simple random sample
A sample $S$ of size $n$ from a population $U$ of size $N$ is called a simple random sample if

1.
it is a sample without replacement, and

2.
the probability of picking this sample is equal to the probability of picking any other sample of size $n$ from the same population $U$.
From the first part of the definition, there are $\left(\genfrac{}{}{0pt}{}{N}{n}\right)$
samples of $n$ items from a population of $N$ items. From the
second part of the definition, the probability of any sample of size
$n$ in $U$ is a constant. Therefore, the probability of picking a
particular simple random sample of size $n$ from a population of
size $N$ is ${\left(\genfrac{}{}{0pt}{}{N}{n}\right)}^{1}$.
Remarks Suppose ${x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}$ are values
representing the items sampled in a simple random sample of size
$n$.

•
The sample mean^{} $\overline{x}=\frac{1}{n}{\sum}_{i=1}^{n}{x}_{i}$ is an unbiased estimator^{} of the true population mean $\mu $.

•
The sample variance ${s}^{2}=\frac{1}{n1}{\sum}_{i=1}^{n}{({x}_{i}\overline{x})}^{2}$ is an unbiased estimator of ${S}^{2}$, where $(\frac{N1}{N}){S}^{2}={\sigma}^{2}$ is the true variance^{} of the population given by
$${\sigma}^{2}:=\frac{1}{N}\sum _{i=1}^{N}{({x}_{i}\overline{x})}^{2}.$$ 
•
The variance of the sample mean $\overline{x}$ from the true mean $\mu $ is
$$\left(\frac{Nn}{nN}\right){S}^{2}.$$ The larger the sample size, the smaller the deviation from the true population mean. When $n=1$, the variance is the same as the true population variance.
Title  simple random sample 

Canonical name  SimpleRandomSample 
Date of creation  20130322 15:13:01 
Last modified on  20130322 15:13:01 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  5 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 62D05 