Probability and Statistics > Sample Size
What is “Sample Size”?
A sample size is a part of the population chosen for a survey or experiment. For example, you might take a survey of dog owner’s brand preferences. You won’t want to survey all the millions of dog owners in the country (either because it’s too expensive or time consuming), so you take a sample size. That may be several thousand owners. The sample size is a representation of all dog owner’s brand preferences. If you choose your sample wisely, it will be a good representation.
When Error can Creep in
When you only survey a small sample of the population, uncertainty creeps in to your statistics. If you can only survey a certain percentage of the true population, you can never be 100% sure that your statistics are a complete and accurate representation of the population. This uncertainty is called sampling error and is usually measured by a confidence interval. For example, you might state that your results are at a 90% confidence level. That means if you were to repeat your survey over and over, 90% of the time your would get the same results.
How to Find a Sample Size in Statistics
A sample is a percentage of the total population in statistics. You can use the data from a sample to make inferences about a population as a whole. For example, the standard deviation of a sample can be used to approximate the standard deviation of a population. Finding a sample size can be one of the most challenging tasks in statistics and depends upon many factors including the size of your original population.
Part One: General Tips on How to Find a Sample Size
Skip to Part Two: Find a Sample Size Given a Confidence Interval and Width (unknown population standard deviation).
Skip to Part Three: Find a Sample Size Given a Confidence Interval and Width (known population standard deviation).
Step 1: Conduct a census if you have a small population. A “small” population will depend on your budget and time constraints. For example, it may take a day to take a census of a student body at a small private university of 1,000 students but you may not have the time to survey 10,000 students at a large state university.
Step 2: Use a sample size from a similar study. Chances are, your type of study has already been undertaken by someone else. You’ll need access to academic databases to search for a study (usually your school or college will have access). A pitfall: you’ll be relying on someone else correctly calculating the sample size. Any errors they have made in their calculations will transfer over to your study.
Step 3: Use a table to find your sample size. If you have a fairly generic study, then there is probably a table for it. For example, if you have a 95% confidence level you can use the table published in this article (scroll to the bottom of the article for the table).
Step 4: Use a sample size calculator, like this one.
Step 5: Use a formula. There are many different formulas you can use, depending on what you know (or don’t know) about your population. If you know some parameters about your population (like a known standard deviation), you can use the techniques below. If you don’t know much about your population, use Slovin’s formula..
How to Find a Sample Size Given a Confidence Interval and Width (unknown population standard deviation)
Part two shows you how to find a sample size for a given confidence interval and width (e.g. 95% interval, 6% wide) for an unknown population standard deviation.
Sample question: 41% of Jacksonville residents said that they had been in a hurricane. How many adults should be surveyed to estimate the true proportion of adults who have been in a hurricane, with a 95% confidence interval 6% wide?
Step 1: Using the data given in the question, figure out the following variables:
- za/2: Divide the confidence interval by two, and look that area up in the z-table:
.95 / 2 = 0.475
The closest z-score for 0.475 is 1.96.
- E (margin of error): Divide the given width by 2.
6% / 2
= 0.06 / 2
- : use the given percentage. 41% = 0.41. If you aren’t given phat, use 50%.
- : subtract from 1.
1 – 0.41 = 0.59
Step 2:Multiply by . Set this number aside for a moment.
0.41 × 0.59 = 0.2419
Step 3: Divide Za/2 by E.
1.96 / .03 = 65.3333333
Step 4: Square Step 3:
65.3333333 × 65.3333333 = 4268.44444
Step 5: Multiply Step 2 by Step 4:
0.2419 × 4268.44444 = 1,032.53671
= 1,033 people to survey.
How to Find a Sample Size Given a Confidence Interval and Width (known population standard deviation)
Part 3 shows you how to determine the appropriate sample size for a given confidence interval and width, given that you know the population standard deviation.
Sample question: Suppose we want to know the average age of an Florida State College student, plus or minus 0.5 years. We’d like to be 99% confident about our result. From a previous study, we know that the standard deviation for the population is 2.9.
Step 2: Multiply step 1 by the standard deviation.
2.58 * 2.9 = 7.482
Step 3: Divide Step 2 by the margin of error. Our margin of error (from the question), is 0.5.
7.482/0.5 = 14.96
Step 4: Square Step 3.
14.96 * 14.96 = 223.8016
That’s it! Like the explanation? Check out our statistics how-to book, with a how-to for every elementary statistics problem type.