Probable Error of Correlation (P.E.)

Probable Error (P.E.) is an important statistical measure used to judge the reliability of Karl Pearson’s coefficient of correlation (r).

In simple words, P.E. helps us decide:

• Whether the calculated value of r is significant, and
• Whether the observed relationship is due to chance or is meaningful.

Meaning of Probable Error

The probable error of correlation indicates the range within which the true value of the population correlation is likely to lie.

It also helps determine:

• If r is reliable
• If correlation is significant

In exams, P.E. is mostly used for interpretation, not computation.

Formula for P.E.

For Karl Pearson’s coefficient of correlation:

P.E. = 0.6745 × (1 − r²) / √n

Where:

• r = Pearson’s correlation coefficient
• n = number of pairs of observations
• 0.6745 = constant (used in statistical estimation)

Importance of Probable Error

P.E. helps in:

• Testing the significance of correlation
• Understanding whether the observed r is due to chance
• Comparing reliability of correlations
• Making business decisions based on true relationships

Rules for Interpretation

Use the following rules in exams (VERY important):

1. If r < P.E.

Correlation is not significant.

• The relationship is weak or unreliable.

2. If r > 6 × P.E.

Correlation is highly significant.

• The relationship is real and dependable.

3. If P.E. is very small

Correlation is more reliable.

4. If r is near zero and P.E. is high

Correlation is meaningless.

Solved Example

Given:

• r = 0.7
• n = 20

Find whether r is significant.

Step 1: Apply Formula

P.E. = 0.6745 × (1 − r²) / √n

r² = 0.7² = 0.49

1 − r² = 0.51

√n = √20 ≈ 4.472

P.E. = 0.6745 × 0.51 / 4.472

P.E. ≈ 0.0769

Step 2: Interpret Significance

Check the rule:

Is r > 6 × P.E. ?

6 × P.E. = 6 × 0.0769 ≈ 0.4614

Since:

r = 0.7 > 0.4614

Conclusion:

The correlation is highly significant. The relationship between X and Y is reliable.

Advantages of Using Probable Error

• Simple method to check significance
• Helps validate correlation results
• Useful when sample size is small
• Eliminates incorrect interpretations

Limitations

• Only applicable to Pearson’s r
• Not accurate for non-linear relationships
• Less precise than modern statistical tests (e.g., t-test)

Summary

• P.E. checks reliability of Pearson’s r.
• Formula: P.E. = 0.6745 × (1 − r²) / √n
• If r < P.E. → Not significant
• If r > 6 × P.E. → Highly significant
• Helps avoid wrong conclusions in business decisions

Quiz Time! 🎯