Question

In statistical analysis of numerical data, which ONE or MORE among the options is/are possible value(s) of correlation coefficient?

• A. 0
• B. 10.3
• C. -0.9
• D. 2

Hint:

The correlation coefficient always lies between -1 and 1; check which options fall outside this range.

Answer 1

The Correct Option is A, C

Step 1: Think about where the formula for r comes from:
The correlation coefficient is built as the covariance of two variables divided by the product of their individual standard deviations.
\[ r = \frac{Cov(X,Y)}{\sigma_X \sigma_Y} \]
This ratio is not arbitrary, it comes from a mathematical inequality, similar in spirit to the Cauchy-Schwarz inequality, which guarantees the numerator can never exceed the denominator in size.

Step 2: See why this forces a fixed boundary:
Because the covariance term can never be larger in magnitude than $\sigma_X \sigma_Y$, the ratio r can never cross beyond 1 on the positive side or below -1 on the negative side.
This is exactly why the bound $-1 \le r \le 1$ holds for every possible dataset, it is not a rule someone chose, it falls directly out of the formula.

Step 3: Use this boundary as a filter on the four numbers given:
0 sits comfortably inside the boundary, since $-1 \le 0 \le 1$ is true, so it can be a valid coefficient.
-0.9 also sits inside the boundary, since it is greater than -1 and less than 1, so it too can be valid.
10.3 and 2 both sit far outside the boundary on the positive side, well beyond what the formula allows, so neither can ever be produced by real data.

Step 4: Conclude which options survive the filter:
Only 0 and -0.9 pass this test, the other two numbers are mathematically impossible outputs of the correlation formula regardless of what dataset is used.

Final Answer:
The values a correlation coefficient can actually take are 0 and -0.9. \[ \boxed{0 \text{ and } -0.9} \]