Sample Size Calculator

Finds the minimum quantity of observations for your test to be statistically significant

What are you doing?

Taking and measuring the in order to be statistically

What is the minimum sample size to state with a confidence level that the true rate is inside the interval between more or less percentage points around the rate of % (a confidence interval between ? and ?)?

Answer: ? observations

Extra stuff

What if...?

Methodology

One Sample Proportion Z-test
test statistic formula for one sample proportion

where

follows a Standard Normal Distribution and the confidence level is found in the middle of the distribution, inside minimum and maximum bounds (two-tailed).
two tailed bell-shaped distribution chart

two tailed bell-shaped distribution chart

Statistical Power

If you perform your test using the answer from this online calculator, you will be statistically correct, meaning the results are going to be statistically significant in accordance with the confidence level you use. Rest assured.
On the other hand, some calculators and methodologies add another requirement on top of the confidence level used by this calculator so far. That is the "statistical power", which increases the minimum sample size significantly. It is related to the risk of Type II error: predicting that the results ARE NOT statistically significant when, in reality, they ARE.

Check the table:

		Reality
		Null hypothesis H₀ Sample comes from a population with a mean rate of ?	Alternative hypothesis H_a Sample comes from a population with a mean rate different from ?
Prediction	[Stat. sig.] Reject that the sample comes from a population with a mean rate of ?	Type I error	✔
Prediction	[Not stat. sig.] Not reject that the sample comes from a population with a mean rate of ?	✔	Type II error

To calculate the Type II error, the alternative hypothesis H_a has to be determined with a value that is not just the opposite of H₀ as it is in the table above. This is where it gets tricky.
You need to set not only the H₀ hypothesis with a numerical value but also the H_a with another numerical value. This brings more complexity to the test design interpretation and calculators rarely ask you to set a H_a assumption. Most methodologies and online calculators solve this by assuming a good H_a for your test is exactly the threshold value to meet the required confidence level, as a simplified approach. They also commonly use 80% as the minimum required power, meaning that, under this alternative hypothesis H_a, if you performed the test several times, 80% of these times you would have correctly rejected H₀, while in the remaining 20% of times, you would have made the Type II error. You can achieve 80% if you increase the sample size, otherwise, if you use the first answer of this calculator, your test is going to have 50% power.
Knowing this, if you still require 80% statistical power to perform your test, the minimum sample size you should use is ? observations.

What is the minimum sample size if you measure a % rate on your sample and want to be able to state with a confidence level that the true rate is %?

Wait! There is no way the rate could be smaller than the reference and still be statistically NOT smaller than it.

Wait! There is no way the rate could be larger than the reference and still be statistically NOT larger than it.

Answer: ? observations

Extra stuff

What if...?

Methodology

One Sample Proportion Z-test
test statistic formula for one sample proportion

where

follows a Standard Normal Distribution and the confidence level is found in the middle of the distribution, inside minimum and maximum bounds (two-tailed).
two tailed bell-shaped distribution chart

at the right side of the distribution (one-tailed).
left tailed bell-shaped distribution chart

at the left side of the distribution (one-tailed).
right tailed bell-shaped distribution chart

Statistical Power

		Reality
		Null hypothesis H₀ Sample comes from a population with a mean rate of ?	Alternative hypothesis H_a Sample comes from a population with a mean rate ?
Prediction	[Stat. sig.] Reject that the sample comes from a population with the mean rate of ?	Type I error	✔
Prediction	[Not stat. sig.] Not reject that the sample comes from a population with the mean rate of ?	✔	Type II error

What is the minimum sample size you need to measure the mean of something with a standard deviation of and be able to state with a confidence level that the true mean is inside the interval between more or less around your sample's measured mean?

Answer: ? observations

Extra stuff

Methodology

One Sample Mean Z-test
test statistic formula for one sample mean

where

follows a Standard Normal Distribution and the confidence level is found in the middle of the distribution, inside minimum and maximum bounds (two-tailed).
two tailed bell-shaped distribution chart

What is the minimum sample size you need to measure the mean of something with a standard deviation of , measure the sample mean of , and be able to state with a confidence level that the true mean is ?

Wait! There is no way the sampled value could be smaller than the reference and still be statistically NOT smaller than it.

Wait! There is no way the sampled value could be larger than the reference and still be statistically NOT larger than it.

Answer: ? observations

Extra stuff

Methodology

One Sample Mean Z-test
test statistic formula for one sample mean

where

follows a Standard Normal Distribution and the confidence level is found in the middle of the distribution, inside minimum and maximum bounds (two-tailed).
two tailed bell-shaped distribution chart

at the right side of the distribution (one-tailed).
left tailed bell-shaped distribution chart

at the left side of the distribution (one-tailed).
right tailed bell-shaped distribution chart

Statistical Power

		Reality
		Null hypothesis H₀ Sample comes from a population with a mean of ?	Alternative hypothesis H_a Sample comes from a population with a mean ?
Prediction	[Stat. sig.] Reject that the sample comes from a population with the mean of ?	Type I error	✔
Prediction	[Not stat. sig.] Not reject that the sample comes from a population with the mean of ?	✔	Type II error

What is the minimum sample size you need to split it , measure a % rate from the first and a % rate from the second (a ? p.p. difference between them) and be able to state with a confidence level that ?

Wait! There is no way the second sample could be smaller than the first and still be statistically not smaller than it.

Wait! There is no way the second sample could be larger than the first and still be statistically not larger than it.

Answer: ?

Extra stuff

Methodology

Two Sample Proportion Z-test with pooled variance
test statistic formula for two sample proportion part 1

test statistic formula for two sample proportion part 2

where

follows a Standard Normal Distribution and the confidence level is found in the middle of the distribution, inside minimum and maximum bounds (two-tailed).
two tailed bell-shaped distribution chart

at the right side of the distribution (one-tailed).
left tailed bell-shaped distribution chart

at the left side of the distribution (one-tailed).
right tailed bell-shaped distribution chart

Statistical Power

		Reality
		Null hypothesis H₀ Samples come from populations with the same rate	Alternative hypothesis H_a ?
Prediction	[Stat. sig.] Reject that the samples come from populations with the same rate	Type I error	✔
Prediction	[Not stat. sig.] Not reject that the samples come from populations with the same rate	✔	Type II error

Wait! There is no way the second sample could be larger than the first and the difference between them be compared with a negative value (meaning second sample rate smaller than first).

Wait! There is no way the second sample could be ? p.p. larger than the first and the difference between them still be statistically not smaller than ? p.p..

Wait! There is no way the second sample could be smaller then the first and the difference between them be compared with a positive value (meaning second sample rate larger than first).

Wait! There is no way the second sample could be ? p.p. (smaller) than the first and the difference between them still be statistically not smaller than ? p.p..

Answer: ?

Extra stuff

Methodology

Two Sample Proportion Z-test with unpooled variance
test statistic formula for two sample proportion with unlooped variance

where

follows a Standard Normal Distribution and the confidence level is found at the left side of the distribution (one-tailed).
right tailed bell-shaped distribution chart

Statistical Power

		Reality
		Null hypothesis H₀ The second sample comes from a population with a rate ? p.p. larger than the first sample	Alternative hypothesis H_a The second sample comes from a population with a rate that is more than ? p.p. larger than the first sample
Prediction	[Stat. sig.] Reject that the difference is ? p.p.	Type I error	✔
Prediction	[Not stat. sig.] Not reject that the difference is ? p.p.	✔	Type II error

What is the minimum sample size you need to split it , measure the sample mean of in the first population which has a standard deviation of , and in the second population which has a standard deviation of (a ? difference between them) and be able to state with a confidence level that the second population true mean is the first one?

Wait! There is no way the second sample could be smaller than the first and still be statistically not smaller than it.

Wait! There is no way the second sample could be larger than the first and still be statistically not larger than it.

Answer: ?

Extra stuff

Methodology

Two Sample Mean Z-test
$test statistic formula for two sample mean. z=\frac{\overline{x_2}-\overline{x_1}}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}$
where

follows a Standard Normal Distribution and the confidence level is found in the middle of the distribution, inside minimum and maximum bounds (two-tailed).
two tailed bell-shaped distribution chart

at the right side of the distribution (one-tailed).
left tailed bell-shaped distribution chart

at the left side of the distribution (one-tailed).
right tailed bell-shaped distribution chart

Statistical Power

		Reality
		Null hypothesis H₀ Samples come from populations with the same mean value	Alternative hypothesis H_a ?
Prediction	[Stat. sig.] Reject that the samples come from populations with the same mean value	Type I error	✔
Prediction	[Not stat. sig.] Not reject that the samples come from populations with the same mean value	✔	Type II error

What is the minimum sample size to split it , measure the sample mean of in the first population which has a standard deviation of , and in the second population which has a standard deviation of (a ? difference between them) and be able to state with a confidence level that the difference between the true means of these populations is not smaller than ?

Wait! There is no way the second sample could be larger than the first and the difference between them be compared with a negative value (meaning second sample rate smaller than first).

Wait! There is no way the second sample could be ? larger than the first and the difference between them still be statistically not smaller than ?.

Wait! There is no way the second sample could be smaller then the first and the difference between them be compared with a positive value (meaning second sample rate larger than first).

Wait! There is no way the second sample could be ? (smaller) than the first and the difference between them still be statistically not smaller than ?.

Answer: ?

Extra stuff

Methodology

Two Sample Mean Z-test
$test statistic formula for two sample mean. z=\frac{\left(\overline{x_2}-\overline{x_1} \right)-d_0}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}$
where

follows a Standard Normal Distribution and the confidence level is found at the left side of the distribution (one-tailed).
right tailed bell-shaped distribution chart

Statistical Power

		Reality
		Null hypothesis H₀ The second sample comes from a population with a mean ? larger than the first sample	Alternative hypothesis H_a The second sample comes from a population with a rate that is more than ? larger than the first sample
Prediction	[Stat. sig.] Reject that the difference is ?	Type I error	✔
Prediction	[Not stat. sig.] Not reject that the difference is ?	✔	Type II error

Q&A:

Where is the α alpha?
It's in the confidence level. A 95% confidence level is the same thing as an α of 5%.

Why do I need to know the rate BEFORE taking the sample?
Because the variability of the sample's rate depend on the level of the rate. You should use the best guess you have about the true rate and then fine tune sample size as the results come in. Some calculators encourage you to use 50% rate, as this is the one that generates the largest variability and, consequently, the largest sample size. But, if you have previous insight about the level of the rate, use it.

Why do I need to know these rates BEFORE taking the sample?
Sample rate: Because the difference between the observed sample rate and a given reference rate is crucial to say if the rates are statistically different or not. If you get a small difference, you'll need a much bigger sample to say they are different.
Reference rate: Because the variability of the sample's rate depend on the level of the rate. Some calculators encourage you to use 50% rate here, as this is the one that generates the largest variability and, consequently, the largest sample size. But, if you do that, be carefull to also adjust the sample rate accordingly, since the sample size will depend on the difference between sample rate and this reference rate.
You should use the best guess you have about the true value of these rates and then fine tune sample size as the results come in.

How do I define the interval?
This represents the level of precision you want to have. Small intervals means high precision, but will also require larger sample sizes. Input an interval that makes sense for what you are testing. Consider that, after the test, this is going to be the level of precision of the rate you'll measure.

Why do other online calculators give different results?
Calculator.net gives the same result, just be careful with the "Margin of Error" parameter because the tooltip recommends inputing the percentage in relative terms, but the math behind it uses absolute terms.
Raosoft and Select statistical services give the same result, but they have an additional parameter, the size of the population. If the size is large, the result is equal to this calculator. They also don't explain well how to input the margin of error, which is the same as this calculator, but they don't say it is treated as absolute terms like percentage points, instead of relative terms.
SurveyMonkey, Qualtrics, CheckMarket and QuestionPro consider the rate to be 50% and don't provide a way to change it. That's probably not a bad assumption for online surveys, given the uncertainty about the answers, but it inflates the sample size, especially if you already know the expected value you are measuring is much different from 50%. These calculators also have the population size parameter and, if the size is large, the result is equal to this calculator.
Sample Size Calculators for designing clinical research gives the same results under the name "Normal approximation to the binomial calculation", and it also gives a higher sample size under "Binomial 'exact' calculation", but there is no explanation about this alternative methodology.
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Why do other online calculators give different results?
Statistics Kingdom and infrrr give the same result, but you input the sample size and they give you the confidence interval or the p-value (the other way around).

Why do other online calculators give different results?
Calculator.net, 365DataScience and Sample Size Calculators for designing clinical research give the same result, but you input the sample size and they give you the confidence interval (the other way around). On the other hand, the latter one provides an improved accuracy for small sample sizes because it uses the student's t distribution instead of the Normal distribution, which is relevant for samples under ~30 observations, making it even more precise than this calculator.
Heads up! Several online calculators like Raosoft, SurveyMonkey, Qualtrics, CheckMarket and QuestionPro assume that your data is binary: each observation is either 1 (one) or 0 (zero), or true/false, or the person answered yes/no to a question, which is OK for measuring what proportion of website visitors makes a purchase, or what is the rate of employment of a population. This calculator also provide that kind of answer, but you have to choose "measuring the rate", rather than "measuring the average value" at the top.
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Why do other online calculators give different results?
365DataScience, Omni Calculator and infrrr give the same result, but you have to input the sample size and they give you the confidence level or the p-value (the other way around). On the other hand, the latter one provides an improved accuracy for small sample sizes because it uses the student's t distribution instead of the Normal distribution, which is relevant for samples under ~30 observations, making it even more precise than this calculator.
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Why do other online calculators give different results?
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