Share on Facebook Share on Twitter Email
Answers.com

sampling distribution

 
Dictionary: sampling distribution

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


Search unanswered questions...
Enter a question here...
Search: All sources Community Q&A Reference topics
Statistics Dictionary: sampling distribution
Top

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 , given by

=1/n (X1+X2+...+Xn)
, is a random variable. The variability of the n values about their mean,
V2=1/n {(X1)2+(X2)2+...+(Xn)2},
is also a random variable. The form of the sampling distributions of 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 is μ and the expectation of V2 is
n-1/n σ2
. This shows that and
S2=n/n-1V2
are unbiased estimators of μ and σ2, respectively. The variance of the sample mean 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
Top

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
Top

In statistics, a sampling distribution is the probability distribution of a given statistic based on a random sample of size n. It may be considered as the distribution of the statistic for all possible samples of a given size. The sampling distribution depends on the underlying distribution of the population, the statistic being considered, and the sample size used. The sampling distribution is frequently opposed to the asymptotic distribution, which corresponds to the limit case n → ∞.

For example, consider a normal population with mean μ and variance σ². Assume we repeatedly take samples of a given size from this population and calculate the arithmetic mean \scriptstyle \bar x for each sample — this statistic is called the sample mean. Each sample will have its own average value, and the distribution of these averages will be called the “sampling distribution of the sample mean”. This distribution will be normal \scriptstyle \mathcal{N}(\mu,\, \sigma^2/n) since the underlying population is normal.

This was an example of a simple statistic taken from one of the simplest statistical populations. For other statistics and other populations the formulas are frequently more complicated, and oftentimes they don’t even exist in closed-form. In such cases the sampling distributions may be approximated through Monte-Carlo simulations, bootstrap method, or asymptotic distribution theory.


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 Statistic Sampling distribution
Normal: \mathcal{N}(\mu, \sigma^2) Sample mean \bar X from samples of size n \bar X \sim \mathcal{N}\Big(\mu,\, \frac{\sigma^2}{n} \Big)
Bernoulli: \operatorname{Bernoulli}(p) Sample proportion of “successful trials” \bar X \bar X \sim \operatorname{Binomial}(n, p)
Two independent normal populations:

\mathcal{N}(\mu_1, \sigma_1^2)  and  \mathcal{N}(\mu_2, \sigma_2^2)

Difference between sample means, \bar X_1 - \bar X_2 \bar X_1 - \bar X_2 \sim \mathcal{N}\! \left(\mu_1 - \mu_2,\, \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} \right)
Any absolutely continuous distribution F with density ƒ Median X(k − 1) from a sample of size n = 2k − 1 f_{X_{(k-1)}}(x) = \frac{(2k-1)!}{(k-1)!^2}f(x)\Big(F(x)(1-F(x))\Big)^{k-1}

External links


 
 

 

Copyrights:

Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. 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 Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Sampling distribution" Read more