Standard Error of Correlation Coefficient Formula

The standard error of the coefficient of correlation measures the expected variation in the correlation coefficient ($r$) when calculated from different samples drawn from the same population. It helps in determining the significance of the observed correlation.

Standard Formula

The widely accepted formula for the standard error of the coefficient of correlation ($SE_r$) for a sample is:

where:

• $r$ is the sample coefficient of correlation.
• $n$ is the number of paired observations in the sample.

Applying the Formula

For the given question, we have 25 paired observations, so the sample size $n = 25$. The formula to calculate the standard error of the coefficient of correlation remains consistent.

Substituting $n=25$ into the formula gives:

Comparing this with the given options, Option B correctly represents the standard formula.

Option Analysis

• Option 1: $\frac{(1 - r^2)}{\sqrt{n}}$ - Incorrect. Lacks the $(n-2)$ denominator adjustment.
• Option 2: $\sqrt{\frac{(1 - r^2)}{n - 2}}$ - Correct. This is the standard formula for the standard error of the correlation coefficient.
• Option 3: $0.6745 \left( \frac{1 - r^2}{\sqrt{n}} \right)$ - Incorrect. The factor $0.6745$ is typically associated with the standard error of the median, not the correlation coefficient.
• Option 4: $\sqrt{\frac{n - 2}{(1 - r^2)}}$ - Incorrect. This is the reciprocal of the standard error formula, squared.

Therefore, the correct formula used to calculate the standard error of the coefficient of correlation for 25 paired observations is $\sqrt{\frac{(1 - r^2)}{n - 2}}$.