Statistics For Economics Chapter 7 Correlation

C.

non-existent

B.

Minus one to plus one

B.

When Y decreases X increases

C.

Independent

C.

Scatter diagram

A.

More accurate than rank correlation coefficient

r is preferred to co-variance as a measure of association because it studies and measures the direction and intensity of relationship among variables.

No, r cannot lie outside -1 and 1 range depending on the types of data. It lies between minus one and plus one. Symobtically.

-1 ≤ r ≥ 1

If in any exercise, r is outside this range it indicates error in calculation.

No correlation does not imply causation. It implies covariation. It should never be interpreted as implying causes and effect relation.

Rank correlation is more precise than simple correlation when the variables cannot be measured meaningfully as in the case of price, income, weight etc. Ranking may be more meaningful when the measurement of the variables are suspect. Ranking may be a better alternative to quantification of qualities.

No, but there is possibility of independence.

No.

Students are suppose of collect the price of any five vegetables from the market everyday for a week. Then they should calculate their correlation of co-efficients.

Impartiality, secularism, beauty, honesty, patriotism etc. are some variables where accurate measurement is difficult.

(i) r as 1 mean that it perfect positive relationship between two variables.

(ii) r as –1 mean that there is perfect negative relationship between two variables.

(iii) r as O mean that there is lack of correlation between two variables.

Because rank correlation co-efficient provides a measure of linear association between ranks assigned to these limits and not their values.

Try yourself.

Calculate of Karl Pearson’s correlation co-efficient.

A.M. of X series =20 A.M. of Y series = 45

Hence, r = 0

Two values X and Y are un-corrected.

There is no linear correlation between them.

Calculation of Correlation:

According to Croxton and Cowden, correlation is defined as “When the relationship is of a quantitative nature, the appropriate statistical tool for discovering and measuring the relationship and expressing it in a brief formula is known as correlation.”

The principal methods are as under:

(i) Scattered Diagram Method.

(ii) Karl Pearson’s Co-efficient of Correlation.

(iii) Spearman’s Rank Correlation Coefficient.

When two variables move in same direction, such a relation is called positive correlation. For example relationship between price and supply. When two variables change in diffdrent directions, it is called negative correlation. For example relationship between price and demand.

(a) Positive and negative correlation.

(b) Linear and non-linear correlation.

(c) Simple and multiple correlation.

–1 < r < + 1.

Positive correlation.

A scattered diagram does not measure the precise extent of correlation. It gives only an approximate idea of the relationship. It is not a quantitative measure of the relationship.

Rank Correlation method is used for the variables whose quantitative measurement is not possible, such as beauty, bravery, wisdom.

Correlation measures the direction and intensity of relationship. It measures covariation and not causation.

The presence of correlation between the variables X and Y simply means that when the value of one variables is found to change in one direction, the value of other variable is found to change either in the same direction (i.e. positive direction) or in the opposite direction (i.e. negative direction) but in a difinate way.

Rank correlation is preferred to Personian co-efficient when extreme values are present.

In general r_k ≤ r.

Spearman’s rank correlation is not as accurate as the ordinary method. This is due to the fact that all the information concerning the data is not utilised.

r and r_k give identical results when the first differences of the values of the items in the series arranged in the order of magnitude are constant.

The use of rank correlation is suitable when data cannot be directly quantitatively measured.

Here r = +1 as the values of X and Y variables move in the same direction.

r = –1, when the values of two variables X and Y move in the opposite directions.

When the change ratio in values of two variables is constant.

Non-liner correlation.

Correlaion between two variables concentrate between +1 and –1.

The scatter diagram given in the question indicates the positive relation.

Negative relation between two variables.

Here r = –1 because the value of X and Y variable move in the opposite direction.

A high value of r indicates linear relationship. Its value is said to be high when it is close to +1 or -1.

The value of r is unaffected by the change of origin and change of scale.

A low value of r indicates a weak linear relation. Its value is said to be low when it is close to zero.

It shows less than proparionate change or there exists a low degree of association between X and Y.

–1 ≤ r ≤ +1.

It implies that X and Y are independent variables.

Karl Pearson’s measure of correlation is given by is given by

According to Croxtion and Cowden, when the relationsip is of a quantitative nature, the appropriate statistical tool for discoverinng and the measuring the relationship and expressing it in a brief formula is knwon as statistics.

(i) When r = 0, it implies that there exists no relationship between two variables X and Y. On other ways, r = 0 shows the absence of relationship.

(ii) r = 1 shows that there is perfect correlation between two variables X and Y.

(iii) r = –1 shows that there is perfect negative relationship between two variables X and Y.

When correlation is perfect and negative then the value of r = –1

(ii) When correlation is perfect and positive then the value of r = +1

(iii) When there is no correlation then the value of r = 0.

Positive Correlation : When two variables move in the same direction, that is when one increases the other also increases and when one decreases the other also decreases then such a relation is called Positive correlation. For example, the relation between price and supply.

Increase in the value of both variables.

X : 10 20 30 40

Y : 100 150 200 250

2. Negative Correlation : When two variables change different directions, it is called negative correlation. For example relationship between price and demand. When prices rises other things remaining constant demand falls and when price falls demand rises.

X : 1 2 3 4

Y : 5 4 2 1

X : 10 20 30 40

Y : 100 150 200 250

Perfect correlation is that where changes in two related variables are exactly proportional. It is of two types:

(i) Positive perfect correlation and (ii) Negative perfect correlation.

There is perfect positive correlation between the two variables of equal proportional changes are in the same direction. It is expressed as +1. If equal proportional changes are in the reverse direction. Then there is negative perfect correlation and it is described as –1.

(a) Example of positive perfect correlation

Price (Rs): 1 2 3 4 5

Supply (units) : 10 20 30 40 50

(b) Example of perfect negative correlaion

Price (Rs): 1 2 3 4 5

Demand (units) : 50 40 30 20 10 Q. 5. Define simple, and partly correlation.

Ans. Simple Correlation : When the relationship between two variables is studied and of these two variables one is independent and the other is dependent, then such correlation is called simple correlation. For example, relationship between income and expenditure.

Partial Correlation : When more than two variables are involved and out of these the relationship between two variables is studied only treating other variables as constant, then such correlation is partial.

Multiple Correlation : When relationship among three or more than three variables is studied simultaneously then such relationship is called multiple correlations. In case of such correlation the entire set of independent variables is studied. For example the effect of rainfall, manure, water etc. to increase the productivity of what are simultaneously studied.

The co-efficient of correlation of Karl Pearson’s is the quantitative measure of the relationship of two variables X and Y. It is represented by r. It is based on arithmetic mean and standard deviation. The co-efflciant of correlation (r) of two variables is obtained by dividing the sum of the products of the corresponding deviations of the various items of the two series from their respective means by the product of their standard deviation and the number of pairs of observations. Symbolically.

Limited degree of correlation coefficient: The value of correlation coefficient lies between minus one and plus one. The calculated result of correlation must valy between -1 to +1. Symbolically.

Rank collection co-efficient and simple correlation coefficient have the same interpretation. Its formula has been derived from simple correlation coefficient where individual values have been replaced by rank. These ranks are used for the calculation of correlation. This coefficient provides a measure of linear association between ranks assigned to units not their values correlation. Following formula is used for calculating rank correlation

In Karl Pearson’s coefficient of correlation, the values of two series are not assigned rank. The correlation coefficient provides a measure of linear association to the values of the variables. Following formula i.e. used for calculating the correlation coefficient.

(i) r as 1 indicates that there is perfect positive correlation between the two values.

(ii) r as –1 indicates that there is perfect negative correlation between the two values.

(iii) r = 0 indicates that there is no relation between the values.

The points in the scatter diagram tendering about a st. line which makes an angle of 30° with the X axis indicate the less than proportionate changes. There exists a low degree of association between X and Y.

(b) No relationship between two variables.

(c) Perfect Positive correlation ( r = +1)

(d) Perfect Negative correlation ( r = –1)

Karl Pearson’s coefficient of correlation : Karl Pearsons’s coefficient of correlation, is used to measure the degree of relationship between two or more variabless. It is represented by r. It is based on arithmetic mean and standard deviation. The coefficient of correlation (r) of two variables is obtained by dividing the sum of the products of the corresponding deviations of the various items of two series from their respective means by the product of their standard deviations and the number of pairs of observaions.

(i) r is always between –1 and +1 that is –1 ≤ r ≤ 1

(ii) If r = +1 or r=–l it means that there is perfect relation between the variables. In other ways the relation between two variables is exact.

Properties of correlation coefficient:Following are main properties of correlation coefficient:

1. r has no unit. It is a pure number.

2. A negative value of r indicates an inverse relation. A change in one variable is associated with change in the other variable in the opposite direction.

3. If r is positive the two variables move in the same direction.

4. If, r = 0, the two variables ate uncorrelated. There in no linear relation between them.

5. If r = 1 r = –, the correlation is perfect. The relation between two variables is extact.

6. A high value of r indicates strong linear relationship. Its value in said to be high when it is close to+1 or –1.

7. A low value of r indicates a weak linear relation. Its value is said to be low when it is close to zero.

8. The value of correlation coefficient lies between minus one and pluse one symbolically:

–1 ≤ r ≤ 1

9. The value of r is unaffected by the change of origin and change of scale.

A scatter diagram : A scatter diagram is a simple visual method for getting some idea about the presence of correlation between two variables. In a scatter diagram, we plot the values of two variables as a set of points on a graph paper, the cluster of points is called scatter diagram.

Drawing of a scatter diagram involves following steps:

1. Writting down the independent variables on X axis.

2. Writting down the dependent variables on Yaxis.

3. Points with the help of given data are marked on the graph paper.

4. When the plotted points show some trend upward or downward, we know that there is some correlation between the variables. When the trend is upward, the correlation is positive. On the other hand, when the trend is downward, the correlation is negative as shown in fig.

Merits of Scatter Diagram:

(i) Scatter diagram is a very simple method of studying correlation between two variables.

(ii) Just a glance of the diagrams is enough to know if the values of the variables have any relation or not.

(iii) Scatter diagram also indicates whether the relationship is positive or negative.

2. Demerits of Scatter Diagram:

(i) A scatter diagram does not measure the precise extent of correlation.

(ii) It gives only an approximate idea of the relationship.

(iii) It is only an qualitative expression of the quantitative change.

Calculation of r between schooling of farmers and annual yields

Let A = 100 H = 10 B = 13.0 R = 100

The table of transformed variables are as follows

Calculation of r between price Index and money supply using step deviation method

		U2	V2	UV
2	1	4	1	2
5	3	25	9	15
9	8	81	64	72
12	10	144	100	120
13	13	169	169	169
ΣU = 41	ΣV = 35	ΣU2 = 423	ΣV2 = 343	ΣUV = 378

Substituting these value in the formula

It shows that is strong positive correlation between price index and money supply

Steps involved in the procedure of calculation of Karl Pearson’s coefficient of correlation by direct method.

1. Calculate mean value x and y.

2. Calculate deviations of values of x series from mean value.

3. Square the deviations.

4. Calculate deviation of values of y series from mean value.

5. Square the deviation.

6. Multiply the square of deviation of X series with the square of deviations of Y series.

7. Use the following formula for calculating correlation coefficient

Calculation of Pearson’s co-efficient of correlation

Steps involved in calculating correlation co-efficient by short cut method.

1. Take any convenient value in X and Y series as assumed mean.

2. Calculate the deviation of the values of individual variable from assumed mean and denote them by dx and dy respectively.

3. Add the deviation of both series seperately and denote them by Σdx and Σdy.

4. Multiply the deviation of the two series and denote them by dx dy.

5. Add the dx dy to obtain Σdxdy.

6. Square the deviation and denote them by dx² and dy².

7. Add dx² and dy² to find out Σdx² and Σdy².

8. Calculate correlation co-efficient by using the following formula :

Here

(i) dX = deviation of X series from the assumed mean (X - A).

(ii) dY = deviation of y series from the assumed mean (Y – A).

(iii) Σ dXdY Sum of the multiple of dX and dY

(iv) ΣdX² sum of square of dX

(v) ΣdY² sum of square of dY

(vi) Σ dX sum of deviation of X series

(vii) Σ dY = sum of deviation of Y series

Calculation of Karl Pearson’s correlation co-efficient

A.M. of X series 69 assumed mean of Y series = 112

Calculate of Karl Pearson’s correlation co-efficient.

A.M. of X series =20 A.M. of Y series = 45

Assumed mean of X = 85 Assumed mean of Y = 80

Steps involved in calculating correlation Co- efficient by step - deviation:

1. Take the assumed mean of X series.

2. Take the assumed mean of Y series.

3. Calculate deviations of X and Y from assumed mean and name them dX and dY respectively.

4. Divide dX and dY by convenient common factor to reduce the calculations. This is also called change of origin and change of scale of the values of X and Y. The value of co- efficient of correlation is unaffected by the change in origin and change of scale if X and Y. After changing the deviations we apply the same formula of assumed mean method.

Step deviation method of calculating correlation co-efficient: This method involves the following steps:

1. Take any convenient value in X and Y series as assumeed value Ax and Ay.

2. With the help of assumed mean of both the series, deviation of the values of individual variable i.e. dx (X-A) and dy (Y-A) are calculated.

3. Now divide dx and dy by some common factor as dx = dx/cl and dy/c² = where c₁ is common factor for series X and C₂ is common factor for series. Y ‘dx’ and dy are step deviations.

4. Multiply the step deviations of the two series and get dx' dy'

5. Add the dx', dy', and get Σdx'edy'

6. Square the dx and dy' and add to find out Σdx 2 and Σdy 2

7. Apply the following formula to calculate co- efficient of correlation.

Calculation of correlation:

Co-efficient by step deviation method

A.M. of X series = 15 A.M. of Y series = 30

There is a perfectly negative correlation between price and quantity demanded.

Calculation of rank correlation:

Calculation of Rank correlation

Substituting these values in formula