Biserial Correlation and Kendall Coefficient of Concordance.
Hilighlights:
- Biserial correlation
- Kendal coefficient of Concordance
Biserial correlation is used to find the relationship between nominal dichotomous variable which has assumption of underline continuity and a continuous variable.(interval/ratio). Such variable are normally psychological construct like attitude, interest, anger, anxiety etc.
Where,
MeanY1 = mean of scores of students who got the item correct
MeanY0 = mean of scores of students who got the item wrong
sdY = standard deviation of the scores of students
P = proportion of students who got the item correct
Q = proportion of students who got the item wrong
Yord = ordinate of the standard normal curve at the point of division between P and Q.
This is similar to the Point Biserial correlation but the only difference is the y-ordinate(yord).
Example: Assuming a researcher is interested in finding the relationship between student anxiety level and achievement in chemistry. Using the data given below
| Anxiety | Chemistry |
| 0 | 19 |
| 1 | 15 |
| 0 | 20 |
| 0 | 14 |
| 0 | 17 |
| 1 | 12 |
| 1 | 8 |
| 0 | 6 |
| 1 | 9 |
| 0 | 10 |
Biserial Correlation and Kendall Coefficient of Concordance
Solution
| Anxiety | Chemistry(Y) | Y-squre |
| 0 | 19 | 361 |
| 1 | 15 | 225 |
| 0 | 20 | 400 |
| 0 | 14 | 196 |
| 0 | 17 | 289 |
| 1 | 12 | 144 |
| 1 | 8 | 64 |
| 0 | 6 | 36 |
| 1 | 9 | 81 |
| 0 | 10 | 100 |
| Total | 1896 | |
Note: we will code anxiety as follows;
Low anxiety = 0
High Anxiety = 1
MeanY1 = (15+12+8+9)/4 = 11
MeanY0 = (19+20+14+17+6+10)/6 = 14.3
P =4/10 = 0.4
Q = 6/10 = 0.6
SdY = 4.54(using raw score formula)
How to Determine Y-ordinate
To find the value of Yord, we find the area and ordinate of Z under the standard normal curve at which 40% is above (upper tail).
Z(40%) 0r 0.4 = 0.253
Go to the area and ordinate of the normal curve. Notice that to get 0.253, since what we have on the standard table is 0.25 only, it means that 0.253 is between 0.25 and 0.26. so we are going to find area and ordinate of 0.25 and 0.26.
| Z | Area | Ordinate |
| 0.25 | 0.9871 | 0.38667 |
| 0.26 | 0.10257 | 0.38568 |
| difference | 0.00386 | 0.00099 |
Note; in ordinate, the lesser z-value has the highest ordinate.
So in the subtraction done in the above table, we took the area of z-value(0.25) form z-value(0.26) and we took the Ordinate of z-value(0.26) from the ordinate of z-value(0.25), reason be that the lesser z-value has the highest ordinate.
So the next step is to divide the value of the ordinate gotten from the subtraction by 3
= 0.00099/3 = 0.00033
So finally, ordinate for 0.253 = 0.38667 – 0.00033 = 0.38634
Yord = 0.38634
Now applying the formula
Yb = (11-14.3)/4.54Sqr(0.40.6/0.38634)
= -0.573
Drawing inference
This shows that there is a high negative relationship between student anxiety level and their achievement in chemistry. This implies that students with high anxiety level tended to achieve high in chemistry and students with low anxiety level tended to achieve low in Chemistry.
Biserial correlation coefficient ranges from -1 to +1 but there are exceptional cases where the Biserial coefficient can take values greater than 1 or -1. This can happen when the point of division of the variable underlying the continuity depreciates appreciably from normal. That means that the point of division between P and Q moves far away from 0.5
Kendall Coefficient of Concordance (w)
This is used when we are interested in finding the relationship between the ranking of more than two judges or raters or scorers. It is used when we have ordinal data. The value of Kendall coefficient of concordance ranges from 0 to 1. Zero represents no agreement and 1 represent a perfect agreement.
The formula
Where, m = number of judges or raters, n = number of students.
Example: Assuming a researcher is interested to determine the relationship among the ranks of 4 judges with the data given below
| S/N | Judge A | JudgeB | Judge C | JudgeD |
| 1 | 3 | 1 | 6 | 7 |
| 2 | 2 | 2 | 8 | 1 |
| 3 | 5 | 3 | 1 | 8 |
| 4 | 7 | 5 | 7 | 5 |
| 5 | 1 | 8 | 3 | 3 |
| 6 | 8 | 4 | 4 | 6 |
| 7 | 4 | 7 | 2 | 2 |
| 8 | 6 | 6 | 5 | 4 |
Solution
| S/N | JA | J B | J C | J D | SR | D | D-sqr |
| 1 | 3 | 1 | 6 | 7 | 17 | -1 | 1 |
| 2 | 2 | 2 | 8 | 1 | 13 | -5 | 25 |
| 3 | 5 | 3 | 1 | 8 | 17 | -1 | 1 |
| 4 | 7 | 5 | 7 | 5 | 24 | 6 | 36 |
| 5 | 1 | 8 | 3 | 3 | 15 | -3 | 9 |
| 6 | 8 | 4 | 4 | 6 | 22 | 4 | 16 |
| 7 | 4 | 7 | 2 | 2 | 15 | -3 | 9 |
| 8 | 6 | 6 | 5 | 4 | 21 | 3 | 9 |
| Total | 144 | 106 |
To get SR(sum of rank) column, the addition is done horizontally for each entries. AR(average rank) is gotten by taking the average of SR.
Average Rank(AR) =144/8 = 18
The ‘D’ known as deviation is gotten column is gotten by subtracting the average rank(AR) from the individual SR column. i.e SR – AR. And the D-square is gotten by squaring the ‘D’ column.
m = 4, n = 8
W = (12106)/844(88 – 1)
= 1272/8064
= 0.16
This shows a low relationship among the ranking of the judges, it implies the judges rarely agree.
