Measures of Variability/Dispersion in Educational statistics

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)	R_K	R_E	D= R_K- R_E	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	8.5	6	2.5	6.25
B	32	49	4	7	-3.0	9.00
C	31	53	6	5	1.0	1.00
D	27	48	10	8	2.0	4.00
E	31	57	6	3	3.0	9.00
F	48	56	3	4	-1.0	1.00
G	52	40	1	10	-9.0	81.00
H	49	61	2	2	0.0	0.00
I	31	62	6	1	5.0	25.00
J	29	42	8.5	9	-0.5	0.25

=1-

NORMAL PROBABILITY CURVE

It is a bell-shaped perfectly bilaterally curve wherein the measures are concentrated closed around the center 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

Tuesday, 13 July 2021

MEASURES OF RELATIONSHIPS