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Variance

4.2 Distribution of Sample Proportion

Sampling Distribution of a Small Proportion

  • Recall that

\text { Sample proportion, }\ \hat{p}=\frac{\text { number in population with attribute }}{\text { population size }}

and can be treated as a random variable.

  • In a small enough population, we are able to list out all the possible samples, the probability of getting each sample, and therefore the sample proportion.
  • Therefore, we are able to construct the probability distribution of \hat{p}.
  • The distribution of a statistic which is calculated from a sample (such as the sample proportion) is called a sampling distribution.
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2.3 Variance and Standard Deviation

Variance

  • The actual outcome will often differ from the mean (or E(X)). Sometimes the difference is large, sometimes it is not (or even sometimes no difference).
  • The variance of a random variable captures the spread of the probability distribution about its mean value. It is defined as

\operatorname{Var}(X)=E\left[(X-\mu)^{2}\right]

and \operatorname{Var}(X) represents the variance of X.

  • \operatorname{Var}(X) can also be denoted as the Greek letter \sigma^{2} (called sigma square).
  • It is also considered as the long-run average value of the square of the distance from X. Also, notice that there is always \operatorname{Var}(X) \geq 0.
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