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sampling distribution

 
Dictionary: sampling distribution
 
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n.

The distribution of a statistic, such as occurs when a number of sample means are calculated for a given population.


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Statistics Dictionary: sampling distribution
 
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Distribution that describes the variation in the values of a statistic over all possible samples. For example, if n values are sampled from a population and if X1, X2,..., Xn, are the random variables representing the individual sample values, then the sample mean X̄, given by

X̄=1/n (X1+X2+...+Xn)
, is a random variable. The variability of the n values about their mean,
V2=1/n {(X1 − X̄)2+(X2 − X̄)2+...+(Xn − X̄)2},
is also a random variable. The form of the sampling distributions of X̄ and V2 will depend on the population, but statements can nevertheless be made about their moments. If the population has mean μ and variance σ2, then, for an infinite population, or for sampling with replacement from a finite population, each of X1, X2,...has mean μ and variance σ2. Consequently the expectation of X̄ is μ and the expectation of V2 is
n-1/n σ2
. This shows that X̄ and
S2=n/n-1V2
are unbiased estimators of μ and σ2, respectively. The variance of the sample mean X̄ is
1/n σ2.


The sample variance is usually taken to be the value of S2, though the value of V2 is sometimes used.


 
Accounting Dictionary: Sampling Distribution
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Giving the probability of each possible value of a statistic. It is computed from a sample of n items, for all possible samples of size n from a particular population. For example, compute a statistic such as the mean, standard deviation, and so on, which will vary from sample to sample. In this manner, a distribution is obtained of a statistic that is its sampling distribution.

 
Wikipedia: Sampling distribution
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In statistics, a sampling distribution is the probability distribution of a given statistic (a numerical quantity calculated from the data values in a sample), based on a random sample. The sampling distribution depends on the distribution of the population, the statistic being considered, and the sample size used.

The sampling distribution may be considered as the distribution of the statistic for all possible samples of a given size.

For example, consider a very large normal population (one that follows the so-called bell curve). Assume we repeatedly take samples of a given size from the population and calculate the sample mean (\bar x, the arithmetic mean of the data values) for each sample. Different samples will lead to different sample means. The distribution of these means is the "sampling distribution of the sample mean" (for the given sample size). This distribution will be normal since the population is normal. (According to the central limit theorem, if the population is not normal but "sufficiently well behaved", the sampling distribution of the sample mean will still be approximately normal provided the sample size is sufficiently large.)

Thus, the mean of the sampling distribution is equivalent to the expected value of any statistic. For the case where the statistic is the sample mean:

\mu_{\bar x} = \mu

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

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

where σ is the standard deviation of the population distribution of that quantity and n is the size (number of items) in the sample.

A very important implication of this formula is that you must quadruple the sample size (4×) to achieve half (1/2) the measurement error. When designing statistical studies where cost is a factor, this may have a factor in understanding cost-benefit tradeoffs.

Alternatively, consider the sample median from the same population. It has a different sampling distribution which is generally not normal (but may be close under certain circumstances).

Examples

Population Sample statistic Sampling distribution
Infinite, X \sim N(\mu, \sigma^2) Sample mean, \bar X \bar X \sim N \left (\mu, \frac{\sigma^2}{n} \right )
Finite (size N), X \sim N(\mu, \sigma^2) Sample mean, \bar X \bar X \sim N \left (\mu, \frac{N - n}{N - 1} \times \frac{\sigma^2}{n} \right )
Infinite, X \sim \operatorname{Binomial}(p) Sample proportion, \bar p \bar p \sim \operatorname{Binomial}(p)
Infinite, X_1 \sim N(\mu_1, \sigma_1^2), X_2 \sim N(\mu_2, \sigma_2^2) Sample difference between means, \bar X_1 - \bar X_2 \bar X_1 - \bar X_2 \sim N \left (\mu_1 - \mu_2, \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}\right )

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Copyrights:

Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved.  Read more
Statistics Dictionary. A Dictionary of Statistics. Second edition revised. Copyright © Oxford University Press, 2008. All rights reserved.  Read more
Accounting Dictionary. Dictionary of Accounting Terms. Copyright © 2005 by Barron's Educational Series, Inc. All rights reserved.  Read more
Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Sampling distribution".  Read more

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