Spread of an average of 25 correlated estimates

Question

You average n = 25 analyst estimates, each with standard deviation \sigma = 6. The estimates move together, with every pair correlated at \rho = 0.3. What is the standard deviation of the average estimate?

Accepted Answer

Analyst estimates that all read the same news are correlated, so averaging them does not cancel error as effectively as the independent rule suggests. Use the correlated-average formula: 	ext{Var}(\bar{X}) = \frac{\sigma^2}{n}\Big[1 + (n-1)ho\Big]. Plug in \sigma^2 = 36, n = 25, ho = 0.3: 	ext{Var}(\bar{X}) = \frac{36}{25}\Big[1 + 24 	imes 0.3\Big] = 1.44 	imes [1 + 7.2] = 1.44 	imes 8.2 = 11.808. So the standard deviation of the average is \sqrt{11.808} \approx 3.44. The independent case would give \sigma/\sqrt{n} = 6/5 = 1.2. A correlation of 0.3 nearly triples the true spread of t…

Spread of an average of 25 correlated estimates

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