**Unit – 2: Correlation and Regression analysis**

**Correlation analysis**

**Q.N.1. Define Correlation analysis. What are its various kinds?**

Ans: - Definition: - Correlation is the degree of the relationship
between two or more variables. It does not explain the cause behind the
relationship. Kinds of correlation may be studied on the basis of:

I. Change in proportion.

II. Number of variation.

III. Change in direction.

(I) Basis of change in proportion:-There are
two important correlations on the basis of change in proportion. They are:

(a) Linear correlation (b) Non-linear
correlation

(a) Linear correlation: - Correlation is said
to be linear when one variable move with the other variable in fixed proportion

(b) Non-linear correlation: - Correlation is
said to be non-linear when one variable move with the other variable in
changing proportion.

(II) On the basis of number of variables: On
the basis of number of variables, correlation may be:

(a) Simple (b) Partial (c) Multiple

(a) Simple correlation: - When only two
variables are studied it is a simple correlation.

(b) Partial correlation: - When more than two
variables are studied keeping other variables constant, it is called partial
correlation.

(c) Multiple correlations: - When at least
three variables are studied and their relationships are simultaneously worked
out, it is a case of multiple correlations

(III) On the basis of Change in direction: On
the basis of Chang in direction, correlation may be

(a)Positive Correlation (b) Negative
Correlation

(a) Positive Correlation: - Correlation is
said to be positive when two variables move in same direction.

(b) Negative Correlation: - Correlation is
said to be negative when two variables moves in opposite direction.

**Q.N.2.What are the uses and limitations of Correlation?**

Ans: - Following are the main advantages of correlation:

1. It gives a precise quantitative value
indicating the degree of relationship existing between the two variables.

2. It measures the direction as well as
relationship between the two variables.

3. Further in regression analysis it is used
for estimating the value of dependent variable from the known value of the
independent variable

4. .The effect of correlation is to reduce the
range of uncertainty in predictions.

1. Extreme items affect the value of the
coefficient of correlation.

2. Its computational method is difficult as
compared to other methods.

3. It assumes the linear relationship between
the two variables, whether such relationship exist or not.

**Q.N.3. What are the different degrees of Correlation?**

Ans: The different degrees of correlation are:

i)
Perfect Correlation: - It two variables vary
in same proportion, and then the correlation is said to be perfect correlation.

ii)
Positive Correlation: - If increase (or
decrease) in one variable corresponds to an increase (or decrease) in the
other, the correlation is said to be positive correlation.

iii)
Negative Correlation: - If increase (or
decrease) in one variable corresponds to a decrease (or increase) in the other,
the correlation is said to be positive correlation.

**iv)**Zero or No Correlation: - If change in one variable does not other, than there is no or zero correlation.

**Q.N.4. What are the different methods of studying correlation?**

**Ans**: - The different methods of studying relationship between two variables are:

i)
Scatter diagram method.

ii)
Graphic method

iii)
Karl Pearson’s coefficient of correlation

iv)
Rank correlation method

**i) Scatter Diagram Method**: - It is a graphical representation of finding relationship between two or more variables. Independent variable are taken on the x-axis and dependent variable on the y-axis and plot the various values of x and y on the graph. If all values move upwards then there is positive correlation, if they move downwards then there is negative correlation.

Merits:

i) It is easy and simple to use and
understand.

ii) Relation between two variables can be
studied in a non-mathematical way.

Demerits:-

i) It is non-mathematical method so the
results are non-exact and accurate.

ii) It gives only approximate idea of the
relationship.

i

**i) Graphic Method**: - This is an extension of linear graphs. In this case two or more variables are plotted on graph paper. If the curves move in same direction the correlation is positive and if moves in opposite direction then correlation is negative. But if there is no definite direction, there is absence of correlation. Although it is a simple method, but this shows only rough estimate of nature of relationship.
Merits: -

i) It is easy and simple to use and
understand.

ii) Relation between two variables can be
studied in a non-mathematical way.

Demerits:-

i) It is non-mathematical method so the
results are non-exact and accurate.

ii) It gives only approximate idea of the
relationship.

**iii) Karl Pearson’s Coefficient of correlation**: - Correlation coefficient is a mathematical and most popular method of calculating correlation. Arithmetic mean and standard deviation are the basis for its calculation. The Correlation coefficient (r), also called as the linear correlation coefficient measures the strength and direction of a linear relationship between two variables. The value of r lies between -1 to +1.

Properties of r:-

i)
r is the independent to the unit of
measurement of variable.

ii)
r does not depend on the change of origin and
scale.

iii)
If two variables are independent to each
other, then the value of r is zero.

Merits:-

i)
The co-efficient of correlation measures the
degree of relationship between two variables.

ii)
It also measures the direction.

iii)
It may be used to determine regression
coefficient provided s.d. of two variables are known.

Demerits:-

i)
It assumes always the linear relationship
between the variables even if this assumption is not correct.

ii)
It is affected by extreme values.

iii)
It takes a lot of time to compute.

**iv) Spearman’s rank Coefficient of correlation**: - This is a qualitative method of measuring correlation co-efficient. Qualities such as beauty, honesty, ability, etc. cannot be measured in quantitative terms. So, ranks are used to determine the correlation coefficient.

Merits:-

i)
It is easy and simple to calculate and
understand.

ii)
This method is most suitable if the data are
qualitative.

Demerits:-

i)
This method cannot be used in case of grouped
frequency distribution.

ii)
Where the number of items exceeds 30 the
calculations become quite tedious and require a lot of time.

**Regression Analysis**

**Q.N.1. Define Regression analysis. What are its Various kinds?**

Ans: -
Definition: -Regression is the measure of the average relationship between two
or more variable in terms of the original units of the data. It is a
statistical tool with the help of which the unknown values of one variable can
be estimated from known values of another variable.

Kinds
of regression may be studied on the basis of:

I.
Change in proportions.

II.
Number of variation.

(I)
Basis of change in proportion:-There are two important regressions on the basis
of change in proportion. They are:

(a)
Linear regression: - Regression is said to be linear when one variable move
with the other variable in fixed proportion

(b)
Non-linear regression: - Regression is said to be non-linear when one variable
move with the other variable in changing proportion.

(II)
On the basis of number of variables: On the basis of number of variables,
regression may be:

(a)
Simple regression: - When only two variables are studied it is a simple
regression.

(b)
Partial regression: - When more than two variables are studied keeping other
variables constant, it is called partial regression.

(c)
Multiple regressions: - When at least three variables are studied and their
relationships are simultaneously worked out, it is a case of multiple
regressions.

**Q.N.2.What are the uses and limitations of Regression?**

Ans: - The
following are main Advantages of regression analysis:

(1)
Helpful to statisticians:- The study of regression helps the statisticians to
estimate the most probable value of one variable of a series for the given
values of the other related variables of the series.

(2)
Nature of relationship: - Regression is useful in describing the nature of the
relationship between two variables.

(3)
Estimation of relationship: - Regression analysis is widely used for the
measurement and estimation of relationship among economic variables.

(4)
Predictions: - Regression analysis is helpful in making quantitative
predictions on the basis of estimated relationship among variables.

(5)
Policy formulation: - The predictions made on the basis of estimated
relationship are used in policy making.

**The following are the main limitation of regression:**

1)
No change in relationship: - Regression analysis is based on the assumption
that while computing regression equation; the relationship between variables
will not change.

(2)
Conditions: - The application of regression analysis is based on certain
conditions like, for existence of linear relationship between the variables;
exact values are needed for the independent variable.

(3)
Spurious relationships: - There may be nonsense and spurious regression
relationships. In such case, the regression analysis is of no use.

**Q.N.3. Distinguish between correlation and regression.**

Ans:-There
are some basis difference between correlation and regression:

(1)
Nature of relationship: - Correlation explains the degree of relationship,
whereas regression explains the nature of the relationship.

(2)
Causal relationship: - Correlation does not explain the cause behind the
relationship whereas regression studies the cause and effect relationship.

(3)
Prediction: - Correlation does not help in making prediction whereas regression
enable us to make prediction.

(4)
Origin and scale: - Correlation coefficient is independent of the change of
origin and scale, whereas regression coefficient is independent of change of
origin but not of scale.

**Q.N.4. What are the characteristics of regression co-efficients?**

Ans: -
Characteristics of good regression coefficient:

**1.**Both regression co-efficients will have the same sign.

**2.**If one regression co-efficient is above unity, then the other regression co-efficient should be below unity.

**3.**If both the regression co-efficient are negative, correlation co-efficient should be negative

**4.**Regression co-efficients are independent of change of origin but not of scale.

**Q.N.5. What are the regression lines? Why do we generally have two regression equations?**

Ans: - A
line of regression by the method of “least square” shows an average
relationship between variables under study. This regression line can be drawn
graphically or derived algebraically. A line fitted by method of least square
is known as the line of best fit. There are two regression lines:-

Regression
line of x on y: - Regression line of x on y is used to predict x for a given
value of y. The regression equation of x on y is x=a+by.

Regression
line of y on x: - Regression line of y on x is used to predict y for a given
value of x. The regression equation of y on x is y=a+bx

Two
regression lines: - We know that there are two lines of regression: - x on y
and y on x. For these lines, the sum of the square of the deviations between
the given values and their corresponding estimated values obtained from the line
is least as compared to other line. One regression line cannot minimise the sum
of squares for both the variables that is why we are getting two regression
lines. (We get one regression line when r = +1 and Two regression lines will be
at right angles when r = 0.)

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