How many independent reads to hit a target precision

Question

Each independent snapshot of order flow gives an unbiased estimate of fair value with a standard deviation of $20$ basis points. You average $n$ such snapshots.

How many snapshots do you need so the averaged estimate has a standard deviation of at most $5$ basis points? What does this say about the cost of precision?

Accepted Answer

Averaging n independent, equally reliable estimates gives each weight 1/n, and the variance of the average is 	ext{Var}(\bar X) = \frac{\sigma^2}{n}, \qquad \sigma{\bar X} = \frac{\sigma}{\sqrt n}. The standard deviation falls with \sqrt n, not n. Setting the target: \frac{20}{\sqrt n} \le 5 \quad\Longrightarrow\quad \sqrt n \ge 4 \quad\Longrightarrow\quad n \ge 16. You need 16 independent snapshots to bring the standard error from 20 down to 5 basis points. The cost of precision. Because error scales as 1/\sqrt n, cutting it in half requires four times the data, and cutting it to a quarter …

How many independent reads to hit a target precision

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