# Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations

### From CFD-Wiki

## Estimators for averaged quantities

Since there can never an infinite number of realizations from which ensemble averages (and probability densities) can be computed, it is essential to ask: *How many realizations are enough?* The answer to this question must be sought by looking at the statistical properties of estimators based on a finite number of realization. There are two questions which must be answered. The first one is:

- Is the expected value (or mean value) of the estimator equal to the true ensemble mean? Or in other words, is yje estimator
*unbiased?*

The second question is

- Does the difference between the and that of the true mean decrease as the number of realizations increases? Or in other words, does the estimator
*converge*in a statistical sense (or converge in probability). Figure 2.9 illustrates the problems which can arise.

## Bias and convergence of estimators

A procedure for answering these questions will be illustrated by considerind a simple **estimator** for the mean, the arithmetic mean considered above, . For independent realizations where is finite, is given by:

| (2) |

Now, as we observed in our simple coin-flipping experiment, since the are random, so must be the value of the estimator . For the estimator to be *unbiased*, the mean value of must be true ensemble mean, , i.e.

| (2) |

It is easy to see that since the operations of averaging adding commute,

| (2) |

(Note that the expected value of each is just since the are assumed identically distributed). Thus is, in fact, an *unbiased estimator for the mean*.

The question of *convergence* of the estimator can be addressed by defining the square of **variability of the estimator**, say , to be:

| (2) |