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When two variables are not related by cause-effect relation and there is lack of linear correlation between them, then such correlation is called spurious correlation. For example: There is spurious correlation between the yearly import of crude oil and the number of marriages during the same time period.

‘Negative correlation’ between the Altitude and amount of Oxygen in the air.

‘Negative correlation’ between the rate of inflation and the purchasing power of common man of country when income of the common man is stable.

r (X, Y) = 0.4. If 5 is added to each observation of X and 10 is subtracted from each observation of y, means r [x + 5, y-10) will also be 0.4, as the value of r is independent of change of origin.

‘Nonsense correlation’ between the annual Import of crude oil and the number of marriages during the same time period.

Correct option is (c) r(x, y) = r (u, v)

Correct option is (a) 0.7

Correct option is (b) – 0.5

Correct option is (c) Scatter diagram

Here, n = 8;

r = 0.5
True difference = 7.

False difference = 3

Now.,  \(r = 1 – \cfrac{6 (\Sigma d^{2}+CF)}{n\left(n^{2}-1\right)}\)

Putting n = 10 and r = 0.5 in the formula.

\(0.5 = 1 – \frac{6Σd^2}{10(10^2−1)}\)

\(∴ \frac{6Σd^2}{10(100−1) }= 1 – 0.5\)

\(∴ \frac{6Σd^2}{990 }= 0.5\)

\(∴ Σd^2 = \frac{0.5×990}{6} = 82.5\)

Corrected Σd² = 82.5 – (False d)² + (True d)²
= 82.5 – (3)² + (7)²
= 82.5 – 9 + 49
= 122.5

Corrected rank correlation coefficient:

\(r = 1 – \cfrac{6( Corrected \,Σd^2)}{n(n^2−1)}\)

Putting n = 10 and corrected Σd² = 122.5 in the formula.

\(r = 1 – \cfrac{6(122.5)}{10(10^2−1)}\)

= \(1 – \frac{735}{990}\)

= 1 – 0.74

= 0.26

Hence, the corrected value of the coefficient of rank correlation obtained is 0.26.

Correct option is (c) 1

The main limitation of scatter diagram method is that this method does not give exact degree of correlation between two variables.

If the value of the covariance is negative, the sign of r will be negative.

n(n² – 1) = 6Σd²

Now, r = 1 –\( \frac{6Σd^2}{n(n^2−1)}\)

Putting n(n² – 1) = 6Σd²

^{\(r = 1 – \frac{6Σd^2}{6Σd^2}\)}

∴ r = 1 – 1 = 0
Hence, the value of r is ‘0’.

Correct option is (b) Covariance of X and Y

r = 1 : There is perfect positive correlation between two variables. When increase / decrease in the value of one variable results in increase / decrease in the value of the other variable in constant proportion then the value of r is 1. In scatter diagram all points lie on the same straight line going in upward direction from left to right.

r = – 1 : There is perfect negative correlation between two variables. When increase / decrease in the value of the variable results decrease / increase in the value of the other variable in constant proportion, then the value of r is – 1. In scatter diagram all point lie on the same straight line going in downward direction from left to right.

r = 0 : There is no linear correlation between two variables, r = 0 shows absence of correlation and hence two variables are said to be linearly uncorrelated. In scatter diagram all points are randomly scattered. When r = 0 we can say only lack of linear correlation between two variables but there may be non-linear correlation between the variables.

When the values of two variables are some arrangement of first n natural numbers, the correlation coefficients obtained by Karl Pearson’s method and Spearman’s method are equal.

Rank Correlation Coefficient: When the measurement of two variables are not in the form of numbers but they are in the form of their ranks, then the coefficient of correlation between X and Y is called rank correlation coefficient. Such measure of correlation coefficient was suggested by Spearman. Hence, it is known as Spearman’s rank correlation coefficient. It is also denoted by r

Merits of Spearman’s rank correlation method : The merits of Spearman’s rank correlation method are as follows:

• This method is easy to understand.
• In calculating correlation coefficient, this method is easier than that of Karl Pearson’s method.
• When the related data is qualitative, this is the only method to find the measure of correlation.
• When there is more dispersion in the related numerical data or the extreme observations are present in the data, Spearman’s method is preferred than Karl Pearson’s method.

Limitations: Following are the limitations of Spearman’s rank correlation method:

• This method does not provide accurate measure of correlation coefficient as compared to Karl Pearson’s method.
• It is tedious to assign ranks when the number of observations is large.
• This method cannot be used for a bivariates frequency distribution.

Correct option is  (b) \(\frac{m^3−m}{12}\)

The assumptions of Karl Pearson’s method are as follows :

• There is linear correlation between two variables.
• There is cause-effect relation between two variables.

If the simultaneous changes occur in the values of the two correlated variables in the same direction and in the same proportion that means if the values of one variable increases / decreased by one unit, then the value of another variable increases / decreases in some constant proportion, the correlation between them is said to be perfect positive correlation.

In case of perfect positive correlation, all the points of a scatter diagram lie on the same line going in upward direction from left to right.

In case of perfect positive correlation, the value of the correlation coefficient r is 1.

For example, the correlation between the number of tickets purchased of a cinema and the price paid is perfect positive correlation. Suppose, 1 ticket cost ₹ 150, we have to pay ₹ 600 for 4 tickets.

If the simultaneous changes occur in the values of the two correlated variable in the opposite direction and in the same proportion that mean if the values of one variable increases / decreases by one unit, then the values of another variable decreases / increases in same constant proportion, the correlation between them is said to be perfect negative correlation.

In case of perfect negative correlation, all the points of a scatter diagram lie on the same line going in the downward direction from left to right.

In the case of perfect negative correlation, the value of correlation coefficient is – 1.

For example; the correlation between the height of a place from sea level and the proportion of oxygen in the air is perfect negative correlation. More and more height of a place from sea level the proportion of oxygen in the air becomes less and less.

If simultaneous changes occur in the values of two related variables due to direct or indirect cause- effect relation, then it can be said that there exists correlation between two variables

When the changes in the values of two correlated variables are nearly in constant proportion means the points of their ordered pairs are on a line or nearer to the line, then it is called linear correlation.

If the points plotted on the graph paper corresponding to ordered pairs of the values of two correlated variables are on a line or nearer to the line, then correlation between to variables is said to be linear correlation.

• Positive Correlation: When the changes in the values of two correlated variables are in the same direction, the correlation between the variables is said to be positive. The correlation between sale and profit of an item is the example of positive correlation.
• Negative Correlation: When the changes in the values of two correlated variables are in opposite direction, the correlation between the variables is said to be negative. The correlation between price and demand of an item is the example of negative correlation.

Correct option is (d) does not have any unit

• Correlation: If simultaneous changes occur in the values of two variables due to direct or indirect cause-effect relation, then it is said that there is correlation between two variables.
• Coefficient of Correlation: A numerical measure showing the strength or degree of a linear correlation between two variables is called the correlation coefficient. It is denoted by r.

Correct option is (a) Perfect Positive Correlation

Correct option is (c) – 1 ≤ r ≤ 1

1. Scatter Diagram Method:

This is a simple method to find the nature or type (positive or negative) of correlation showing the values of variable X on X-axis and that of variable Y on Y-axis by an appropriate scale, the points corresponding to n ordered pairs (x₁, y₁), (x₂, y₂), …. (x_n, y_n) are plotted on the graph paper. The graph showing the plotted points is called a scatter diagram. The pattern of points on a scatter diagram shows the nature or type of correlation and the strength of correlation up to some extent.

2. Karl Pearson’s Product Moment Method:

The method of finding the measure of the strength of linear correlation between two variables X and Y using the n pairs (x₁, y₁), (x₂, y₂), …….. (x_n, y_n) of observations on two variables X and Y is called Karl Pearson’s product moment method.

3. Spearman’s Rank Correlation Method:

When n pairs of observations (x₁, y₁), (x₂, y₂), …, (x_n, y_n) on two variables X and Y are given, then the method by which the correlation coefficient computed using the ranks assigned to the values of two variables in descending order of magnitudes is called Spearman’s rank correlation method.

Correct option is (b) 1 or – 1

Correct option is (d) Partial Negative Correlation

• The maximum value of correlation coefficient r is 1 and the minimum value is – 1. Thus, – 1 ≤ r ≤ 1.
• Correlation coefficient r (x, y) between variables X and Y and the correlation coefficient r (y, x) between variables Y and X are equal. Thus, r (x, y) = r (y, x).
• The value of r is not changed by the change of origin and scale. This means r (x, y) = r (u, v); where u = \(\frac{x−A}{C_x}\) and v = \(\frac{y−B}{C_y}\); Cx >0, Cy > 0 and A, B, Cx, Cy are constants.
• The value of r is independent of units of measurements of variables X and Y.
• Coefficient correlation r is an absolute measure.

The properties of correlation coefficient are as follows :

• The value of correlation coefficient r lies in the interval – 1 to 1. i.e., – 1 ≤ r ≤ 1.
• The correlation coefficient r is free from unit of measurement, i.e., it does not have any unit of measurement.
• The correlation coefficient between variables X and Y is same as that of between Y and X, i.e., r (x, y) = r (y, x).
• The value of correlation coefficient r does not change with the change of origin and scale, i.e., r (x, y) = r (u, v)

where, u = \(\frac{x−A}{C_x}\);

v = \(\frac{ y−B}{C_y}\). C_x > 0, C_y > 0

and A, B, C_x, C_y are constant.

• The correlation coefficient r is an absolute measure.
• If the sign of any one of two variables is changed then the sign of the correlation coefficient also changes, i.e., r(-x, y) = -r(x, y); r (x, – y) = – r (x, y)
• If the signs of both the variables are changed then the sign of the correlation remain unchanged, i.e., r(- x, – y) = r(x, y).

Merits of scatter diagram are as follows :

• Scatter diagram is a simple method to know the nature of correlation between two variables.
• It requires only the knowledge of plotting the points and the mathematical knowledge is required less.
• Up to some extent it gives idea about the strength of correlation between the variable.
• The patterns of the points on a scatter diagram. suggests whether the correlation is linear or not.
• The presence of some pairs of extrem observations does not arise any difficulty in judging the nature of correlation between the variable.

The limitations of scatter diagram are as follows:

• It does not give exact degree of correlation between two variables.
• It is not useful method of studying correlation between classified bivariate data.

Here Σ(x – x̄) (y – ȳ) = – 65, S_x = 3, S_y = 4 and n = 10.

Now, \( r = \frac{Σ(x−\bar x)(y−\bar y)}{n⋅Sx⋅Sy}\)

\(= \frac{−65}{10×3×4}\)

\(= \frac{−65}{120}= -0.54\)

Here, n = 10;

x̄ = 160;

ȳ = 55;

Σxy = 90000;

S_x = 25 and S_y = 10.

Now, \(r= \frac{\sum xy-n\bar x\bar y}{n.S_x.S_y}\)

\(=\frac{90000-10\times160\times55}{10\times25\times10}\)

\(=\frac{90000-88000}{2500}\)

\(=\frac{2000}{2500}=0.8\)

Hence, the correlation coefficient between height and weight obtained is 0.8.