Standard Error of Statistic

The term standard error has already been introduced very briefly in pervious tutorials while discussing about the sampling distribution of means. In this tutorial we will try to make some more comments on standard error.

In the broader sense of the term standard errors of mean, median, standard deviation, coefficient of correlation, regression coefficients etc. The mean, the median etc., are all refer to the sample and therefore referred to as sample statistic. The standard error of mean is very common and is widely used.

The standard error of any statistic may be determined by first drawing all possible samples (with replacement) of size nfrom the given population, then for each of the sample the required statistic (may be mean, median or standard deviation etc.) is computed. We therefore have a number of values of that statistic (one value for each sample). These values form the so called distribution of the sample statistic (e.g., the distribution of sample mean). If we calculate the standard deviation of such a distribution it will be referred to as the standard error of that statistic.

This method of determining the standard error is sometimes impractical. Fortunately we have relationships (which can be empirically verified) between the population standard deviation, the standard error and the sample size. The standard error of some important statistic are presented in table given below, where \sigma is the standard deviation of the population and n is the sample size.

Standard Errors
\sigma /\sqrt n
\sigma /\sqrt {\frac{\pi }{{2n}}}
\sigma /\sqrt {2n}
\frac{{{\sigma _X}}}{{{\sigma _Y}}}\sqrt {\frac{{1 - {\gamma ^2}}}{n}}
\sqrt {\frac{{1 - {\gamma ^2}}}{n}}

From the list of formulae given above it can be seen that the standard error depends, in most of the cases, on two factors, the sample size and the standard deviation of population. It measures the precision of the concerned statistic in estimating the parameter. In the above formulae, if the population is large and the population standard deviation \sigma is unknown, then \sigma can be replaced by S (the sample standard deviation) defined in pervious tutorial without much affecting our estimate of standard error specially when the sample size is large is reasonably large.