The sampling distribution is a distribution of a sample statistic. When using a procedure that repeatedly samples from a population and each time computes the same sample statistic, the resulting distribution of sample statistics is a sampling distribution of that statistic. To more clearly define the distribution, the name of the computed statistic is added as part of the title. For example, if the computed statistic was the sample mean, the sampling distribution would be titled “the sampling distribution of the sample mean.”
What is it?
The sampling distribution is a distribution of a sample statistic. When using a procedure that repeatedly samples from a population and each time computes the same sample statistic, the resulting distribution of sample statistics is a sampling distribution of that statistic. To more clearly define the distribution, the name of the computed statistic is added as part of the title. For example, if the computed statistic was the sample mean, the sampling distribution would be titled “the sampling distribution of the sample mean.”
Suppose a population consists of the first ten integers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and a process takes a random sample without replacement of size N=3 from this population. The random sampling might generate sets that look like { 8, 3, 7}, { 2, 1, 5}, { 6, 3, 5}, { 10, 7, 5}… If the mean () of each sample is found, the means of the above samples would appear as follows: 6, 2.67, 4.67, 7.33… If this process was continued indefinitely, then the infinite number of sample means would form a sampling distribution of the mean. Graphically, this sampling distribution of the mean would appear as follows:
Every statistic has a sampling distribution. For example, suppose that instead of the mean, mediansmedians The score value that cuts the distribution in half, such that half the scores fall above the median and half fall below it. A measure of central tendency. (Md) were computed for each sample. That is, within each sample the scores would be rank ordered and the middle score would be selected as the median. Using the samples above, the medians would be: 7, 2, 5, 7… The infinite number of medians would be called the sampling distribution of the median and could be graphically shown as follows:
It is possible to make up a new statistic and construct a sampling distribution for that new statistic. For example, by rank ordering the three scores within each sample and finding the mean of the highest and lowest scores a new statistic could be created. Let this statistic be called the mid-mean and be symbolized by. For the above samples the values for this statistic would be: 5.5, 3, 4.5, 7.5… and the sampling distribution of the mid-mean could be graphically displayed as follows:
Just as the population distributions can be described with parametersparameters Variables within the model that must be set before the model is completely specified. Variables that change the shape of the probability model., so can the sampling distribution. The expected value and variance of any distribution can be represented by the symbols m (mu) and s2 (Sigma squared), respectively. In the case of the sampling distribution, the m symbol 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, that of the median by , and so on. The value of can be thought of as the theoretical mean of the distribution of means. In a similar manner the value of, is the theoretical mean of a distribution of medians.
The square root of the variance of a sampling distribution is given a special name, the standard errorstandard error The theoretical standard deviation of a sampling distribution.. In order to distinguish different sampling distributions, each has a name tagged on the end of “standard error” and a subscript on the s symbol. The theoretical standard deviation standard deviation A measure of variability. The positive square root of the variance. of the sampling distribution of the mean is called the standard error of the mean and is symbolized by. Similarly, the theoretical standard deviation of the sampling distribution of the median is called the standard error of the median and is symbolized by.
In each case the standard error of the sampling distribution of a statistic describes the degree to which the computed statistics may be expected to differ from one another when calculated from a sample of similar size and selected from similar population models. The larger the standard error of a given statistic, the greater the differences between the computed statistics for the different samples. From the example population, sampling method, and statistics described earlier, we would find = = = 5.5 and =1.46, = 1.96, and =1.39.
Why is the sampling distribution important?
Properties of statistics
Statistic have different properties as estimators of a population parameters. The sampling distribution of a statistic provides a window into some of the important properties. For example if the expected value of a statistic is equal to the expected value of the corresponding population parameter, the statistic is said to be unbiased. In the example above, all three statistics would be unbiased estimators of the population parameter.
Consistency is another valuable property to have in the estimation of a population parameter, as the statistic with the smallest standard error is preferred as an estimator estimator A statistic used to estimate a model parameter.of the corresponding population parameter, everything else being equal. Statisticians have proven that the standard error of the mean is smaller than the standard error of the median. Because of this property, the mean is generally preferred over the median as an estimator of.
Selection of distribution type to model scores
The sampling distribution provides the theoretical foundation to select a distribution for many useful measures. For example, the central limit theorem describes why a measure, such as intelligence, that may be considered a summation of a number of independent quantities would necessarily be distributed as a normal (Gaussian) curve.
Hypothesis testing
The sampling distribution is integral to the hypothesis testing procedure. The sampling distribution is used in hypothesis testing to create a model of what the world would look like given the null hypothesis was true and a statistic was collected an infinite number of times. A single sample is taken, the sample statistic is calculated, and then it is compared to the model created by the sampling distribution of that statistic when the null hypothesis is true. If the sample statistic is unlikely given the model, then the model is rejected and a model with real effects is more likely. In the example process described earlier, if the sample {3, 1, 4} was taken from the population described above, the sample mean (2.67), median (3), or mid-mean (2.5) can be found and compared to the corresponding sampling distribution of that statistic. The probability of finding a sample statistic of that size or smaller could be found for each e.g. mean (p< .033), median (p<.18), and mid-mean (p<.025) and compared to the selected value of alpha (α). If alpha was set to .05, then the selected sample would be unlikely given the mean and mid-mean, but not the median.
How can sampling distributions be constructed?
Mathematically
Using advanced mathematics statisticians can prove that under given conditions a sampling distribution of some statistic must be a specific distribution. Hogg and Tanis (1997, p. 256) prove the following theorem:
If X1, X2, …, Xn are observations of a random sample of size n from the normal distribution N(µ, d2),
and
then
is χ2(n-1)
The given conditions describe the assumptions that must be made in order for the distribution of the given sampling distribution to be true. For example, in the above theorem, assumptions about the sampling process (random sampling) and distribution of X (a normal distribution) are necessary for the proof.
Of considerable importance to statistical thinking is the sampling distribution of the mean, a theoretical distribution of sample means. A mathematical theorem, called the Central Limit Theorem, describes the relationship of the parameters of the sampling distribution of the mean to the parameters of the probability model and sample size.
Monte Carlo Simulations
It is not always easy or even possible to derive the exact nature of a given sampling distribution using mathematical derivations. In such cases it is often possible to use Monte Carlo simulations to generate a close approximation to the true sampling distribution of the statistic. For example, a non-random sampling method, a non-standard distribution, or may be used with the resulting distribution not converging to a known type of probability distribution. When much of the current formulation of statistics was developed, Monte Carlo techniques, while available, were very inconvenient to apply. With current computers and programming languages such as Wolfram Mathematica (Kinney, 2009), Monte Carlo simulations are likely to become much more popular in creating sampling distributions.
Summary
The sampling distribution, a theoretical distribution of a sample statistic, is a critical concept in statistical thinking. The sampling distribution allows the statistician to hypothesize about what the world would look like if a statistic was calculated an infinite number of times.
David W. Stockburger
US Air Force Academy
USAF Academy, CO 80840-6200
References
Hogg, R. V. and Elliot, A. T. (1997) Probability and Statistical Inference, Fifth Edition. Upper Saddle river, NJ: Prentice Hall.
Kinney, J. J. (2009) A Probability and Statistics Companion. Hoboken, NJ: John Wiley & Sons, Inc.
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