Properties of Sampling Distributions

Sampling Distributions and Inferential Statistics

Sampling distributions are important for inferential statistics. In practice, one will collect sample data and, from these data, estimate parameters of the population distribution. Thus, knowledge of the sampling distribution can be very useful in making inferences about the overall population.

For example, knowing the degree to which means from different samples differ from each other and from the population mean would give you a sense of how close your particular sample mean is likely to be to the population mean. Fortunately, this information is directly available from a sampling distribution. The most common measure of how much sample means differ from each other is the standard deviation of the sampling distribution of the mean. This standard deviation is called the standard error of the mean.

Standard Error

The standard deviation of the sampling distribution of a statistic is referred to as the standard error of that quantity. For the case where the statistic is the sample mean, and samples are uncorrelated, the standard error is:

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

Where $s$ is the sample standard deviation and $n$ is the size (number of items) in the sample. An important implication of this formula is that the sample size must be quadrupled (multiplied by 4) to achieve half the measurement error. When designing statistical studies where cost is a factor, this may have a role in understanding cost-benefit tradeoffs.

If all the sample means were very close to the population mean, then the standard error of the mean would be small. On the other hand, if the sample means varied considerably, then the standard error of the mean would be large. To be specific, assume your sample mean is 125 and you estimated that the standard error of the mean is 5. If you had a normal distribution, then it would be likely that your sample mean would be within 10 units of the population mean since most of a normal distribution is within two standard deviations of the mean.

More Properties of Sampling Distributions

1. The overall shape of the distribution is symmetric and approximately normal.
2. There are no outliers or other important deviations from the overall pattern.
3. The center of the distribution is very close to the true population mean.

A statistical study can be said to be biased when one outcome is systematically favored over another. However, the study can be said to be unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated.

Finally, the variability of a statistic is described by the spread of its sampling distribution. This spread is determined by the sampling design and the size of the sample. Larger samples give smaller spread. As long as the population is much larger than the sample (at least 10 times as large), the spread of the sampling distribution is approximately the same for any population size