Q.The correlation coefficient between the price of a commodity (X) and its quantity demanded (Y) is computed as $r=-0.85$. Interpret this result, and state whether it proves that a change in price causes the change in demand.

$r$ is invariant under a change of origin, so $r(x+0.2,y+0.2)=r(x,y)=0.7$.

A key property of Karl Pearson's correlation coefficient is that it is independent of change of origin and scale. Adding a constant to each value only shifts the origin; the deviations $(x-\bar x)$ and $(y-\bar y)$ are unchanged, so $r$ is unchanged.

$r(x+0.2,\,y+0.2)=r(x,y)=0.7.$

By its properties, Karl Pearson's correlation coefficient always satisfies $-1 \le r \le 1$ (inclusive), so the correct range is option (c).

In the GSEB Class-12 Statistics Linear Correlation chapter, one key property of $r$ is that it is a pure (unit-free) number lying in the closed interval $[-1, 1]$.

• $r = +1$ : perfect positive correlation.
• $r = -1$ : perfect negative correlation.
• $r = 0$ : no linear correlation.

Since the endpoints are attainable, the correct statement uses $\le$ signs, not strict inequalities:

$-1 \le r \le 1$

The correlation coefficient always satisfies $-1 \le r \le 1$; the endpoints are included because $r=\pm 1$ denotes perfect correlation.

Reasoning. By its definition $r = \dfrac{\text{Cov}(x,y)}{\sigma_x\,\sigma_y}$, and it can be shown algebraically that $|r| \le 1$. The value $r=+1$ occurs for perfect positive correlation and $r=-1$ for perfect negative correlation, so both endpoints are achievable and must be included.

Correlation is unchanged by scale/origin but flips sign when one variable's sign is reversed; here $r(x,-y)=\mathbf{-0.5}$.

Given: $r(-x,\,y) = -0.5$.

Step 1 — property. Reversing the sign of one variable multiplies $r$ by $-1$:

$r(-x,\,y) = -\,r(x,\,y).$

So $-0.5 = -\,r(x,y)$, giving $r(x,y) = 0.5$.

Step 2 — apply again for $r(x,-y)$.

$r(x,\,-y) = -\,r(x,\,y) = -0.5.$

$r$ is bounded: $-1 \le r \le 1$. Only $-1.07$ lies outside this range, so option (b) is impossible.

A fundamental property of Karl Pearson's coefficient of correlation is that its value can never go beyond the limits $-1$ and $+1$:

$-1 \le r \le +1$

Checking each option:

• (a) $0.99$ — lies in $[-1, 1]$, possible.

• (b) $-1.07$ — lies outside $[-1, 1]$ (since $-1.07 < -1$), impossible.

• (c) $-0.85$ — lies in $[-1, 1]$, possible.

• (d) $0$ — lies in $[-1, 1]$, possible (indicates no linear correlation).