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simple random sample (Definition)

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 $ \binom{N}{n}$ 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 $ \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 $ 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}{n-1} \sum_{i=1}^{n}(x_i-\overline{x})^2$ is an unbiased estimator of $ S^2$, 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 $ n=1$, the variance is the same as the true population variance.



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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:
AMS MSC62D05 (Statistics :: Sampling theory, sample surveys)

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