Statistics How To

Sample Size in Statistics: How to Find it

Probability and Statistics > Sample Size

Watch the video or read the article below:

Links mentioned in the video:
95% CI Table
calculator.

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.

A census is where every member of a population is surveyed, not just a sample.

A census is where every member of a population is surveyed, not just a sample.

How to Find a Sample Size in Statistics

sample size
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).
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How to Find a Sample Size in Statistics: Steps

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
    = 0.03
  • phat: use the given percentage.  41% = 0.41. If you aren’t given phat, use 50%.
  • qhat :  subtract phatfrom 1.
    1 – 0.41 = 0.59

Step 2:Multiply phatby qhat. 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 1: Find z a/2 by dividing the confidence interval by two, and looking that area up in the z-table:
.99/2 = 0.495.  The closest z-score for 0.495 is 2.58.

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.

Sample Size in Statistics: How to Find it was last modified: July 26th, 2016 by Andale

15 thoughts on “Sample Size in Statistics: How to Find it

  1. David

    In my research class my professor used the formula Z alpha – Z beta = d(square root of n) d is cohen’s d and n is the sample

  2. Casey

    What if I were trying to standardize some time, for instance. So I want to determine the amount of time that it takes someone to lift up a milk jug. I want to say this time with 95% confidence. I’m struggling with the concept in this type of scenario.

  3. Andale Post author

    Hello, Casey. Please post your question on our forum (. Without any data (like the mean and standard deviation) though, it would be impossible to determine anything :)

  4. farideh

    how can we determine the sample size when the total population is over 80,000. the confidence level is 95%. the data of the survey will be nominal.

  5. John

    Step 3 of “How to Find a Sample Size Given a Confidence Interval and Width (known population standard deviation)” should read “Divide Step 2” not “Divide Step 1”.

  6. Andale Post author

    I’ve found a couple of references stating that you need to add a couple of programs (NPROP and NMEAN) to your TI 83. That said, I can’t find a location for these programs. They don’t seem to be available on TI-Calc.org or anywhere else I can see. IF you find them, please let me know!

  7. Zymoni

    hey…. alright so how do i find the sample size if phat and qhat are unknown? kinda lost here and would be helpful if a solution was added to the website for others to find as well.

  8. Andale Post author

    If phat is unknown, use 50% (0.5). I’ll add it to the article. Thanks for your suggestion!

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