Sample Size Calculator

Q: How to calculate sample size?

To calculate sample size, use the sample size formula : n = (Z² × p × (1 − p)) / e² . Where: n — sample size Z — z-score (1.96 for 95% confidence) p — estimated percentage of people who’ll answer a certain way (use 0.5 if you’re not sure) e — margin of error (e.g., 0.05 for 5%)

Q: What sample size is too small?

In most cases, fewer than 30 samples is considered too small to provide statistically reliable results . This is the case especially if you're working with a large population. With such a small sample, you risk high margins of error and low confidence in the results.

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Steven Wooding is a physicist by training with a degree from the University of Surrey specializing in nuclear physics. He loves data analysis and computer programming. He has worked on exciting projects such as environmentally aware radar, using genetic algorithms to tune radar, and building the UK vaccine queue calculator. Steve is now the Editorial Quality Assurance Coordinator here at Omni Calculator, making sure every calculator meets the standards our users expect. In his spare time, he enjoys cycling, photography, wildlife watching, and long walks. See full profile

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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.

What is sample size?

Sample size refers to the number of observations or data points collected from a population for a study or survey. It is a critical factor in ensuring the accuracy and reliability of research results, which is known as statistical significance. A correctly chosen sample size helps to reflect the true characteristics of the population, reducing sampling error.

Before looking at the sample size formula and calculation, there are a few statistical terms that are useful to note:

Population: The entire group you want to study (like all your customers or a city’s residents).
Margin of Error: A measure of how much your results could differ from the truth.
Confidence Level: How sure you are that your results reflect the real picture (usually 95%).
Standard Deviation: A number that tells you how spread out your data is.
Study Power: The chance your study will spot real differences or effects if they exist.

Sample size formula and calculation

The equation that our sample size calculator uses is:

n_1 = Z^2\cdot p \cdot \frac{1-p}{\mathrm{ME}^2}

where:

$Z$ — The z-score associated with the confidence level you chose. Our statistical significance calculator calculates this value automatically, but if you want to learn how to calculate it by hand, take a look at the instructions of our confidence interval calculator.
$\mathrm{ME}$ — Margin of error: This value tells you that you can be sure (with a given confidence level, for example, 95%) that the real value doesn't differ from the one that you obtained by more than this margin of error. You can learn more about it at our margin of error calculator.
$p$ — 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%.
$n_1$ — 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_2 = \frac{n_1}{1+\frac{n_1}{N}}

where:

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

How to calculate sample size: an example

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 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 essential for the canteen to know the result, with a margin of error of $2\%$ maximum.
Decide on your confidence level. We can assume you want to be $99\%$ 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 $30\%$ . 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 $25,000$ .
All you need to do now is input all this data into our sample size calculator. It finds the sample size required for the result to be statistically significant is $3,051$ . You need to ask that many students the same question… Are you sure you can't settle for a $95\%$ confidence level? 😀

Factors affecting sample size and how to choose the right one

Here are some factors to consider to ensure reliable results from your sample size:

Population Size
For large populations (over 10,000), a sample of around 384 is typically sufficient for a 95% confidence level with a ±5% margin of error. For smaller populations (under 1,000), you need to apply a finite population correction (FPC) to adjust the sample size accordingly.
Confidence Level and Margin of Error
A higher confidence level or a smaller margin of error (e.g., ±3% instead of ±5%) requires a larger sample to achieve greater precision.
Study Power
A higher study power requires a larger sample size to reduce the risk of false negatives.
Variability in the Population
A larger sample is needed to capture that variability if the population is very diverse. When unsure, a conservative estimate of 50% variability is often used to calculate sample size.
Type of Study
Different studies require different sample sizes. Exploratory surveys might need fewer participants, while studies like clinical trials or those on sensitive topics require larger samples for accuracy. If analyzing subgroups (e.g., age or gender), ensure your sample size accounts for each group.

💡 How do I choose a good sample size? Define your study goals first, then estimate the variability and the population size. For most studies, a sample size of 100 to 400 is sufficient. For more precision, aim for 1,000 or more. Beyond that, increasing sample size yields diminishing returns. Again, this highly depends on your project and the factors listed above. Remember that larger samples don’t always guarantee better results, so avoid oversampling unless necessary.

Other useful tools beyond the sample size calculator

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, population mean, and standard deviation to find the probability that your sample's 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.

FAQs

What is a reliable sample size?

A reliable sample size gives accurate results and adequately represents your target population. For most large populations, a sample size between 300 and 400 is typically enough to achieve a 95% confidence level with a 5% margin of error. However, if your population is small or the stakes of the study are high (like in clinical trials), you may need a larger proportion or more precise calculation.

How to calculate sample size?

To calculate sample size, use the sample size formula:
n = (Z² × p × (1 − p)) / e².
Where:
n — sample size
Z — z-score (1.96 for 95% confidence)
p — estimated percentage of people who’ll answer a certain way (use 0.5 if you’re not sure)
e — margin of error (e.g., 0.05 for 5%)

What sample size is too small?

In most cases, fewer than 30 samples is considered too small to provide statistically reliable results. This is the case especially if you're working with a large population. With such a small sample, you risk high margins of error and low confidence in the results.

How does sample size affect confidence interval?

The bigger your sample size, the narrower your confidence interval. That means your estimate is more precise and statistically significant.

On the contrary, a small sample size gives you a wide confidence interval, which adds more uncertainty to your results. So, more samples usually mean more reliable data.