance:
 = 0.01
V -
Statistical Method Used:
Kendall’s Coefficient of Concordance W
VI -
Rejection Region
Df = n – 1
VII-
Df =7
X2
= 18.475
Computation
W = 0.94
VIII-
Df = 8 – 1
The computed X2
= 19.74
Decision
Since the computed value of W
= 0.94. the verbal interpretation is very high
correlation
and
the
computed
X2
= 19.74
is greater than
the
2
X
= 18.475 at 0.01 level of significance, therefore accept the alternative hypothesis.
IX-
Conclusion
There is an agreement among the three judges in rating the eight judges.
KENDALL’S COEFFICIENT OF
CONCORDANCE
Jegie 179
KENDALL’S COEFFICIENT OF CONCORDANCE ()
Definition
❖ If we wish to determine the relationship among three or more sets
of ranks, KENDALL’S COEFFICIENT OF CONCORDANCE ()
is used.
Formula

12  D2
=
m2 (n) ( n2 - 1 )
Where:
m
n
D
X
12
=
=
=
=
=
number of judges
number of individuals to be judged
 (Rank per row - x)
is the average
is constant
To check:
Total number of ranks = m ( n ) ( n + 1 )
2
Note:
Perfect agreement is indicated by
by
 = 0.
 = 1 and lack of agreement by
To test the significance:
Formula:
X2 = m (2-1)
Where:
m
n

Degrees of Freedom

= number of judges
= number of projects being rated
= Kendall’s coefficient of concordance
df = n - 1
KENDALL’S COEFFICIENT OF
CONCORDANCE
Jegie 180
Problem
Calculate the value of the coefficient of concordance by using the data consisting of the
ranking of 10 projects by five judges.
Individual Project
Project
Project
Project
Project
Project
Project
Project
Project
Project
Project
1
2
3
4
5
6
7
8
9
10
Judge 1
2
1
3
5
4
7
6
8
9
10
Judges Ranks
Judge 2
Judge 3
1
3
4
5
2
8
6
7
10
9
2
1
4
5
6
3
8
7
10
9
Judge 4
Judge 5
3
2
1
5
7
4
6
8
9
10
Solution
I
Specific Problem
Is there an agreement among these five judges in the ranking of the 10 projects?
II
Null Hypothesis
There is no agreement among these five judges in the ranking of the 10 projects.
III
Alternative Hypothesis
There is an agreement among these five judges in the ranking of the 10 projects.
IV
Level of Significance
 = 0.05
4
2
3
1
6
7
5
9
8
10
KENDALL’S COEFFICIENT OF
CONCORDANCE
Jegie 181
V
Statistical Method Used
Kendall Coefficient of Concordance
VI
Rejection Region
Degrees of freedom
(df)
(df)
=
=
=
n - 1
10 - 1
9
The critical value at 5 % level of significance:
VII
Individual
ProJect
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
X2
0.05
= 16.919
Computation
Judges Ranks
J1
J2
2
1
3
5
4
7
6
8
9
10
1
3
4
5
2
8
6
7
10
9
J3
2
1
4
5
6
3
8
7
10
9
J4
3
2
1
5
7
4
6
8
9
10
J5
4
2
3
1
6
7
5
9
8
10
Sum
of
Ranks
(1)
( Difference)
D
(4)
12
9
15
21
25
29
31
39
46
48
12-27.5 =15.5
9-27.5 =18.5
15-27.5 =12.5
21-27.5 = 6.5
25-27.5 = 2.5
29-27.5 = 1.5
31-27.5 = 3.5
39-27.5 =11.5
46-27.5 =18.5
48-27.5 =20.5
 Ranks
= 275
(2)
n
= 10
Get the Sum of Ranks per project or per row.
P1
P2
= 2+1+2+3+4
= 1+3+1+2+2
=
=
12
9
D2
(5)
240.25
342.25
156.25
42.25
6.25
2.25
12.25
132.25
342.25
420.25
 D2
=1,696.50
Procedure
1.
(D x D)
( Ranks – Mean)
etc.
KENDALL’S COEFFICIENT OF
CONCORDANCE
Jegie 182
)
2.
Sum all the ranks of the projects (  Ranks)
 Ranks = 12 + 9 + 15 + 21 + 25 + 29 + 31 + 39 + 46 + 48
 Ranks = 275
3.
Find the mean by dividing (  Ranks) by the total number of projects
Mean
 Ranks
=
Total Number of Projects being rated
4.
5.
Mean
=
275
10
Mean
=
27.50
Get the difference (D) by subtracting the Mean from the total ranks
per project as shown in Column with number (4). (Disregard the negative sign. )
D
=
Sum of Ranks - Mean
D1
D2
D3
Etc.
=
=
=
12 – 27.50
9 – 27.50
15 – 27.50
=
=
=
15.50
18.50
12.50
Square the difference ( D2 ) as shown in column with number (5).
(D1)2
=
(15.50) (15.50) =
240.25
(D2)2
=
(18.50) (18.50) =
342.25
(D3)2
=
(12.50) (12.50) =
156.25
KENDALL’S COEFFICIENT OF
CONCORDANCE
Jegie 183
6.
Find the total of D2 (  D2 ) .
 D2 = 240.25 + 342.25 + 156.25 + 42.25 + 6.25 + 2.25 + 12.25 + 132.25
+ 342.25 + 420.25
 D2 = 1,696.50
7.
Substituting the computed values to the given formula
 =
12 D2
m (n) ( n2 - 1)
2
Where:
m
n
D2
= number of judges
= no. of projects to be judged
= 1696.50
 =
12 ( 1 696.50 )
(5)2 ( 10 ) ( 10 2 - 1 )
 =
12 ( 1 696.50 )
25 ( 10 ) ( 100 -1 )
 =
20 358
25 (10) (99)
 =
20 358
24 750
 =
0.82
=
=
=
=
5
10
20 358
25 (10) (99)
0.82
It means there is a high agreement among these five judges in the ranking of
the ten projects since 0.82 falls between  0.71 to  0.90 as shown on page
27 which means high correlation or marked relationship.
8. Check your solution by getting the Total sum of ranks
To check:
Total sum of ranks = m ( n ) ( n + 1 )
2
Total sum of ranks = (5) (10) (10 + 1)
2
KENDALL’S COEFFICIENT OF
CONCORDANCE
Jegie 184
Total sum of ranks = (5) (10) (11)
2
Total sum of ranks =
Total sum of ranks = 275
✓
Note:
page 26.
550
2

is same as rs in the analization of the interpretation. Please refer to
❖ To test the significance:
Formula:
X2 = m (n-1)

m
n
number of judges
number of projects being rated
Kendall’s coefficient of concordance
Where:

=
=
=
=
=
=
5
10
0.82
9. Substitute the values to the given formula.
X2 = m (n-1) W
X2 = (5) (10-1) (0.82)
X2 = (5) (9) (0.82)
X2 = 36.90
VIII
Decision
Since the computed value X2 = 36.90 is greater than X2 0.05 = 16.919, therefore,
accept the alternative hypothesis that it is significant and there exist relationship in
the judging of the ten projects by the five judges.
IX
Conclusion
There is a significant agreement among these five judges in the ranking of the ten
projects.
POST TEST- Problem
Kendall’s Coefficient of Concordance (W)
Jegie
185
Problem
Four judges (parole board members) rank eight convicts on “parole readiness.” By
using the coefficient of concordance, indicate the degree of consistency of the judges.
Convict
1
2
3
4
5
6
7
8
1
Judge
2
3
4
1
2
3
4
5
6
7
8
1
4
3
2
6
5
7
8
1
3
2
4
5
6
8
7
1
2
4
3
5
7
6
8
Solve the problem by getting the following requirements:
1).
Specific Problem
2).
Null Hypothesis
3).
Alternative Hypothesis
4).
Statistical Method Used
5).
Level of Significance
6).
Region of Rejection
7).
Computation of the value of
8).
Decision
9).
Conclusion

POST TEST-Answers
Kendall’s Coefficient of Concordance (W)
I -
Jegie
186
Specific Problem
Is there any consistency in the ranking of the four judges in the parole readiness of the
eight convicts?
II -
Null Hypothesis
There is no consistency in the ranking of the four judges in the parole readiness of the
eight convicts.
III -
Alternative Hypothesis
There is consistency in the ranking of the four judges in the parole readiness of the eight
convicts.
IV -
Level of Significance:
Assume  = 0.01
V -
Statistical Method Used:
VI -
Rejection Region
Use Kendall Coefficient of Concordance
Degrees of freedom
(df) = n – 1 = 8 – 1 = 7
The critical value at 0.01 level of significance X2 0.01 = 18.475.
Appendix D
VII-
Computation
 = 0.92
VIII-
Please refer to
X2 obs
= m(n–1)(w)
= 4 ( 8 – 1 ) ( 0.92 )
= 25.76
Decision
Since the observed absolute value of chi-square x 2 obs = 25.76 is greater than the critical
value at 0.01 level of significance x2 0.01 = 18.475, therefore accept the alternative
hypothesis.
IX –
Conclusion
There is a significant relationship on consistency in the ranking of the four judges in the
parole readiness of the eight convicts.
PRACTICE Problem
Kendall’s Coefficient of Concordance (W)
Jegie
187
Three judges rated eight essays with the following results. Test if there is a significant
relationship in judging the eight essays by the three judges. Level of significance is 0.01.
Essay
1
2
3
4
5
6
7
8
JUDGE
A
B
C
8
6
4
1
3
2
5
7
7
5
6
2
3
1
4
8
8
6
5
1
2
3
4
7
Please be guided by the nine steps in solving the problem.
1). Specific Problem
2). Null Hypothesis
3). Alternative Hypothesis
4). Statistical Method Used
5). Level of Significance
6). Region of Rejection
7). Computation of the Value of w
8). Decision
9). Conclusion
PRACTICE Problem - Answers
Kendall’s Coefficient of Concordance (W)
I -
Jegie
188
Specific Problem
Is there an agreement among these three judges in the ranking of the eight essays?
II -
Null Hypothesis
There is no significant agreement among the three judges in the ranking of the eight
essays.
III -
Alternative Hypothesis
There is an agreement among the three judges in the ranking of the eight essays.
IV -
Level of Significance:
assume  = 0.05
V -
Statistical Method Used
Use Kendall Coefficient of Concordance
VI -
Rejection Region
Degrees of freedom (df) = n - 1
= 8 –1
=7
The critical value at 5% level of significance X2
VII-
= 3.355 for two-tailed test
Computation
 = 0.94
VIII-
0.05
x2 = 19.74
Decision
Since the computed value of x2 = 19.74 is greater than x20.05 = 14.067, therefore reject
the null hypothesis and accept the alternative hypothesis.
IX –
Conclusion
There is an agreement among the three judges in the ranking of the eight essays.