Sampling Distributions

Terms P arameter - a number (usually unknown) that describes a p opulation. S tatistic – a number that can be computed from s ample data. A statistic is used to estimate an unknown parameter. Sampling Variability – Due to random chance, results from different samples will vary. *Remember* - There is always error in sampling.

Example Suppose that 25% of all US adults are either single or divorced..25 is a parameter. We take a random sample of 50 US adults and determine the proportion of them who are divorced or single. We repeat this 5 times. Our results are:.29,.32,.21,.24,.28. These values are statistics. Note that these values vary around the true value.

How to Handle Sampling Variability Take a large number of samples of the same size. Calculate the desired statistic for each sample. Make a histogram of the values of the statistic. Examine the histogram for the center, shape, outliers, and the spread. If one were able to do this an infinite amount of times, the histogram would approach a smooth curve containing all possible samples of that size. This curve is known as a Sampling Distribution.

Definition The Sampling Distribution of a statistic is the probability distribution of values (means, proportions, etc.) taken by the statistic in all possible samples of the same size from the same population. A Sampling Distribution may be comprised of means, proportions, slopes, etc. We will start with means. They are described using C(center), U(unusual), S(shape), S(spread). For means, we will speak of the “sampling distribution of x-bar”.

Bias of a Statistic The bias of a statistic concerns the center of the distribution. A statistic is unbiased if: That is, if the sampling distribution mean is equal to the true mean of the parameter, for all possible samples of a given size.

Variability of a Statistic The variability of a statistic is described by the spread of the sampling distribution. The spread is determined by the sampling design and by the size of the sample. Larger samples produce less variation (less error in sampling). For identical sample sizes, as long as the population is at least 10 times the sample size, variability won’t be affected by the size of the population.

How Bias and Variability Interact

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