Just as the population distribution can be described with certain parameters, so can the sampling distribution. The expected value (mean) of any distribution can be represented by the symbol $$\mu$$ (mu). In the case of the sampling distribution, the mean $$\mu$$ is often written with a subscript to indicate which sampling distribution is being described. For example, the expected value of the sampling distribution of the mean is represented by the symbol $$\mu_{\bar x}$$. The value of $$\mu_{\bar x}$$ can be thought of as the theoretical mean of the distribution of sample means.

If we pick a large enough number of samples (of the same size) from a population and calculate their means, then the mean ($$\mu_{\bar x}$$ ) of all these sample means will approximate the mean ($$\mu$$) of the population. That is why the sample mean $$\bar x$$, is called an estimator of the population mean $$\mu$$. Thus, the mean of the sampling distribution is equal to the mean of the population:

$\mu_{\bar x} = \mu$

The standard deviation of a sampling distribution is given a special name, the standard error. The standard error of the sampling distribution of a statistic, denoted as $$\sigma_{\bar x}$$, describes the degree to which the computed statistic may be expected to differ from another one calculated from a sample of similar size and selected from similar population model. The larger the standard error of a given statistic, the greater the differences between the computed statistics for the different samples (Lovric 2010).

Note: The standard error, $$\sigma_{\bar x}$$, is not equal to the standard deviation, $$\sigma$$, of the population distribution (unless $$n = 1$$).

The standard error is equal to the standard deviation of the population divided by the square root of the sample size:

$\sigma_{\bar x} = \frac{\sigma}{\sqrt{n}}$

The equation holds true only when the sampling is done either with replacement from a finite population, or with or without replacement from an infinite population. This corresponds to the condition that the sample size $$(n)$$ is small in comparison to the population size $$(N)$$. The sample size is considered to be small compared to the population size, if the sample size is equal to or less than 5 % of the population size, so if

$\frac{n}{N} \le 0.05.$

If this condition is not satisfied, the following equation is used to calculate $$\sigma_{\bar x}$$:

$\sigma_{\bar x} = \frac{\sigma}{\sqrt{n}}\sqrt{\frac{N-n}{N-1} }$

In most practical applications, however, the sample size is small compared to the population size.

Citation

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Hartmann, K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.