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Creative Research Systems Sample Size Calculator

This free interactive web-based sample size calculator can be used to understand the relationship between number of samples, variation in indicators and confidence in survey results.

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Sample Size Calculator

Sample size calculation formula, how to calculate sample size: an example, other useful tools beyond the sample size calculator.

If you're conducting research and wonder how many measurements you need so that it is statistically significant , this sample size calculator is here to help you. All you need to do is ask yourself these three questions before you use it:

• How accurate should your result be? (margin of error)
• What level of confidence do you need? (confidence level)
• What is your initial estimate ? (proportion estimate)

Read on to learn how to calculate the sample size using this tool, and what do all the variables in the sample size calculation formula mean.

The equation that our sample size calculator uses is:

• Z Z Z — The z-score associated with the confidence level you chose. Our statistical significance calculator calculates this value automatically (shown in Advanced mode ), but if you want to learn how to calculate it by hand, take a look at the instructions of our confidence interval calculator .
• M E \mathrm{ME} ME — Margin of error, also known as the confidence interval . It tells you that you can be sure (with a probability of confidence level, for example, 95%), that the real value doesn't differ from the one that you obtained by more than this percentage. You can learn more about it at our margin of error calculator .
• p p p — Your initial proportion estimate. For example, if you are conducting a survey among students trying to find out how many of them read more than 5 books last year, you may know a result of a previous survey — 40%. If you have no such estimate, use the conservative value of 50%.
• n 1 n_1 n 1 ​ — Required sample size.

If your population is finite — for example, you are conducting a survey among students of only one faculty — you need to include a correction in the following form:

• N N N — Total population size.
• n 2 n_2 n 2 ​ — Size of the sample taken from the whole population that will make your research statistically significant.

We will analyze a survey case step-by-step, so you can get a clear picture of how to use our sample size calculator . You are planning to conduct a survey to find out what is the proportion of students on your campus who regularly eat their lunch at the campus canteen.

Decide how accurate you want your result to be. Let's say that it is important for the canteen to know the result, with a margin of error of 2 % 2\% 2% maximum.

Decide on your confidence level . We can assume you want to be 99 % 99\% 99% sure that your result is correct.

Do you have an initial proportion guess ? Let's say you accessed a similar survey from 10 years ago and the proportion was equal to 30 % 30\% 30% . You can assume it as your initial estimate.

Is the total population of students so high that you can assume it's infinite ? Probably not. You need to find the current data for the number of students on the campus — let's assume it is 25 , 000 25,000 25 , 000 .

All you need to do now is to input all this data into our sample size calculator. It finds the sample size required for the result to be statistically significant is 3 , 051 3,051 3 , 051 . You need to ask that many students the same question… Are you sure you can't settle for a 95 % 95\% 95% confidence level? 😀

Now that you know how to calculate sample size, you can go beyond and use it to calculate other statistics of interest in your research:

Sampling error calculator : sample size is the most influential feature when predicting the sampling error. Use it to calculate the error of your sample.

Normal probability calculator for sampling distributions : use your sample size, along with the population mean and standard deviation, to find the probability that your sample mean falls within a specific range.

Sampling distribution of the sample proportion calculator : use your sample size and the population proportion to find the probability that your sample proportion falls within a specific range.

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Sample Size Calculator

This resource can be used to calculate the sample size for an audit based on the total number of items in the collection and the confidence interval (margin of error) and confidence level (%) you wish to have with your sample.

Click the link on the right-hand side to view the resource on the Creative Research Systems website.

Date created: 2012

Author: Creative Research Systems

Publisher: Creative Research Systems

Sample Size Calculator

Find out the sample size.

This calculator computes the minimum number of necessary samples to meet the desired statistical constraints.

Find Out the Margin of Error

This calculator gives out the margin of error or confidence interval of observation or survey.

Related Standard Deviation Calculator | Probability Calculator

In statistics, information is often inferred about a population by studying a finite number of individuals from that population, i.e. the population is sampled, and it is assumed that characteristics of the sample are representative of the overall population. For the following, it is assumed that there is a population of individuals where some proportion, p , of the population is distinguishable from the other 1-p in some way; e.g., p may be the proportion of individuals who have brown hair, while the remaining 1-p have black, blond, red, etc. Thus, to estimate p in the population, a sample of n individuals could be taken from the population, and the sample proportion, p̂ , calculated for sampled individuals who have brown hair. Unfortunately, unless the full population is sampled, the estimate p̂ most likely won't equal the true value p , since p̂ suffers from sampling noise, i.e. it depends on the particular individuals that were sampled. However, sampling statistics can be used to calculate what are called confidence intervals, which are an indication of how close the estimate p̂ is to the true value p .

Statistics of a Random Sample

The uncertainty in a given random sample (namely that is expected that the proportion estimate, p̂ , is a good, but not perfect, approximation for the true proportion p ) can be summarized by saying that the estimate p̂ is normally distributed with mean p and variance p(1-p)/n . For an explanation of why the sample estimate is normally distributed, study the Central Limit Theorem . As defined below, confidence level, confidence intervals, and sample sizes are all calculated with respect to this sampling distribution. In short, the confidence interval gives an interval around p in which an estimate p̂ is "likely" to be. The confidence level gives just how "likely" this is – e.g., a 95% confidence level indicates that it is expected that an estimate p̂ lies in the confidence interval for 95% of the random samples that could be taken. The confidence interval depends on the sample size, n (the variance of the sample distribution is inversely proportional to n , meaning that the estimate gets closer to the true proportion as n increases); thus, an acceptable error rate in the estimate can also be set, called the margin of error, ε , and solved for the sample size required for the chosen confidence interval to be smaller than e ; a calculation known as "sample size calculation."

Confidence Level

The confidence level is a measure of certainty regarding how accurately a sample reflects the population being studied within a chosen confidence interval. The most commonly used confidence levels are 90%, 95%, and 99%, which each have their own corresponding z-scores (which can be found using an equation or widely available tables like the one provided below) based on the chosen confidence level. Note that using z-scores assumes that the sampling distribution is normally distributed, as described above in "Statistics of a Random Sample." Given that an experiment or survey is repeated many times, the confidence level essentially indicates the percentage of the time that the resulting interval found from repeated tests will contain the true result.

Confidence Interval

In statistics, a confidence interval is an estimated range of likely values for a population parameter, for example, 40 ± 2 or 40 ± 5%. Taking the commonly used 95% confidence level as an example, if the same population were sampled multiple times, and interval estimates made on each occasion, in approximately 95% of the cases, the true population parameter would be contained within the interval. Note that the 95% probability refers to the reliability of the estimation procedure and not to a specific interval. Once an interval is calculated, it either contains or does not contain the population parameter of interest. Some factors that affect the width of a confidence interval include: size of the sample, confidence level, and variability within the sample.

There are different equations that can be used to calculate confidence intervals depending on factors such as whether the standard deviation is known or smaller samples (n<30) are involved, among others. The calculator provided on this page calculates the confidence interval for a proportion and uses the following equations:

Within statistics, a population is a set of events or elements that have some relevance regarding a given question or experiment. It can refer to an existing group of objects, systems, or even a hypothetical group of objects. Most commonly, however, population is used to refer to a group of people, whether they are the number of employees in a company, number of people within a certain age group of some geographic area, or number of students in a university's library at any given time.

It is important to note that the equation needs to be adjusted when considering a finite population, as shown above. The (N-n)/(N-1) term in the finite population equation is referred to as the finite population correction factor, and is necessary because it cannot be assumed that all individuals in a sample are independent. For example, if the study population involves 10 people in a room with ages ranging from 1 to 100, and one of those chosen has an age of 100, the next person chosen is more likely to have a lower age. The finite population correction factor accounts for factors such as these. Refer below for an example of calculating a confidence interval with an unlimited population.

EX: Given that 120 people work at Company Q, 85 of which drink coffee daily, find the 99% confidence interval of the true proportion of people who drink coffee at Company Q on a daily basis.

Sample Size Calculation

Sample size is a statistical concept that involves determining the number of observations or replicates (the repetition of an experimental condition used to estimate the variability of a phenomenon) that should be included in a statistical sample. It is an important aspect of any empirical study requiring that inferences be made about a population based on a sample. Essentially, sample sizes are used to represent parts of a population chosen for any given survey or experiment. To carry out this calculation, set the margin of error, ε , or the maximum distance desired for the sample estimate to deviate from the true value. To do this, use the confidence interval equation above, but set the term to the right of the ± sign equal to the margin of error, and solve for the resulting equation for sample size, n . The equation for calculating sample size is shown below.

EX: Determine the sample size necessary to estimate the proportion of people shopping at a supermarket in the U.S. that identify as vegan with 95% confidence, and a margin of error of 5%. Assume a population proportion of 0.5, and unlimited population size. Remember that z for a 95% confidence level is 1.96. Refer to the table provided in the confidence level section for z scores of a range of confidence levels.

Thus, for the case above, a sample size of at least 385 people would be necessary. In the above example, some studies estimate that approximately 6% of the U.S. population identify as vegan, so rather than assuming 0.5 for p̂ , 0.06 would be used. If it was known that 40 out of 500 people that entered a particular supermarket on a given day were vegan, p̂ would then be 0.08.

Sample Size Terminology

Factors that affect confidence intervals.

Sample Size Calculator

Published: March 1, 2010 by Jessica Harper

Not sure what sample size you need? Try out this calculator from Creative Research Systems.

About the Author

Jessica Harper

Sample Calculator

How many people do you need to talk to to get a margin of error you are comfortable with? Consult our handy calculator and find out!

Listed below are few terms you will need to understand before using the sample size calculator:

Confidence Level: The percentage value that tells how confident a researcher can be about being correct. A 95% confidence level is a generally acceptable level of confidence and is most typically used and accepted within the research industry. Which means that if a study were conducted 100 times, answers would be within the margin of error 95 out of 100 times.

Margin of Error / Confidence Interval: Assumes that you have a random sample. The margin of error for a typical survey is +/- 5%. This means that results may vary as much as five percent in either direction. The margin of error for sub-samples (i.e. men versus women), is based on the number in that sub-sample.

Population Size: The population size is the universe from which you are taking your sample. If the population size is very large or unknown, leave this field blank.

DIRECTIONS:

• Choose a confidence level
• Enter an acceptable margin of error (between .1 and 50 – leave off % sign)
• Enter population (if known) Click on “calculate”
• This calculator requires Internet Explorer 3.0 or later or Netscape 3.0 or later or a compatible browser. Leave the population box blank, if the population is very large or unknown.

CAUTION: Sampling error is only one source of bias. Contact us to discuss other possible sources of bias (i.e. question bias). ***************

(This sample size calculator was created by Creative Research Systems .)

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The Quantitative Research Sample Size Calculator

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We may love our in-depth qualitative research tools, but we know the value of integrating both qual and quant methods. That’s why we have built our DIY quantitative research sample size calculator. From some basic information, this tool displays the recommended sample size required for your research to be statistically significant.

Use the calculator to work out how many people you need to complete your survey or poll to be confident in the accuracy of your results.

Not sure what values to use? This brief guide explains the terms used in our sample size calculator, in addition to providing recommended values for optimum results.

Sample Size:   Your sample size is the amount of consumers in your target population that you will be researching. This calculator provides a recommended sample size – i.e. the minimum amount of consumers you need to research for your results to be statistically significant within your defined parameters.

Population Size:   The population size is the approximate amount of consumers in the group that you want to research. For example, if you want to understand the internet usage habits of the entire UK population, your population would be all UK consumers ( 64.6 million  at the latest estimate).

Confidence Level:   A confidence level is defined as the statistical probability that the value of a parameter falls within a specified range of values. Therefore a confidence level of N% means you can be N% sure that your results contain the true mean average of the designated population.

In market research, the most commonly used confidence level is 95%. A higher confidence level indicates a higher probability that your results are accurate, but increasing it can dramatically increase the required sample size. Finding a balance between confidence and an achievable research goal is crucial.

In this calculation, each confidence level is translated to a   z-score. A z-score is a statistical method for rescaling data that helps researchers draw comparisons easier. The following table details the z-score generated from each confidence level:

Margin of Error:   The margin of error is the maximum acceptable difference in results between the population and sample. On a basic level, if a poll were to ask 1,000 people if they drive a car and 70% of people were to answer yes – a margin of error of +/- 5% would indicate that in the total population, between 66.5% and 73.5% would answer in the same way.

The smaller the margin of error, the more representative of the total population the results will be. However, decreasing the margin of error will also result in a sharp increase in sample size. We recommend using a 5% margin of error as standard, which should never be increased above 10%.

The DIY Sample Size Calculation

Want to work out your required sample size by hand? Our free calculator uses the following equation. Simply follow the steps below to work out how many research participants you need to complete your research.

S = (z 2  (d(1 - d))/ e 2 ) / 1 + (z 2  (d(1 - d)) / e 2 )

S   = sample size |  P   = population size  |  z   = z-score  |  e   = margin of error  |  d  = standard deviation

Please note that our calculator assumes a standard deviation of 0.5. Use the manual equation to change the standard deviation.

Now that you have calculated the recommended amount of participants required to make your results statistically significant, the next step is to invite participants. In our experience, a typical panel based survey will yield a 15%-20% response rate without an incentive, but approximately a 30% response rate when a relevant incentive is offered. There are a multitude of factors that can affect survey response rate, from length to design, acessibility to relevance.

However, by using an estimate of 30% survey completion and the minimum sample provided by our calculator, it is possible to work out how many consumers you'll need to reach. For example, if you need to reach a minimum sample of 1,000 consumers - you'll need to invite approximately 3,334 consumers within your target population   (S (1 - 0.7) . 

What are your top tips for improving survey response rates? Let us know in the comments below and start a discussion.

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How to calculate sample size

A research is made up of various steps that allow to achieve the desired result, within this methodology there is an element that is of vital importance for the research itself: how to calculate sample size.

What is a population sample?

A population sample is a set of elements that represent the total universe, that is, they are a fraction of the total number of individuals to be evaluated.

Establishing the size of said sample is an important process in any investigation since it will allow a viable and credible study to be carried out always delimited by the objectives of the study and the different characteristics of each population.

Determining the size of a sample saves both financial and human resources, in addition to considerably reducing the time to carry out the research you are carrying out, which can be of all kinds, such as opinion studies or some market research .

How to calculate the sample size you need for your research?

There are different ways to obtain your sample size depending on the data available, for example, if we have the number of people to whom we will carry out the study (for example, the number of inhabitants in X city) , It is said that there is a finite universe, this time we will address this class of universes and how to obtain the ideal sample size, to achieve the above, the following formula proposed by Murray and Larry (2005) is used: .

Where: n = is the size of the population sample to obtain. N = is the size of the total population. σ = represents the standard deviation of the population. If this data is not known, it is common to use a constant value that is equivalent to 0.5 Z = is the value obtained through confidence levels. Its value is a constant, generally there are two values depending on the desired degree of confidence, 99% being the highest value (this value equals 2.58) and 95% (1.96) the minimum value accepted to consider the investigation as trustworthy. e = represents the acceptable limit of sampling error, generally ranging from 1% (0.01) to 9% (0.09), with 5% (0.5) being the standard value used in research.

Once the appropriate values have been established, the values are substituted and the formula applied to obtain the size of the population sample corresponding to the determined finite universe.

Once the mathematical process has been completed, the sample will be obtained, which, as mentioned at the beginning, will help us to carry out a valid and complete research.

If you want to know more about sampling, we recommend you read our article: Types of sampling .

If you feel a little lost or this seems very confusing, don't worry, we have something for you. We have created a tool to help you in this process, use our Sample Size Calculator !

Now that you know how to calculate the sample size, it's time for the surveys!

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Web Survey Software Packages

Software for e-mail surveys, web page surveys, online surveys.

Creative Research Systems offers web survey software, live online reports and full-service web survey hosting .

There are two ways to do online surveys using the Web/Internet or an intranet:

• E-mail surveys
• Web page surveys

When using email surveys you send a survey to a list of respondents, and they reply with their answers. When using Web page surveys, you post your questionnaire on a Web page. People then visit that page in response to an email invitation or a link on a site and answer your questions. The Survey System's Web/Internet Survey Software Module includes both types of surveys. Email surveys are described on a separate page. Web page surveys are described here. Working sample online surveys are presented below.

Since people take Web surveys on many different devices, you can create fully responsive surveys that adapt by using different fonts and layouts on different devices. You can also create mobile-first surveys, that are optimized for smartphones, but look fine on PCs.

As part of a comprehensive online survey software package, The Survey System's Web page questionnaire software offers sophisticated questionnaire logic features not found in most "Web survey" software packages. These features mean higher quality data. After you collect your survey data, you can produce more impressive tables, charts, and text reports than you can with other packages. You can even produce reports online.

Another benefit of this comprehensive Web survey software solution is that you enter your question and answer choice text and any logic instructions only once. Then you can use them for Web page surveys, email surveys, paper questionnaires, and telephone interviews.

Web Survey Software Appearance Features

• Create responsive design surveys that adapt to PCs, smartphones and tablets
• Put all your questions on a single page or divide them into as many pages as desired
• Use background colors or images and logos
• Vary fonts and colors
• Set left and right margins
• Show pictures and video
• Play sound files
• Use option buttons, checkboxes, drop-down lists, and text fields
• Use graphical radio buttons and checkboxes
• Use stars to collect ratings
• Make buttons and boxes larger than standard size
• Highlight choices as the mouse moves over them
• Click on labels to select choices (makes your surveys easier to take!)
• Show related questions in grid format
• Select parts of a picture by clicking (ideal for testing advertisements)
• Select words in a paragraph by clicking
• Use sliders to enter numbers
• Rank items by dragging images or text
• Show all or part of your questionnaire as a data entry form
• Display questions side-by-side
• Show a progress bar
• Use CSS buttons
• Use your own CSS file to customize your pages' appearance
• Use your site's page template
• Include links that pop-up other sites or pages
• Save your appearance choices in a file to reuse as needed
• And many more...

Questionnaire Logic Features

• Use individual or overall passwords to control who completes your questionnaire
• Let people start a survey one time and complete it later
• Skip questions based on answers to other questions (branching)
• Reveal questions that were initially hidden on a page
• Show previous answers in the text of current questions (piping)
• Only show answer choices that were picked (or were not picked) in a previous question
• Randomize answer choice order
• Randomize question order
• Calculate values
• Require answers to some or all questions
• Show a running total of a series of related answers and check the result (constant sum)
• Check that answers are not higher or lower numbers than allowed
• Make sure answers such as "None" cannot be picked together with another choice
• Use heat maps to see where people click on an image
• Create menu pricing tests
• Combine Web survey answers with previous information from the same people
• Establish quotas to achieve a balance of different kinds of people
• Redirect completes, over-quotas, and not-qualifieds to a sample supplier

Email Integration & Invitations

• Email invitations to people to take your survey
• Include individual names and passwords
• Include passwords in a link to the start of a survey - recipients do not have to enter them
• Email reminder messages to those who have not completed a survey or thank-you messages to those who have
• Send email automatically from Web surveys to thank respondents or to alert management to dissatisfied clients
• Maintain a do not email list

Sample Web/Internet Questionnaires

These sample web questionnaires are designed to illustrate some features of our software. They are not intended as model questionnaires per se. Please try them all. They illustrate some different appearance choices and logic features.

• Mobile First Survey - a survey designed especially for smartphones, but which also works well on PCs
• Responsive Design Survey - a survey with labels as buttons, that adapts to PCs, smartphones and tablets
• Animated Button Sample - a survey that includes branching, piping, randomizations and a constant sum
• Restaurant Sample - a survey using rating stars, side-by-side questions, piping and other logic
• Employee Sample - a survey with a grid question, checkmarks in checkboxes and a rank-by-dragging-images question
• Lifestyle Sample - a one-page survey that includes section breaks and a colored grid
• Advertisement Testing Sample - a survey that asks viewers to watch a video, click on parts of a picture and to click on words in a paragraph
• Product Sample - a survey that illustrates the password feature and how it can bring in information you already have about people. It also uses sliders and a drop-down grid to enter choices and shows a progress bar.
• Form and Branding Sample - a registration form. This example shows how The Survey System can insert a questionnaire into a site's template page, making a survey conform to a site's look
• Menu Test - a survey pricing test example
• Heat Map Example - a survey that records where people click on an image. It also shows how the answer to one question can make another question appear on the page. You can view the heat map report here.

Mobile Optimized Web Surveys

Web survey hosting.

Web page surveys must be hosted on a Windows-based Web server. A major advantage of using Windows to host surveys is that you can test the actual survey using your browser on your own PC before uploading it to the Web server.

If your site is not on a Windows server, we offer Web survey hosting. The pages can look like part of your site.

To enhance security, the instructions and data can be stored in a folder that is not set up to be accessible from the Web.

Request a free quote from Creative Research Systems on our web/internet survey software.

• Evaluation Edition
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• Types of Questions
• Survey Logic
• Mixed Mode Surveys
• Web Surveys
• Interviewing ( CATI )
• Mobile Surveys
• Advanced Statistics
• Voice Recording
• Indexes/Benchmarks
• Telephone and Online Panel Management
• Online Backup
• Price/Ordering
• Upgrade Pricing
• Specifications/Support

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