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Distribution of Sample Proportions

Distribution of Sample Proportions

Author: Ryan Backman
Description:

Determine the distribution of sample proportions in a given scenario.

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Hi, this tutorial covers the distribution of sample proportions. So let's start with a definition. Distribution of sample proportions is the distribution of all possible sample proportions for a certain size n. All right, so to kind of envision what a distribution of sample proportions look like, consider this.

Suppose a coin is flipped 10 times, so in this case, n is equal to 10, and the number of heads is recorded. OK, so let's say that we got 6 heads out of 10. The distribution of sample proportions would be the distribution of all possible proportions of heads when a coin is flipped 10 times. So if we got 6 heads, it would be 6 out of 10, so sample proportion-- remember, we call that p hat-- would equal 0.6. That would be one of the possible values on that sampling-- on that distribution of sample proportions.

OK, so for any distribution, it's important to consider the distribution's center, spread, and shape, so we're going to dress each of those things as it relates to the distribution of sample proportions. So if we start with the center, the mean of the distribution of sample proportions is equal to the population proportion. So the mean of the distribution of sample proportions, so if it's the mean of the distribution of sample proportions, we'd notate it as mu sub p hat. That's equal to the population proportion.

Now, this population parameter could be known, for example, if there was a binomial question type and a census of responses could be obtained. So if you're able to determine the population proportion, we know that that's going to equal the mean of the distribution of sample proportions. So this is an important relationship here, mu sub p hat equals p.

All right, so that is measuring center. So now, if we think about the next one, the spread. Let's, first of all, define the standard deviation of the distribution of sample proportions. And it is defined as a measure calculated by taking the square root of the quotient of p times 1 minus p and n.

So the standard deviation of the distribution of sample proportions is going to be sigma sub p hat. That's going to equal the square root of the quotient-- quotient, again, means division-- of p times 1 minus p over n. So this will give us a way of measuring the spread of the distribution of sample proportions.

So we have the center of the spread, and then the last thing we'd like to talk about is the shape. And like always, we want the shape to be normal. So the distribution of sample proportions is approximately normal as long as the sample size is sufficiently large. The larger the sample size, the closer the distribution of sample proportions is to the normal distribution.

So to recap, remember that the mean of the distribution of sample proportions is just equal to the population proportion. The standard deviation of the distribution of sample proportions is the square root of p times 1 minus p over n, where p is your population proportion, n is your sample size. And then the shape will be approximately normal as long as n is sufficiently large. All right, that has been your tutorial on the distribution of sample proportions. Thanks for watching.

Terms to Know
Distribution of Sample Proportions

A distribution where each data point consists of a proportion of successes of a collected sample. For a given sample size, every possible sample proportion will be plotted in the distribution.

Standard Deviation of a Distribution of Sample Proportions

The square root of the product of the probabilities of success and failure (p and q, respectively) divided by the sample size.

Standard Error

The standard deviation of the sampling distribution of sample proportions.

Formulas to Know
Mean of a Distribution of Sample Proportions

mu subscript p with overbrace on top end subscript space equals space p

Standard Deviation of a Distribution of Sample Proportions

sigma subscript p with overbrace on top end subscript space equals space square root of fraction numerator p q over denominator n end fraction end root