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Correlation: B.Com. II Notes

 Correlation

In previous blog we discussed about the measure central tendency and Dispersion to use to study the variable. the correlation is a statistical concept that allows us to measure and understand the relationship between two or more variables. it provided a valuable information about that variables. e.g. price and demand of commodity, income and expenditure of family, height and weight of group of persons. their we use the relation   between this two variables. in above examples we see the one variable increases other variable is also changes in same or opposite direction. 

definition: Correlation is statistical tool which study the relationship between two or more variables. for analysis of correlation various method and techniques are used. 

example: i. Demand and supply of product                                                                                                                       ii. price and demand                                                                                                                                           iii. Income and expenditure

Use Of Correlation:

the correlation analysis is widely used in economic, business and other fields. 

i. To Predict: if we know the relation between two variables, we can estimate the value of them when value of other is known. e.g. we know the correlation between height and weight then we calculate the weight for known value of height.

ii. To control: The correlation also enables us to control our activity. e.g. we know the correlation between fertilizer and crop, then we control the yield and life of the crop, (use of more fertilizer is hazard to crop. 

iii. To Plan: the knowledge of correlation help to planning. e.g. if we know the relation between the rainfall and yield of crop, then we know the rainfall we calculate the yield of crop, depend on the yield of crop we  plan for import and export.

Types of Correlation:

i. Positive and Negative Correlation

ii. Linear and non-liner Correlation.

iii, simple, Multiple and  Partial Correlation.

Positive correlation: If both the variables changes in same direction i.e. if one variable increases other variable also increases or if the one variable decreases then other variable also decreases, the correlation is said to be positive correlation. e.g. i. height and weight mean height is increases weight also increases.

Negative Correlation: if the both variables are changes in opposite direction i.e. if one variable increases other variable decreases or if the one variable decreases then other variable also increases. e.g. price and demand mean the price increases demand is decreases.

the difference between the positive and negative correlation depends on the direction of change of two variables.

when change in one variable is not affected on other variable it is no correlation between two variables.

Linear Correlation:  this type of correlation is based on the nature of the graph of two variables. if the graph is straight line the correlation is Linear correlation.

if the graph is not a straight line but is curve is called Non-Linear Correlation.

Simple Correlation: we study only two variable say price and demand it is simple correlation.

Multiple Correlation: we study more than two variables is called multiple correlation.

Partial Correlation: we study the more than two variables but correlation is studied between two variables only, and the effect of the other variable is assumed as constant.

Methods of studying Correlation:

I. Graphical method : Scatter Diagram

II. Mathematical Method:  

i. Karl Pearson Coefficient of correlation 'r'

                                                   ii. Spearman's Rank Correlation Coefficient 'R'

i. Scatter diagram: scatter diagram is a graph showing correlation between two variable. the N pair of values (x1,y1), (x2,y2) .....(xn,yn) of two variables x and y are plotted on the graph or XY plane we get the points or dot's on graph and they are generally scattered so it is called scatter diagram or dot diagram.

the scatter point show the direction and degree of correlation between x and y or any two variable the direction of correlation is denoted as + or -sign and degree by r. 

from the  scattered diagram we interpret the degree and direction of correlation. 

Interpretation of scatter diagram :

i. if the graph shows all the point lie on a rising straight line towards the right, the correlation is perfect positive and r = +1 e.g. correlation between age and height of children , their is strong correlation between  age and height , as children grow older they get taller and this relationship is quite linear in their growth phases.  we see the graph 





ii. if the graph shows all the point lie in falling straight line towards the right, the correlation s perfect negative and r=-1.e.g. the correlation between price and demand if price increases demand decreases.


iii. if all the points lies in narrow strip which rise towards right.  Then the correlation is high degree positive correlation, i.e. 08<r>1.





iv. if all the points lie in narrow strip which rise towards left, then the correlation is high –ve correlation it is -1<r>-0.8





v. if all point rise on broad line or strip rising towards the right the correlation is low degree positive correlation.





vi. if all point rise on broad line or strip rising towards the left the correlation is low degree positive correlation.



vii. if all point are scattered without any pattern, then there is no correlation, i.e. r=0.



Merit and demerit:

Merit:

i.                    It is very easy to draw.

ii.                  It is easy to understand.

iii.                It gives quick idea about the correlation between variables.

Demerits:

i.                    It is not used for mathematical calculation.

ii.                  It no measure the degree of correlation.

iii.                It is not suitable when values are large.

Example 1. To draw a scatter diagram & indicate whether correlation is positive or negative.

Price:

17

18

19

21

25

26

30

Supply:

30

37

36

42

50

51

54

 

To draw scattered diagram we plot the price on x –axis and supply on y-axis. Then we get a scatted diagram.



From this diagram we see the the point are rising hence it is high positive correlation.

Karl Pearson's Coefficient of correlation:

    Karl Pearson's gives the mathematical method to measure the correlation between two variables. based on following assumptions. 

i. there is linear relationship between two variables.

ii. there is cause and effect relationship between two variables.

definition: Karl Pearson's Coefficient of correlation between two variables X and Y, denoted as corr(X,Y) or "r" and it is given as      

\(r = \frac{{\text{Cov}(X, Y)}}{{\sigma_X \sigma_Y}}\)

Where:

  • \( \text{Cov}(X, Y) \) is the covariance of \(X\) and \(Y\), calculated as: \[ \text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) \]
  • \( \sigma_X \) is the standard deviation of \(X\).
  • \( \sigma_Y \) is the standard deviation of \(Y\).

The correlation coefficient formula can also be written as:

\[ r = \frac{\sum_{} xy}{\sqrt{x^2} \cdot \sqrt{y^2}} \]

Where:

  • \( x = (x - \bar{x}) \)
  • \( y = (y - \bar{y}) \)

Properties of \( r \):

  • 1. \( r \) always lies between -1 to +1.
  • 2. \( r \) is unaffected by a change of origin and scale.
  • 3. \( r \) is equal to the square root of the product of two regression coefficients, i.e., \( r = \sqrt{b_{xy} \cdot b_{yx}} \).
  • 4. \( r \) is free from unit.

Interpretation of \( r \):

  • If \( r = +1 \), there is a perfect positive correlation between two variables.
  • If \( r = -1 \), there is a perfect negative correlation between two variables.
  • If \( r = 0 \), there is no correlation between two variables.
  • If \( 0.8 < r \leq 1 \), there is a high degree of positive correlation between two variables.
  • If \( -1 \leq r \leq -0.8 \), there is a high degree of negative correlation between two variables.
  • If \( 0.4 \leq r \leq 1 \), there is a low degree of positive correlation between two variables.
  • If \( -0.4 \leq r \leq -1 \), there is a low degree of negative correlation between two variables.
  • The type of correlation depends on the sign of \( r \): if \( r \) has a negative sign, it means negative correlation, and if \( r \) has a positive sign, it indicates positive correlation.

Merits and Demerits:

Merits:

  • \( r \) gives the numerical value of correlation.
  • It is also useful for estimation.

Demerits:

  • The value of \( r \) is not affected by extreme values, so it may not provide proper results in some cases.
  • Sometimes, it is difficult to calculate.

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