Sample Proportion in Statistics: Definition & Formula

Sample Proportion in Statistics: Definition & Formula
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  • 0:01 Sample and Population…
  • 1:34 Example
  • 3:04 Is the Sample…
  • 3:46 How to Determine the…
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Lesson Transcript
Instructor: David Britz

David has taught Developmental Mathematics and Statistics and has M.S. degrees in Math Education and Statistics

This lesson talks about the definition, formula, and use of the sample proportion. We also see a brief intro into the concept of margin of error and selection of sample size. After completing the lesson, take a short quiz to see what you have learned.

Sample and Population Proportions

Mr. Smith is a poll worker on the campaign of Bill Jones. He wants to predict the percentage of the state's citizens who will vote for Mr. Jones. But, there are millions of voters in the state, and he cannot poll every single voter. So, he must estimate the proportion of the population by taking a sample (polling).

Proportion is the decimal form of a percentage, so 100% would be a proportion of 1.000; 50% would be a proportion of 0.500, etc. The proportion of the population voting for Mr. Jones is symbolized by the symbol p. The proportion has the following formula:

p = (number of favorable outcomes) / (number of outcomes in the population)

If we are talking about the proportion of a sample rather than a population, then we would use this slightly modified formula:

sample proportion formula

In this second formula, the symbol on the left is called p-hat and is used to refer to the proportion of a sample of the population as opposed to the proportion of the whole population. While the formulas look very similar, the difference is very important. In many situations, polling or sampling the entire population is difficult, way too expensive, or impossible.

Also note that in statistics, the word 'favorable' simply refers to the outcomes we are studying and is not always what we would consider a 'good' outcome.

Example

Mr. Smith polls a sample of n = 1,000 likely voters in the state. 520 of those polled say that they intend to vote for Mr. Jones. This means that we have 520 favorable outcomes out of the 1,000 polled and 520 / 1000 = .520.

Since this is only an estimate of the true proportion voting for Jones, it is likely different from p. In statistics, we call this sampling error. Error in this case does not mean a mistake. It refers to a difference between the sample and population that results from not having access to the entire population.

To explore sampling further, let's take a hypothetical situation that in actuality 54% of the population will vote for Mayor Jones, so p = 0.540. We could simulate our sampling process by polling 1,000 voters, measuring the proportion of 'yes' votes, and then repeating that process 100 times, each time with a new sample of 1,000 voters. We list the first 10 results:

Sample # Proportion of 1,000
Voting for Jones
Percent of 1,000
Voting for Jones
1 .518 51.8%
2 .519 51.9%
3 .541 54.1%
4 .557 55.7%
5 .561 56.1%
6 .535 53.5%
7 .533 53.3%
8 .558 55.8%
9 .551 55.1%
10 .542 54.2%

A histogram graph of all 100 samples of 1,000 voters can be seen here:

Proportion_Histogram

The horizontal axis shows the sample proportion; the vertical axis shows the number of times that sample proportion occurred. Every one of our 100 samples resulted in between 51% and 57% voting for Jones, and as expected, the graph is centered on 54% since that is the population proportion.

Is the Sample Representative of the Population?

As you can see, the variation is very little even though we only sampled 1,000 voters. As surprising as this may seem, if you want to be able to estimate the proportion within plus or minus 0.03, or 3%, you only need a sample of about 1,100 regardless of the population size.

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