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.

## 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).

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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:*

**z**: Divide the confidence interval by two, and look that area up in the z-table:_{a/2}

.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**- : 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 Z_{a/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.

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

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.

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 :)

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.

Hello, Farideh,

Can you post your question on our homework help forum:

can we calculate sample size if we don’t know the exact population?

Yes…see part 2 (unknown population standard deviation)

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”.

Thanks for catching that. It’s now fixed :)

how do you do this (minimum sample size) on a ti-83?

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!

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.

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

how do you calculate required sample for 95% confidence level when the sample is stratified?

Use this calculator for one stratum at a time. Then add up all of your samples to get the total.

A census reports states that a city records a population of 23,000 in 2006

The statistical agency projects that by 2011, the city will hit a population of 34,000

1. How can we calculate what the population may have been in 2007, 2008, 2009, and 2010

2. How can we calculate the percentage of increase yearly?

3. How can we estimate the population in 2012, 2013, 2014, 2015 and 2016 in actual figures?

NB: I do not know the model or linear algorithm used by the statistical agency to arrive at that projection. I am simply interested in estimating or maybe I should say extrapolating what the population should be in 2016 to enable me build up some assumptions for a sample survey frame. I am carrying out a social research in the field

I just need some ideas on how to do this statistically. I want to have this fair knowledge so that I can use this to do an extrapolation of some sort into 2016. I want to use this to have a rough estimate of what the population may be this year and then find a mean population with which to help me find an appropriate sample size for that population

Thank you

I’d start by trying regression analysis (say, for example, simple linear regression). You can use the results for projections. That said, it’s going to be a very rough estimate as you only have two data points!