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PRESENTATION ON:CORRELATION AND RANK

CORRELATIONSUBMITTED TO:DIBYOJOTI BHATTACHARJEE

SUBMITTED BY:

NABONITA DAS (49)SOMHITA CHAKRABORTY (66)ARPITA DUTTA (50)

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Meaning Of Correlation

Correlation is a statistical measure for finding out degree(or strength) of association between two(or more) variables. If the change in one variable effects a change in other variable then these variables are said to be correlated.

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Types of correlation

Correlation may be of different types. Some of the most important types are:

• Positive and negative correlation.• Linear and nonlinear correlation.• Simple, partial and multiple correlation.• Spurious or Non-sense Correlation

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• Positive and Negative correlation:- Correlation is said to be Positive when the variables move in

the same direction. It means that when the value of one variable increases, the value of the other variable also increases and vise versa.

If the movement of the two variables are in the opposite direction, i.e., one variable is increasing and other is decreasing, then the correlation is negative.

Positive—*Price & Supply. *Advertisement exp and sales Negative- *Price & demand. *Interest rate on loan and loans taken by the public.

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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 the other variable, the correlation is said to be linear.

If the amount of change in one variable does not bear a constant ratio to the amount of change in other variable, the correlation is said to be non-linear or curvilinear.

example: Linear-amount of rainfall and reading in the rain gauge. Non-Linear-Income and expenditure of a person.

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Simple, partial and multiple correlation The simple correlation studies the relationship between two variables. When relationship is measured between three or more variables then it is a case of multiple or partial correlation. Multiple correlation studies the extent in which a variable is affected by the combined influence of a group of other variables. The study of relationship between two variables, keeping the affects of other variable constant is known as partial correlation.Example:Simple-Price and supply of a commodity.Multiple-Yield of crop with the combined effect of rainfall,temperature,use of fertilizers, humidity etc.Partial-Yield of crop and rainfall keeping the effect of temperature, use of fertilizers, humidity etc.

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Spurious or Non-Sense correlation

It is sometimes seen that though there is no cause and effect relationship between two variables, but still the value of the correlation coefficient calculated on numerical basis may prove to be significant. Such type of correlation is called spurious correlation or non-sense correlation.

Example: The correlation between GDP of India and

rainfall in Japan for the last ten years.

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Uses of correlation

• Correlation coefficient measures the degree of relationship between the variables.

• It may be used to determine the regression coefficient provided that standard deviations of the two variables are known.

• In educational and psychological measurement, it is used in the problems of reliability and validity of tests.

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Methods of correlation

G Graphic

Scatter diagram co-variance Rank Concurrent deviation method

Algebric

Graph

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Karl Pearson’s correlation coefficient

• A mathematical method for measuring the intensity or the magnitude of linear relationship between two variables was suggested by Karl Pearson. It is also known as Pearsonian correlation coefficient between two variables X and Y and usually denoted by rxy or r(x,y) or

simply ‘r’.

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Interpretation of r

If r=+1, implies that there is a perfect positive correlation between the variables.

If r=-1, implies that there is perfect negative correlation between the variables.

If r=0, the variables are uncorrelated. A value of r which is very near to +1 or - 1 means

that the values are highly correlated. A value of r which is very near to 0 means that

the correlation between the variables is very low.

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Formula For Correlation Cofficient

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Calculate the correlation coefficient from the following data:

X: -10, -5, 0, 5, 10 Y: 5, 9, 7, 11, 13

Solution:

x y x2 y2 xy

-10 5 100 25 -50

-5 9 25 81 -45

0 7 0 49 0

5 11 25 121 55

10 13 100 169 130

∑x = 0 ∑y = 45 ∑x2 = 250 ∑y2 = 445 ∑xy = 90

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We know,

Karl Pearson’s coefficient of correlation is

rxy

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Importance of Correlation in Business Decision Making

Correlation and regression analysis can help business to investigate the determinants of key variables such as their sales. Variations in a companies sales are likely to be related to variation in product prices,consumers,incomes,tastes and preference's, correlation analysis can be used to test the goodness of it.

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Rank

• The Karl Pearson’s correlation coefficient cannot be used in cases where the direct quantitative measurement of the phenomenon under study is not possible, e.g., efficiency, honesty, intelligence, satisfaction, etc. In such cases one may rank the different items in order of preference and apply the Spearman method of rank differences for finding out the degree of correlation.

Uses of rank correlation coefficient

• 1. Rank correlation coefficient are used when association between attributes are to be studied. Since, attributes cannot be expressed in terms of quantities so in such cases it is not possible to apply the Karl Pearson’s method of calculating the correlation.

• 2. Since, the Karl Pearson’s method of calculating the correlation is difficult so this method is often preffered to get a quick understanding about the degree of association between variables.

• 3. In some cases it is beneficial to express things in terms of rank instead of actual figures.

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)12(

26

1

nn

d

The formula is

Where, ‘P’ denotes coefficient of rank correlation.

‘d’ denotes the difference between the paired rank. ‘n’ denotes the number of pairs.

‘

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Calculation of Rank correlation coefficient

Q) Calculate Spearman’s rank correlation coefficient from the marks scored by 12 students in two subjects and interpret your result.

Mathematics: 60 34 40 50 45 41 22 43 42 66 64 46

Statistics: 75 32 34 40 45 33 12 30 36 72 41 57 19

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Marks in Maths (X) Rank in Maths (x) Marks in Stat (Y) Rank in Stat (Y) d = x-y d2

60 3 75 1 2 4

34 11 32 10 1 1

40 10 34 8 2 4

50 4 40 6 -2 4

45 6 45 4 2 4

41 9 33 9 0 0

22 12 12 12 0 0

43 7 30 11 -4 16

42 8 36 7 1 1

66 1 72 2 -1 1

64 2 41 5 -3 9

46 5 57 3 2 4

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)12(

261

nn

d

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The value of rank correlation coefficient is +0.83, which is nearer to + 1.Thus it appears that there is a high degree of correlation between the marks obtained by the students in the two subjects.

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Thank you