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simple random sample

(Definition)

A sample of size from a population of size is called a simple random sample if

it is a sample without replacement, and
the probability of picking this sample is equal to the probability of picking any other sample of size from the same population .

From the first part of the definition, there are $\binom{N}{n}$ samples of

items from a population of

items. From the second part of the definition, the probability of any sample of size

is a constant. Therefore, the probability of picking a particular simple random sample of size

from a population of size

is $\binom{N}{n}^{-1}$ .

Remarks Suppose $x_1,x_2,\ldots,x_n$ are values representing the items sampled in a simple random sample of size

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 $s^2=\frac{1}{n-1} \sum_{i=1}^{n}(x_i-\overline{x})^2$ is an unbiased estimator of , where $(\frac{N-1}{N})S^2=\sigma^2$ is the true variance of the population given by
$\displaystyle \sigma^2:=\frac{1}{N}\sum_{i=1}^{N}(x_i-\overline{x})^2.$
The variance of the sample mean $\overline{x}$ from the true mean $\mu$ is
$\displaystyle \left(\frac{N-n}{nN}\right) S^2.$
The larger the sample size, the smaller the deviation from the true population mean. When , the variance is the same as the true population variance.

"simple random sample" is owned by CWoo.

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Cross-references: variance, sample variance, mean, unbiased estimator, sample mean, size
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This is version 2 of simple random sample, born on 2005-04-28, modified 2005-04-29.
Object id is 6980, canonical name is SimpleRandomSample.
Accessed 7762 times total.

Classification:

62D05 (Statistics :: Sampling theory, sample surveys)

Pending Errata and Addenda

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