This document discusses different methods for measuring relationships between variables, including cross tabulation, percentage difference, Gamma, and Spearman's rank order correlation coefficient. It provides examples of each method and explains how to calculate and interpret them. Cross tabulation involves creating a two-way table to examine relationships between categorical variables. Gamma and Spearman's rank order correlation examine relationships between ordinal variables by comparing pairs of ranks. Pearson's correlation coefficient measures both the direction and strength of linear relationships between variables.

1. Measures of relationships
•Cross tabulation and percentage difference
•Gamma
•Spearman’s rank order correlation co-efficient(Rho)

2.  This section examines the relationship
between variables.
E.g.
 Children grow, their weight increases
 Demand increases if price decreases
 Application of fertilizers increases crop yield
There may be,
 Direct-indirect relationship
 Positive-negative relationship

3. In examining the relationship between variables, we
have to consider certain questions:
 Is there relationship between variables?
 What is the degree and direction of their
relationship?
 Is the relationship a casual one?
 Is the relationship statistically significant?

4. Cross tabulation and percentage
difference
 It is used for Nominal variables which are
purely qualitative and can be categorized
only.
 E.g.Age, income, brand, religion, language
etc..

5. Procedure,
 Two way table is prepared.
 The value of one variable are put along one side
of the table.
 And the values of the other variable along the
other side of the table
 Each variable is categorized into two or more
categories; and the variables are cross-tabulated
for those sub categories.
 The percentages are computed for them.
 On the basis of the percentages, conclusions are
drawn.

6. Preference for brand X and Age
(Hypothetical)

8.  The strength of relationship is determined
by the pattern of differences between the
values of variables.

9. Gamma
 The ordinal level variables have rank
order differences .
 The most common ordinal variables are
Gamma & Spearman’s rank order
correlation coefficient.
 Gamma or G or ϒ
G= (ns – nd )/ (ns + nd )
 ns the number of similar pairs
 Nd the number of dissimilar pairs

10.  It is based on pair by pair comparison
 It uses information about one variable to
tell us something about a second one.
 E.g. suppose two respondents X andY are
asked to rank the product characteristics
that they would consider while buying a
scooter.

11.  After counting the number of similar and
dissimilar pairs, the level of association
between the two sets of ranks is
compared by the ratio of the
preponderant type of pairs.
 This value is ϒ

12.  .Characteristics Respondent X Respondent Y
Fuel efficiency -F 1 1
Price - P 2 2
Mechanical Efficiency - M 3 3
Riding comfort - R 4 4
Style- S 5 5
F 1 5
P 2 4
M 3 3
R 4 2
S 5 1
F 1 1
P 2 2
M 3 3
R 4 5
S 5 4

14.  ns= 10; nd= 0
 ns= 0; nd= 10
 ns= 08; nd= 02
 G= (10 - 0) / (10 + 0) = +1
 G= (0- 10) / (0 + 10) = -1
 G= (8 – 2) / (8 + 2 ) = +.60

15.  A coefficient of +1 indicates the perfect
positive association between the variables
in terms of pair by pair comparison.
 G= -1 indicates the perfect inverse
association.
 G= + 0.60 combination of similar &
dissimilar ones.

16. Spearman’s rank order correlation
co-efficient Rho or e
 This is the oldest of the frequently used
measures of ordinal associations.
 It is a measure of the extent of agreement
or disagreement between two sets of ranks.
 It is nonparametric measures and so it dose
not require the assumption of a bivariate
normal distribution.
 Its value ranges between -1 (perfect negative
association) and +1 (perfect positive
association)

17.  Rho or e = 1- [ (6∑D2 ) / n(n2 – 1) ]
 D= difference between X,Y ranks
assigned to the object.
 n= number of observation

19.  e= 1- [ (6∑D2 ) / n(n2 – 1) ]
=1-[(6*40)/ 5(52- 1) ]
= -1.0
This value – 1 indicates that there is perfect
negative association between the two sets
of ranks.

20. Correlation analysis
It involves three main aspects:
1. Measuring the degree of association
between two variables
2. Testing whether the relationship is
significant
3. Establishing the cause and effect
relationship if any.

21.  When two variables move in same
direction, their association is termed as
positive correlation.
 When they move in opposite direction,
their association is termed as negative
correlation.

22. Karl pearson product moment
correlation coefficient r
 Most common measure
 This measure expresses both strength and
direction of linear correlation.
 This measure expresses both strength and
direction of linear correlation.

24. •Examine the relationship between
period of education (X) and religious
prejudice (Y) for a sample of six
respondents.
X Y
3 1
6 7
8 3
9 5
10 4
2 2
38 22

25. X Y X2 Y2 XY
3 1 9 1 3
6 7 36 49 42
8 3 64 9 24
9 5 81 25 45
10 4 100 16 40
2 2 4 4 4
38 22 294 104 158

26.  r= 0.53
 The correlation coefficient ranges from -
1.0 to +1.0.
 -1.0 indicates perfect negative correlation
& +1 indicated perfect positive
correlation.
 + or - 0.5 moderate positive or negative
correlation.