## Sample Size in Statistics: How to Find it

### 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.

### 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: .95 / 2 = 0.475 The closest z-score for 0.475 is_{a/2}**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.

*Sample Size in Statistics: How to Find it was last modified: November 10th, 2016 by Andale*