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Correlation Analysis | Business Statistics Notes | B.Com Notes Hons & Non Hons | CBCS Pattern

BUSINESS STATISTICS NOTES
B.COM 2ND AND 3RD SEM NEW SYLLABUS (CBCS PATTERN)
Correlation analysis

Correlation Analysis Meaning

Correlation analysis is simply the degree of the relationship between two or more variables under consideration. If two or more quantities vary in such a way that movements in one are accompanied by movement in the other quantity, these quantities are said to be correlated. For example, there exist some relationship between prices of the product and quantity demanded, rainfall and crops etc. Correlation analysis measures the degree of relationship the variables under consideration.

In the words of Simpson & Kafka “Correlation analysis deals with the association between two or more variables.”

Correlation Coefficients Meaning

Correlation Coefficient is a numerical measure of degree of relationship between two variables. Correlation coefficient of two variables ranges from -1 to +1.

Table of Contents

1. Meaning of Correlation analysis and Correlation Coefficients

2. Significance and Limitations of Correlation Analysis

3. Various Types of Correlation

4. Different Degrees of Correlation

5. Different Methods of Correlation analysis (Along with merits and demerits)

a) Scatter Diagram Method

b) Graphic Method

c) Karl Pearson’s Coefficient of Correlation

d) Spearmen’s Rank Correlation

6. Correlation and Causation

7. Probable Error

COMING SOON

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Significance and Limitations of Correlation Analysis

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. The prediction based on correlation analysis is likely to be more variable and near to reality.

5. Correlation analysis also helps in studying the causes of economic disturbance and suggests measures through which stabilizing forces may become effective.

Following are the main 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.

4. Correlation helps in determining the degree of relationship between two variables but it does not tell us anything about cause and effect relationship.

Various Types of Correlation

Kinds of correlation may be studied on the basis of:

A. On the Basis of change in proportion: There are two important correlations on the basis of change in proportion. They are:

(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.

B. On the basis of number of variables: On the basis of number of variables, correlation may be:

(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.

C. On the basis of Change in direction: On the basis of Chang in direction, correlation may be

(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.

Different degrees of Correlation

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.

Different methods of studying correlation

The different methods of studying relationship between two variables are:

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 this method.

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

iii) It is not influenced by the extreme items.

Demerits:

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

ii) It gives only approximate idea of the relationship.

ii) 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)        The coefficient of correlation lies between -1 and +1.

ii)       The co-efficient of correlation is independent to the unit of measurement of variable.

iii)     The co-efficient of correlation is independent the change of origin and scale.

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

v)      The coefficient correlation is the geometric mean of two regression coefficients.

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)     Great care must be exercised in interpreting the value of Karl Pearson’s coefficient of correlation as very often the coefficient is misinterpreted.

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.

Features of Spearman’s rank correlation:

i) The sum of the differences of ranks between two variables shall be zero.

ii) Spearmen’s correlation coefficient is distribution-free or non-parametric.

Merits:

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

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

iii)     This is the only method that can be used where ranks are given and not the actual data.

iv)     If actual values are given, than rank method can be applied for ascertaining correlation.

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.

Correlation and Causation

Correlation helps in determining the degree of relationship between two variables but it does not tell us anything about cause and effect relationship exists between variables. Correlation shows only the direction in which both variables move which may be same or opposite or no movement at all. Even if the degree of correlation between two variables is very high, it does not mean that change in one variable affects the value of another variable. It is also possible that correlated variables may be influenced by one or more other variables.

For example: suppose the correlation of teacher’s salaries and the consumption of liquor over a period of year comes out to be 0.8, this does not prove that every teacher drink; nor does it prove that liquor sale increases teacher’s salaries. Instead, both variables move together because both are influenced by a third variable – long-rum growth in national income and population. This type of study between variables that cannot be casually related is called spurious or nonsense correlation.

Correlation indicates only the mathematical result but while studying the correlation between two variables we should reach a conclusion based on logical reasoning and intelligent investigation of significantly related matters. This can be done only with the help of causation. Causation indicates that one event is the result of occurrence of other event i.e., there is a casual relationship between two variables. Correlation should be used with causation to derive effective results. In the above mentioned example, we find that there is correlation between teacher’s salaries and sale of liquor but no causation exist between the two variables. So, we can say that correlation does not imply causation. Similarly, causation between two variables does not imply correlation between variables.

Probable error

Correlation coefficients are calculated from sample data and there are chances of errors. In order to interpret the value of correlation coefficient probable error is used. With the help of probable error it is possible to determine the reliability of the value of the coefficient so far as it depends on the condition of random sampling. The probable error of the coefficient of correlation is obtained as follows:

P.E. (r) = 0.6745(1-r2)/√n

Where r is the coefficient of correlation and n is the number of pairs of observation. If r < P.E. (r) then there is no evidence of correlation. On the other hand if r > 6P.E. (r), then coefficient of correlation is practically certain. By adding and subtracting the value of probable error from the coefficient of correlation, we gent respectively the upper and lower limits within which coefficient of correlation can be expected to lie. Symbolically, rho (⍴) = r+ P.E.

But the measure of probable error can be properly used only when the following three conditions exist:

1. The data must approximate a normal frequency curve.

2. The statistical measure for which the P.E. is computed must have been calculated from a sample.

3. The sample must have been selected in an unbiased manner and the individual items must be independent.

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