$r=\dfrac{\text{Cov}(x,y)}{\sigma_x\sigma_y}$ measures the degree and direction of linear relationship, $-1\le r\le 1$.

Karl Pearson's coefficient of correlation between two variables $X$ and $Y$ is defined as the ratio of their covariance to the product of their standard deviations:

$r=\frac{\text{Cov}(x,y)}{\sigma_x\,\sigma_y}=\frac{\Sigma(x-\bar x)(y-\bar y)}{\sqrt{\Sigma(x-\bar x)^2\,\Sigma(y-\bar y)^2}}.$

It is a pure number lying between $-1$ and $+1$; its sign shows the direction (positive/negative) and its magnitude shows the strength of the linear relationship.

The correlation coefficient always lies between $-1$ and $+1$.

Karl Pearson's coefficient of correlation $r$ measures the strength and direction of the linear relationship between two variables and is a pure number with no unit. It is bounded:

$-1\le r\le +1.$

• $r=+1$: perfect positive correlation,

• $r=-1$: perfect negative correlation,

• $r=0$: no linear correlation.

Karl Pearson's formula is $r=\dfrac{\text{cov}(x,y)}{S_x\,S_y}$, so the numerator is the covariance of $X$ and $Y$. Option (B).

The Karl Pearson coefficient of correlation is defined as

$r=\frac{\text{cov}(x,y)}{S_x\cdot S_y}=\frac{\tfrac{1}{n}\sum (x-\bar x)(y-\bar y)}{S_x\,S_y}.$

Here the denominator is the product of the standard deviations $S_x$ and $S_y$, while the numerator is the covariance $\text{cov}(x,y)$ between the two variables.

Opposite-direction movement of two variables indicates negative correlation.

The direction of movement decides the sign of correlation:

• Same direction (both increase or both decrease together): positive correlation.
• Opposite directions (one increases as the other decreases): negative correlation.

Since the two variables here move in opposite directions, the correlation is negative.

Income and expenditure move in the same direction, an example of positive correlation.

A positive correlation exists when both variables increase (or decrease) together. Examine the options:

• (a) Repayment period and EMI: a longer period lowers the EMI — opposite direction (negative).
• (b) Income and expenditure: as income rises, expenditure usually rises too — same direction (positive).
• (c) Weight and Income: no logical relationship — essentially no correlation.
• (d) Price and demand: as price rises, demand falls — opposite direction (negative).

Only income and expenditure move together, giving positive correlation.

If two variables move together in the same direction — both increasing or, as here, both decreasing — the correlation between them is positive.

The sign of correlation is decided by the direction in which the variables move together:

• Positive correlation: both variables change in the same direction (both increase together, or both decrease together).
• Negative correlation: the variables change in opposite directions (one increases while the other decreases).

Here both variables move in the decreasing direction together — i.e. the same direction — so the correlation is positive.