The Basics  A population is the entire group on which we would like to have information.  A sample is a smaller group, selected somehow from the population, on which we do have information.

The Basics  A parameter is a number (such as the mean or standard deviation) which describes a population  A statistic is a number (such as the mean or standard deviation) which describes a sample –

Sampling Distributions  Suppose you select a gazillion different random samples from the same population  From each sample, you compute a certain statistic  The distribution of these gazillion different statistics is called a sampling distribution

Sampling variability  Take a large number of samples from the same population.  Calculate the sample statistic for each sample.  Make a histogram of the values of the statistic.  Examine the histogram’s shape, center, and spread.

Sampling distributions  The sampling distribution of a statistic is the distribution of values that statistic would take if you sampled all possible samples of size N.  For example, suppose 60% of all Americans don’t like to shop for clothes.  If we draw a random sample of 100 Americans, how many of them don’t like to shop for clothes?

Sampling distribution for p = 0.60 and N = % 66% 63% 59% 54% 61% 60% 57% 61% 63% 59% 62%

Sampling distribution for p = 0.60 and N = % 66% 62% 59% 61% 56% 54% 58% 68% 62% 67% 56% 55% 51% 63% 58% 61% 70% 65% 61% 56% 60% 57% 59% 64% 56% 63% 56% 50% 63% 60% 65% 62% 56% 51% 62% 59% 65% 53% 57% 55% 64% 66% 61% 56% 63% 57% 59% 61% 54% 52% 60% 55% 63% 60% 67% 50% 56% 60% 55% 58% 57% 60% 63% 59% 61% 64% 58% 60% 61% 69% 61% 56% 64% 70% 61% 62% 65% 61% 56% 60% 57% 62% 58% 59% 58% 56% 53% 49% 56% 58% 53% 72% 63% 61% 60% 51% 69% 57% 65% 62% 65% 51% 57% 56% 58% 53% 58% 56% 57% 61% 65% 58% 57% 66% 62% 58% 54% 65% 62% 59% 56% 59% 61% 53% 60% 61% 59% 62% 58% 59% 58% 61% 67% 59% 61% 58% 65% 63% 59% 54% 61% 60% 57% 61% 63% 59% 64% 61% 60% 61% 56% 62% 60% 64% 56% 61% 56% 60% 61% 56% 58% 68% 67% 56% 55% 62% 64% 52% 58% 53%

AFTER A THOUSAND SAMPLES: Sampling distribution for p = 0.60 and N = 100 Mean = 59.97% St Dev = 4.74%

AFTER A GAZILLION SAMPLES: Sampling distribution for p = 0.60 and N = 100 Mean = 60%Standard Deviation = 4.899%

The bias of a statistic  A statistic is unbiased if the mean of its sampling distribution is expected to be equal to the population parameter it is estimating.  But if the statistic over-estimates or under-estimates the parameter, then that statistic is biased.

The variability of a statistic The spread of the sampling distribution shows how variable the statistic is, from one sample to another. Usually, the spread of the sampling distibution is smaller for large samples, and larger for small samples.