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The law of large numbers
- As the number of randomly-drawn observations (n) in a
- sample increases, the mean of the sample gets closer and closer to the population mean.
Sampling distribution of a statistic
- the distribution of all possible values taken by the
- statistic when all possible samples of a fixed size n are taken from the population. It is a theoretical idea—we do not actually build it. The sampling distribution of a statistic is
- the probability distribution of that statistic.
For any population with mean m and standard deviation s :
- The mean, or center of the sampling distribution of (X bar), is equal to the population
- mean m.
- There is no tendency for a sample mean to fall systematically above or below m, even if the distribution of the raw data is skewed. The mean of the sampling distribution is an unbiased estimate of the population mean m —it will be “correct on average”
- in many samples.
The standard deviation of the sampling distribution is s/√n, where n is the sample size.
- The standard deviation of the sampling distribution measures how much the sample
- statistic varies from sample to sample. It is smaller than the standard deviation of the population by a factor of √n.Averages are less variable than individual observations.
For normally distributed populations
When a variable in a population is normally distributed, then the sampling distribution of x bar for all possible samples of size n is also normally distributed.
If the population is N(m,s), then the sample means distribution is N(m,s/√n).
The central limit theorem:
- When randomly sampling from any population with mean
- m and standard deviation s, when n is large enough, the sampling distribution of x bar is approximately normal N(m,s/√n).