__Regression analysis:____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.

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

(b) Non-linear regression

(b) Non-linear regression

(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

(b) Partial

(c) Multiple

(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.

__Uses and limitations of Regression:__

__Uses of Regression:__(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.

__Llimitations 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.

__Correlation analysis____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.

II. Number of variation.

III. Change in direction.

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

__Basis of change in proportion__:(a) Linear correlation

(b) Non-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

(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 Chang in direction, correlation may be

__On the basis of Change in direction:__(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.

__Uses and limitations of Correlation:__

__Uses 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.

__Limitations of correlation:__

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.

__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.

__Degrees of Correlation:__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.

__Characteristics of regression co-efficients:__**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.

__Different methods of studying correlation:__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.

__What are the regression lines? Why do we generally have two regression equations?__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.

__+1__and Two regression lines will be at right angles when r = 0.)

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