Standard Error

Quite simply, the standard error is the standard deviation of the sampling distribution of a statistic. The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate. For example, the sample mean is the usual estimator of a population mean. However, different samples drawn from that same population would in general have different values of the sample mean. The standard error of the mean (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all possible samples (of a given size) drawn from the population. Secondly, the standard error of the mean can refer to an estimate of that standard deviation, computed from the sample of data being analyzed at the time.

In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the underlying errors.

Standard Error of the Mean

As mentioned, the standard error of the mean (SEM) is the standard deviation of the sample-mean's estimate of a population mean. It can also be viewed as the standard deviation of the error in the sample mean relative to the true mean, since the sample mean is an unbiased estimator. SEM is usually estimated by the sample estimate of the population standard deviation (sample standard deviation) divided by the square root of the sample size (assuming statistical independence of the values in the sample):

$\displaystyle SE_\bar{x} = \frac{s}{\sqrt{n}}$

where:

• $s$ is the sample standard deviation (i.e., the sample-based estimate of the standard deviation of the population), and
• $n$ is the size (number of observations) of the sample.

This estimate may be compared with the formula for the true standard deviation of the sample mean:

$\displaystyle SD_\bar{x} = \frac{\sigma}{\sqrt{n}}$

The standard error and the standard deviation of small samples tend to systematically underestimate the population standard error and deviations. This is due to the fact that the standard error of the mean is a biased estimator of the population standard error. Decreasing the uncertainty in a mean value estimate by a factor of two requires acquiring four times as many observations in the sample. Or decreasing standard error by a factor of ten requires a hundred times as many observations.

Standard Error Versus Standard Deviation

The standard error and standard deviation are often considered interchangeable. However, while the mean and standard deviation are descriptive statistics, the mean and standard error describe bounds on a random sampling process. Despite the small difference in equations for the standard deviation and the standard error, this small difference changes the meaning of what is being reported from a description of the variation in measurements to a probabilistic statement about how the number of samples will provide a better bound on estimates of the population mean. Put simply, standard error is an estimate of how close to the population mean your sample mean is likely to be, whereas standard deviation is the degree to which individuals within the sample differ from the sample mean.

Correction for Finite Population

The formula given above for the standard error assumes that the sample size is much smaller than the population size, so that the population can be considered to be effectively infinite in size. When the sampling fraction is large (approximately at 5% or more), the estimate of the error must be corrected by multiplying by a "finite population correction" to account for the added precision gained by sampling close to a larger percentage of the population. The formula for the FPC is as follows:

$\displaystyle \text{FPC} = \sqrt{\frac{N-n}{N-1}}$

The effect of the FPC is that the error becomes zero when the sample size $n$ is equal to the population size $N$.

Correction for Correlation In the Sample

If values of the measured quantity $A$ are not statistically independent but have been obtained from known locations in parameter space $x$, an unbiased estimate of the true standard error of the mean may be obtained by multiplying the calculated standard error of the sample by the factor $f$:

$\displaystyle f=\sqrt{\frac{1+\rho}{1-\rho}}$

where the sample bias coefficient $\rho$ is the widely used Prais-Winsten estimate of the autocorrelation-coefficient (a quantity between $-1$ and $1$) for all sample point pairs. This approximate formula is for moderate to large sample sizes and works for positive and negative $\rho$ alike.

Relative Standard Error

The relative standard error (RSE) is simply the standard error divided by the mean and expressed as a percentage. For example, consider two surveys of household income that both result in a sample mean of $50,000. If one survey has a standard error of $10,000 and the other has a standard error of $5,000, then the relative standard errors are 20% and 10% respectively. The survey with the lower relative standard error has a more precise measurement since there is less variance around the mean. In fact, data organizations often set reliability standards that their data must reach before publication. For example, the U.S. National Center for Health Statistics typically does not report an estimate if the relative standard error exceeds 30%.