Kendall’s Coefficient of Concordance

W

Kendall’s W only gives the degree of association or agreement among the ranks assigned by different judges or respondents on different objects or attributes.

From the data above:

D2 = 354m = 9N = 8 W=0.94

Three Judges rated eight essays with the following results. Calculate the coefficient of Concordance for these data.

EXAMPLE

ESSAY

ABCDEFGH

Judge 1

Judge 2

Judge 3

8 7 86 5 64 6 51 2 13 3 22 1 35 4 47 8 7

R

23171548613

22108

D

9.53.51.59.55.57.50.5

8.5

D2

90.2512.252.2590.2530.2556.250.2572.25

S =354

D2=∑(R-Ā)2SUM OF RANKS R-A

Ā=108 / 8=13.5

FORMULA for Kendall’s W Coefficient:

𝑊=12𝑆

𝑚2(𝑁 )(𝑁 2−1)WHERE: S=Sum of squares of the R- from the Ā m=Number of judges or respondents ranking the objects or attributes.

N=Number of attributes or objects that is evaluated by judges or respondents.

From the data above:

S = 354m = 3N = 8 W=0.94

From the data above:

D2 = 354m = 9N = 8 W=0.94

Therefore, The size of this coefficient of concordance indicates that there is a very high agreement among these three judges in the ranking of the eight essays.

NOTE: Kendall’s W only gives the

degree of association or agreement among the ranks assigned by different judges or respondents on different objects or attributes.

However, the significance of this W should be tested through either critical X2 or F values.

From the data above:

D2 = 354m = 9N = 8 W=0.94

Five consumers of similar profile rated the eight different colors of packages for biscuit to find out the most preferred one. Calculate the coefficient of concordance for these data.

EXAMPLE

From the data above:

D2 = 354m = 9N = 8 W=0.94

The null hypothesis there is no significant agreement among the judges (or respondents) in the ranking of different color packages. The alternative hypothesis there is a significant agreement among the judges (or respondents) in the ranking of different color packages.

Formulate a null hypothesis and an alternate hypothesis:

COLOR CONSUMERS R D D2

1 2 3 4 5 SUM OF RANKS R-A D2=∑(R-Ā)2

red 1 2 1 1 2 7 15.5 240.25orange 2 1 2 3 4 12 10.5 110.25Yellow 3 4 3 2 6 18 4.5 20.25green 4 3 6 6 7 26 3.5 12.25blue 5 6 4 4 3 22 0.5 0.24pink 6 5 8 7 8 34 11.5 132.25violet 7 8 7 5 5 32 9.5 90.25black 8 7 5 8 1 29 6.5 42.25

∑=180 S=648 Ā=180/8=22.5

From the data above:

D2 = 354m = 9N = 8 W=0.94W=0.62

From the data above:

S = 648m = 5N = 8

From the data above:

D2 = 354m = 9N = 8 W=0.94

The size of this coefficient of concordance indicates that there is a moderate agreement among these five judges in ranking the eight package colors.

Interpretation of W.

From the data above:

D2 = 354m = 9N = 8 W=0.94

Find out the critical value:We can find out the critical value by

referring to the table 1, which gives values of “S” for W’s significance of 0.05 and 0.01 levels. Please note that this table is applicable only when ‘m’ ranges from 3 to 20 and ‘n’ ranges from 3 to 7.In case of large samples, that is, when the number of objects or attributes to be ranked are evaluated is greater than 7, we have the option of finding out the critical value either through computation of X2 distribution values or F-distribution values.

From the data above:

D2 = 354m = 9N = 8 W=0.94

In the case of X2 distribution the value will be:

Where: m = number of judges respondentsN = number of objects or attributes being rankedW = Kendall’s coefficient of concordance

From the data above:

D2 = 354m = 9N = 8 W=0.94

𝑿𝟐=𝒎 (𝑵−𝟏 )𝑾

X2=5(8-1)0.62

X2=21.7

From the data above: W = 0,62 m = 5 N = 8

From the data above:

D2 = 354m = 9N = 8 W=0.94

Locate the critical chi value at 0.05 level:

Get the degrees of freedom: N-1 -> 8-1=7

1 2.706 3.841 5.024 2 4.605 5.991 7.378 3 6.251 7.815 9.348 4 7.779 9.488 11.143 5 9.236 11.070 12.833 6 10.645 12.592 14.449 7 12.017 14.067 16.013 8 13.362 15.507 17.535 9 14.684 16.919 19.023 10 15.987 18.307 20.483

0.90 0.95 0.975

V=14.07

From the data above:

D2 = 354m = 9N = 8 W=0.94

Compare the calculated value and critical value:

X2=21.7 V=14.07

Therefore, the null hypothesis that there is no significant agreement among the judges (or respondents) in the ranking of different color packages is rejected.

From the data above:

D2 = 354m = 9N = 8 W=0.94

We can safely conclude that there exists a considerable but significant agreement among the judges as to the ranking of different colored packages is concerned. Thus, it is evident that at least one colored package is ranked significantly higher than the other.

Compare the calculated value and critical value:

W=0.94

ENDThank You

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Dr. Nasie Seneriches, PHD

Reporter:Zhajarah S. Mama