MEASURES OF RELATIONSHIPS
If we have two sets of scores from the
same group of students, it is often desirable to know the degree to which the
scores are related. A teacher may be interested in the relationship between:
1. The
Mathematics test scores and the scores in other school subjects;
2. The
ability of a student to play a game and his scholastic achievement;
3. His
physical fitness and the scholastic achievement;
4. His
scholastic achievement and intelligence;
5. The
shape of his head and intelligence;
6. His
speed in reading and ability for abstract thinking;
7. His
ability to memorise and chronological age; and so on.
The relationship among abilities can be
studied by the method of correlation. It is a method of summarizing the
relationship between two sets of data.
A coefficient of correlation is a single
number indicating the going togetherness of two variables.
Nature of Relationships (Types)
(A). If the
relative position of each student in a sample is exactly the same in one test
as in the other, the relationship is perfect
and positive and the coefficient of correlation is 1.00.Here, the two
variables vary together in the e same direction.
Examples: 1. Relationship
between the diameter and the circumference of a circle;
2. Relationship between the temperature
and the height of the mercury column in a
thermometer;
3. Relationship between speed and the
distance covered in a fixed time.
(B). If the
relative position of each student in a sample is exactly in the reverse order
in one test as in the other, the relationship is perfect but negative and the coefficient of correlation is
-1.00.
Here, the two variables vary together
in the opposite direction. Other examples of a perfect negative relationship
are:
1. Relationship
between the pressure and volume of a gas at a constant temperature;
2. Relationship
between the number of men doing a work and the time required to do it;
3. Relationship
between the selling price and loss, if the cost price is constant.
(C). If in
two tests, there is no correspondence between the two sets of the scores
achieved by students in a sample, there is no systematic relationship and the
coefficient of correlation is Zero.
Examples of Zero relationship are:
1. The
diameter of a cylinder and its height;
2. The
number of pages in a book and the quality of the content;
3. The
height of the bottle and the price of the medicine it contains.
Interpreting the Correlation Coefficient
The following classification may be useful
in interpreting a coefficient of correlation.
1. r from 0.00 to + 0.20 --------à Denotes an indifferent, negligible relationship;
2. r from + 0.20 to + 0.40 -----à Denotes a definite but
low, slight relationship;
3. r
from + 0.40 to + 0.70
-----à
Denotes a substantial, marked relationship;
4. r
from + 0.70 to + 0.99 -----à Denotes a high to very high relationship;
5. r
of + 1.00 ---------------à
Denotes a perfect relationship.
This classification is broad and
tentative, and can only be accepted as a general guide with certain
reservations. A correlation coefficient is always to be judged with reference
to the conditions under which it is obtained and the objectives of the
experiment.
Use of Correlation Coefficient
1. It
is useful in validating a test; e.g., a group intelligence test.
2. It
is useful in determining the reliability of attest.
3. It
gives us an indication of the degree of the objectivity of a test.
4. It
can answer the validity of arguments for or against a statement or a belief;
e.g., Men are more intelligent than women.
5. It
indicates the nature of the relationship between two variables.
6. It
predicts the value of one variable from another. We can often predict from a
battery of aptitude tests the probable success of an individual who plans to
enter a given trade or profession. Advice on such a basis is measurably better
than subjective judgement.
7. A
knowledge of it is helpful in educational and vocational guidance, prognosis,
in the selection of workers in office or factory, and in educational
decision-making.
Methods of Correlation
There are several methods of finding the
coefficient of correlation between two series of measures. It the data exist in
raked (ordinal) form, it is easier to use the Spearman Rank-difference
Correlation Coefficient rho (ᵖ). In this method, if the data are in the form of
raw scores, they are translated into rank order.
Procedure:
1. Rank
the individuals from the highest to the lowest in the first variable. If two or
more students get the same score, give them the average rank, e.g., the two
individual D and E get the same score of 59. Each is ranked 9.5 instead of 9
and other 10. Call these as ranks R1.
2. Similarly,
rank the second variable and call them as R2.
3. Find
the difference between each pair of ranks. D=R1-R2. Check that the algebraic
sum of D is always zero.
4. Square
each D to find D2. Sum up D2 and apply the formula.
Example-1: Finding coefficient of
correlation by Spearman’s Rank difference method:
|
Students |
Marks in History (X) |
Marks in Geography(Y) |
Rank in History (R1) |
Rank in Geography (R1) |
D=R1-R2 |
D2 |
|
A |
72 |
62 |
6 |
7.5 |
- 1.5 |
2.25 |
|
B |
77 |
76 |
5 |
3 |
2 |
4.00 |
|
C |
79 |
77 |
4 |
2 |
2 |
4.00 |
|
D |
59 |
52 |
9.5 |
9 |
0.5 |
0.25 |
|
E |
59 |
62 |
9.5 |
7.5 |
2 |
4.00 |
|
F |
92 |
81 |
2 |
1 |
1 |
1.00 |
|
G |
97 |
67 |
1 |
5 |
- 4 |
16.00 |
|
H |
68 |
44 |
7 |
10 |
-3 |
9.00 |
|
I |
91 |
74 |
3 |
4 |
-1 |
1.00 |
|
J |
60 |
65 |
8 |
6 |
2 |
4.00 |
N = 10 ∑D=0 ∑D2=45.50
ᵖ = 1 - 6 ∑D2 = 1
- 6X45.50 = 1 – 273 =
1 – 0.275 = 0. 725
N (N2-1) 10 (100-1) 990
Interpretation
Scores in History and scores in Geography
have high positive relationship, which indicates that students scoring high
marks in the History test would also, generally scores high marks in the Geography
test.
2) Compute coefficient of correlation by
rank difference method for the following data and interpret the result.
|
Marks in science |
Marks in Mathematics |
R1 |
R1 |
D=R1-R2 |
D2 |
|
92 |
81 |
2 |
1 |
1 |
1 |
|
77 |
76 |
5 |
3 |
2 |
4 |
|
68 |
44 |
7 |
10 |
-3 |
9 |
|
59 |
62 |
9.5 |
7.5 |
2 |
4 |
|
91 |
74 |
3 |
4 |
-1 |
1 |
|
60 |
65 |
8 |
6 |
2 |
4 |
|
97 |
67 |
1 |
5 |
-4 |
16 |
|
59 |
52 |
9.5 |
9 |
0.5 |
0.25 |
|
79 |
77 |
4 |
2 |
2 |
4 |
|
72 |
62 |
6 |
7.5 |
-1.5 |
2.25 |
∑D2 = 45.5
ᵖ = 1 - 6 ∑D2 = 1 – 6 X 45.5/10 X 99 = 1 - 273 =
1 – 0.275 = 0. 725
N (N2-1)
990
ᵖ=0.725
Interpretation: The obtained
value is 0.725, hence the coefficient of correlation is positive and it denotes
high correlation with variables.
3) Compute coefficient of correlation by
rank difference method for the following data and interpret the result.
|
Marks in kannada (K) |
Marks in English(E) |
RK |
RE |
D= RK- RE |
D2 |
|
93 |
81 |
2 |
2 |
0 |
0 |
|
49 |
60 |
5 |
5 |
0 |
0 |
|
68 |
72 |
3 |
3 |
0 |
0 |
|
97 |
82 |
1 |
1 |
0 |
0 |
|
35 |
40 |
6 |
6 |
0 |
0 |
|
54 |
65 |
4 |
4 |
0 |
0 |
∑D2= 0
ᵖ = 1 - 6 ∑D2 = 1- 6 (0)/6 (35) = 1-0= 1
N (N2-1)
ᵖ=+1 (It is perfectly positive correlation)
4).Compute coefficient of correlation by
rank difference method for the following data and interpret the result.
|
Marks in Mathematics |
Marks in science |
Rank in Science (R1) |
Rank in Mathematics(R2) |
D=R1-R2 |
D2 |
|
88 |
80 |
1 |
2 |
1 |
1 |
|
70 |
55 |
3 |
4 |
1 |
1 |
|
60 |
68 |
4 |
3 |
1 |
1 |
|
45 |
35 |
5 |
5 |
0 |
0 |
|
75 |
85 |
2 |
1 |
1 |
1 |
∑D2 =
4
ᵖ =
1 - 6 ∑D2 = 1- 6 X 4/ 5 X 24 = 1- 24/120 = 1- 0.2 =
0.8
N (N2-1)
ᵖ=0.8
Interpretation: Therefore the
coefficient of correlation between the Mathematics and Science marks is
Positively High.
5).Compute coefficient of correlation by
rank difference method for the following data and interpret the result.*
|
Students |
Marks in Science |
Marks in History |
R1 |
R2 |
D=R1-R2 |
D2 |
|
A |
29 |
51 |
|
|
|
|
|
B |
32 |
49 |
|
|
|
|
|
C |
31 |
53 |
|
|
|
|
|
D |
27 |
48 |
|
|
|
|
|
E |
31 |
57 |
|
|
|
|
|
F |
48 |
56 |
|
|
|
|
|
G |
52 |
40 |
|
|
|
|
|
H |
49 |
61 |
|
|
|
|
|
I |
31 |
62 |
|
|
|
|
|
J |
29 |
42 |
|
|
|
|
NORMAL PROBABILITY CURVE
It is a bell-shaped perfectly bilaterally
curve wherein the measures are concentrated closed around the centre and taper
off from this central high (crest) to the left and right.
Properties of NPC
1. It
is perfectly bilaterally symmetrical about the mean, median and mode; i.e., all
these coincide exactly in the middle of the NPC.
2. From
the maximum point at the mean of NPC, the height of the curve declines as we go
in either direction from the mean. This falling-off is slow at first, then
rapid and then slow again.
3. Theoretically,
the curve never touches the base line. Hence the range is unlimited.
4. The
points of inflection (the points where the curvature changes its direction) are
each + 1 SD from the mean ordinate.
5. The
height of the curve at a distance of 1 SD, 2 SD, 3 SD from the mean on both
sides is 60.7%, 13.5 % and 1.1% respectively of the height at the mean.
6. In
a normal distribution, the mean equals the median exactly and the skewness is
zero.
SKEWNESS:
A distribution is said to be skewed
when----------
(i)
The mean and the median fall at different points
in the distribution;
(ii)
The balance or centre of gravity is shifted to
one side or the other.
When the dispersion or scatter of the
scores in a series is greater on one side of the point of a central tendency
than on the other, the distribution is skewed.
Positively Skewed Distribution
When
most of the scores pile up at the low end (or left) of the distribution and
spread out more gradually towards the high end of it, the distribution is said
to be positively skewed.
In a positively skewed distribution, the
mean falls to the right of the median.
i.e., Mean>Median
Negatively Skewed Distribution
When most of the scores pile up the high
end (or right) of the distribution, and spread out more gradually towards the
low end of it, the distribution is said to be negatively skewed.
In a negatively skewed distribution, the
mean falls to the left of the median.
i.e., Mean < Median
No comments:
Post a Comment