Sampling Distributions & Central Limit Theorem

The sampling distribution of a statistic is the distribution of values that statistic would take in all possible samples of the same size from a population.

The sampling distribution of x
If x is the mean of an SRS of size n taken from a reasonably large population with mean µ and standard deviation σ, then the sampling distribution of x will have a mean of µ, and a standard deviation of ^σ/_√n .

Because the sampling distribution of x has a standard deviation of ^σ/_√n, x is always going to be less variable than individual observations from the population (which has standard deviation σ.)

x is called an unbiased estimator of µ because its prediction of µ tends to be accurate, that is, it doesn't have a systematic tendency to overestimate or underestimate µ.

If a population has a normal distribution then so will its sample mean.

Central Limit Theorem
For any population with a finite standard deviation greater than 0, the sampling distribution of x will always be approximately normal when n is large. How large n needs to be depends on how close to normal the population distribution is, but the sampling distribution of x is always more normal for greater n.

Because of the above, any variable whose values are composites of many small random influences will always have an approximately normal distribution. That's why its so common to encounter data with a normal distribution.

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