Correlation analysis is a statistical method used to measure the strength of the linear relationship between two variables. A high correlation indicates a strong relationship, while a low correlation means the variables are weakly related. Researchers use correlation analysis in market research to identify relationships, patterns, and trends between variables. There are three types of correlation - positive, negative, and no correlation. Methods for studying correlation include scatter diagrams and Karl Pearson's coefficient of correlation. Spearman's rank correlation coefficient is used when variables are qualitative rather than quantitative.

1. Correlation Analysis

2. ⚫Correlation analysis in research is a statistical
method used to measure the strength of the linear
relationship between two variables and compute
their association.
⚫ Correlation analysis calculates the level of change
in one variable due to the change in the other.

3. ⚫A high correlation points to a strong relationship
between the two variables, while a low correlation
means that the variables are weakly related.
⚫When it comes to market research, researchers use
correlation analysis to analyze quantitative data to
identify the relationship, patterns, significant
connections, and trends between two variables or
datasets.

4. Advantages of correlation analysis
⚫ Observe relationships: A correlation helps to identify the
absence or presence of a relationship between two
variables. It tends to be more relevant to everyday life.
⚫ A good starting point for research: It proves to be a good
starting point when a researcher starts investigating
relationships for the first time.
⚫ Uses for further studies: Researchers can identify the
direction and strength of the relationship between two
variables and later narrow the findings down in later
studies.
⚫ Simple metrics: Research findings are simple to classify.
The findings can range from -1.00 to 1.00. There can be
only three potential broad outcomes of the analysis.

5. Types of Correlation

6. ⚫ Correlation between two variables can be either a positive correlation, a negative correlation,
or no correlation.
Let's look at examples of each of these three types:
Positive correlation: A positive correlation between two variables means both the variables move
in the same direction. An increase in one variable leads to an increase in the other variable and
vice versa.
For example, spending more time on a treadmill burns more calories.
Negative correlation: A negative correlation between two variables means that the variables
move in opposite directions. An increase in one variable leads to a decrease in the other
variable and vice versa.
For example, increasing the speed of a vehicle decreases the time you take to reach your
destination.
Weak/Zero correlation: No correlation exists when one variable does not affect the other.
For example, there is no correlation between the number of years of school a person has
attended and the letters in his/her name.
Strong correlation does not suggest that X1 causes X2 or X2 causes X1. We
say "correlation does not imply causation."

9. Linear and non-linear correlation
⚫ If the amount of change in one variable tends to bear a constant
ratio to the amount of change in other variable, then the
correlation is said to be linear.
⚫ Correlation is non linear, if the amount of change in one
variable does not bear a constant ratio to the amount of change
in the other variable.

10. Simple, Partial & Multiple Correlation
⚫ Simple Correlation – When we consider only two variables and check
the correlation between them it is said to be Simple Correlation.
For example, radius and circumference of a circle., Price and quantity
demanded
⚫ Multiple Correlation – When we consider three or more variables for
correlation simultaneously, it is termed as Multiple Correlation.
For example
Price of Cola Drink, Temperature, Income and Demand for Cola.
when we study the relationship between the yield of rice per acre and
both the amount of rainfall and the amount of fertilizers used, it is a
problem of multiple correlation.
⚫ Partial Correlation – When one or more variables are kept constant
and the relationship is studied between others, it is termed as Partial
Correlation.
For example, If we keep Price of Cola constant and check the correlation
between Temperature and Demand for Cola, it is termed as Partial Correlation.

11. Methods of studying Correlation

12. Scatter Diagram
⚫This is the simplest method of studying correlation
between two variables. The two variables x and y are
taken on the X and Y axes of a graph paper. Each pair
of x and y value we mark a dot and we get as many
points as the number of pairs of observation. By
looking through the scatter of points, we can form an
idea as whether the variables are related or not.

13. Scatter Diagram
⚫ If all the plotted points lie on a straight line rising from the
lower left hand corner to the upper right hand corner,
correlation is said to be perfectly positive.
⚫ If all the plotted points lie on a straight line falling from the
upper left hand corner to the lower right hand corner of the
diagram, correlation is said to be perfectly negative.
⚫ If all the plotted points fall in a narrow line and the points are
rising from the lower left hand corner to the upper right hand
corner of the diagram, there is degree of positive correlation
between variables.
⚫ If the plotted points fall in a narrow bank and the points are
lying from the upper left hand corner to the right hand corner,
there high degree of negative correlation.
⚫ If the plotted points lie scattered all over the diagram, there is
no correlation between the two variables.

15. Merits and limitations of Scatter diagram
⚫Merits
It is simple and non-mathematical method of studying
correlation between variables. Making a scatter diagram
usually is the first step in understanding the relationship
between two variables.
⚫Limitations
In this method we cannot measure the exact degree of
correlation between the variables.

16. Karl Pearson’s coefficient of correlation
( Covariance method)
⚫The degree to which the change in 1 continuous
variable is associated with a change in another
continuous variable can mathematically be described
in terms of the covariance of the variables.
⚫ Covariance is similar to variance, but whereas
variance describes the variability of a single variable,
covariance is a measure of how 2 variables vary
together.

17. KARL PEARSON’S CO-EFFICIENT OF
CORRELATION
⚫It is a measure of the strength of a linear
association between two variables and is denoted
by r or rxy
⚫This method of correlation attempts to draw a line
of best fit through the data of two variables,
⚫the value of the Pearson correlation coefficient, r,
indicates how far away all these data points are to
this line of best fit.

18. Assumptions
While calculating the Pearson‘s Correlation Coefficient,
we make the following assumptions:
⚫ There is a linear relationship (or any linear component
of the relationship) between the two variables.
⚫We keep Outliers either to a minimum or remove them
entirely.

19. KARL PEARSON’S CO-EFFICIENT OF
CORRELATION

20. Properties of the Pearson’s Correlation
Coefficient
1. r lies between -1 and +1, or –1 ≤ r ≤ 1, or the
numerical value of r cannot exceed one (unity)
2. The correlation coefficient is independent of the
change of origin and scale.
3. Two independent variables are uncorrelated but the
converse is not true.

25. SPEARMAN’S RANK CORRELATION
COEFFICIENT
⚫Instead of taking the values of the variables the ranks
(or order) of the observations are considered and
Pearson‘s coefficient of correlation for the ranks is
calculated.
⚫ The correlation coefficient so obtained is called rank
correlation coefficient.
⚫ This measure is useful in dealing with qualitative
characteristics such as intelligence, beauty, morality,
honesty etc.

26. SPEARMAN’S RANK CORRELATION
COEFFICIENT

28. Calculation of Spearman’s Rank correlation
when equal or repeated ranks occur
⚫While assigning rank, if two or more items have equal
values (i.e., if there occur a tie), they may be given
mid rank. Thus, if two items are on the fifth rank,
each may ranked as 5 + 6 /2 = 5.5 and the next item in
the order of size would be ranked seventh. When two
or more ranks are equal, the following formula is used
for computing rank correlation.

31. Merits and demerits of Rank
Correlation Merits
Merits
1. It is easy to compute and understand
2. It is highly useful when the data are of a qualitative nature like
intelligence, beauty etc.
3. When the ranks of different item-values are given, this is the only
method for finding the degree of correlation.
Demerits
1. This method cannot be employed for finding out correlation in a
grouped frequency distribution.
2. It is difficult to calculate rank correlation, if we have more than
30 items of observation as ranking them requires much labour.
3. Compared to Pearson method, rank correlation is not precise.

32. Revision
1. The correlation coefficient for X and Y is known to be zero. We then
can conclude that:
a. X and Y have standard distributions
b. the variances of X and Y are equal
c. there exists no relationship between X and Y
d. there exists no linear relationship between X and Y
e. none of these

33. Revision
2. What would you guess the value of the correlation coefficient to be
for the pair of variables: "number of man-hours worked" and
"number of units of work completed"?
a. Approximately 0.9
b. Approximately 0.4
c. Approximately 0.0
d. Approximately -0.4
e. Approximately -0.9
3. In a given group, the correlation between height measured in feet
and weight measured in pounds is +.68. Which of the following
would alter the value of r?
a. height is expressed centimeters.
b. weight is expressed in Kilograms.
c. both of the above will affect r.
d. neither of the above changes will affect r

34. Revision
4. Under a "scatter diagram" there is a notation that the coefficient of
correlation is .10. What does this mean?
a. plus and minus 10% from the means includes about 68% of the cases
b. one-tenth of the variance of one variable is shared with the other
variable
c. one-tenth of one variable is caused by the other variable
d. on a scale from -1 to +1, the degree of linear relationship between the
two variables is +.10