hypothesis testing how to calculate sample size

Hypothesis Testing Calculator

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The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

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11.8: Effect Size, Sample Size and Power

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• Page ID 8245

• Danielle Navarro
• University of New South Wales

In previous sections I’ve emphasised the fact that the major design principle behind statistical hypothesis testing is that we try to control our Type I error rate. When we fix α=.05 we are attempting to ensure that only 5% of true null hypotheses are incorrectly rejected. However, this doesn’t mean that we don’t care about Type II errors. In fact, from the researcher’s perspective, the error of failing to reject the null when it is actually false is an extremely annoying one. With that in mind, a secondary goal of hypothesis testing is to try to minimise β, the Type II error rate, although we don’t usually talk in terms of minimising Type II errors. Instead, we talk about maximising the power of the test. Since power is defined as 1−β, this is the same thing.

power function

Let’s take a moment to think about what a Type II error actually is. A Type II error occurs when the alternative hypothesis is true, but we are nevertheless unable to reject the null hypothesis. Ideally, we’d be able to calculate a single number β that tells us the Type II error rate, in the same way that we can set α=.05 for the Type I error rate. Unfortunately, this is a lot trickier to do. To see this, notice that in my ESP study the alternative hypothesis actually corresponds to lots of possible values of θ. In fact, the alternative hypothesis corresponds to every value of θ except 0.5. Let’s suppose that the true probability of someone choosing the correct response is 55% (i.e., θ=.55). If so, then the true sampling distribution for X is not the same one that the null hypothesis predicts: the most likely value for X is now 55 out of 100. Not only that, the whole sampling distribution has now shifted, as shown in Figure 11.4. The critical regions, of course, do not change: by definition, the critical regions are based on what the null hypothesis predicts. What we’re seeing in this figure is the fact that when the null hypothesis is wrong, a much larger proportion of the sampling distribution distribution falls in the critical region. And of course that’s what should happen: the probability of rejecting the null hypothesis is larger when the null hypothesis is actually false! However θ=.55 is not the only possibility consistent with the alternative hypothesis. Let’s instead suppose that the true value of θ is actually 0.7. What happens to the sampling distribution when this occurs? The answer, shown in Figure 11.5, is that almost the entirety of the sampling distribution has now moved into the critical region. Therefore, if θ=0.7 the probability of us correctly rejecting the null hypothesis (i.e., the power of the test) is much larger than if θ=0.55. In short, while θ=.55 and θ=.70 are both part of the alternative hypothesis, the Type II error rate is different.

What all this means is that the power of a test (i.e., 1−β) depends on the true value of θ. To illustrate this, I’ve calculated the expected probability of rejecting the null hypothesis for all values of θ, and plotted it in Figure 11.6. This plot describes what is usually called the power function of the test. It’s a nice summary of how good the test is, because it actually tells you the power (1−β) for all possible values of θ. As you can see, when the true value of θ is very close to 0.5, the power of the test drops very sharply, but when it is further away, the power is large.

Effect size

Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned with mice when there are tigers abroad – George Box 1976

The plot shown in Figure 11.6 captures a fairly basic point about hypothesis testing. If the true state of the world is very different from what the null hypothesis predicts, then your power will be very high; but if the true state of the world is similar to the null (but not identical) then the power of the test is going to be very low. Therefore, it’s useful to be able to have some way of quantifying how “similar” the true state of the world is to the null hypothesis. A statistic that does this is called a measure of effect size (e.g. Cohen 1988; Ellis 2010). Effect size is defined slightly differently in different contexts, 165 (and so this section just talks in general terms) but the qualitative idea that it tries to capture is always the same: how big is the difference between the true population parameters, and the parameter values that are assumed by the null hypothesis? In our ESP example, if we let θ 0 =0.5 denote the value assumed by the null hypothesis, and let θ denote the true value, then a simple measure of effect size could be something like the difference between the true value and null (i.e., θ−θ 0 ), or possibly just the magnitude of this difference, abs(θ−θ 0 ).

Why calculate effect size? Let’s assume that you’ve run your experiment, collected the data, and gotten a significant effect when you ran your hypothesis test. Isn’t it enough just to say that you’ve gotten a significant effect? Surely that’s the point of hypothesis testing? Well, sort of. Yes, the point of doing a hypothesis test is to try to demonstrate that the null hypothesis is wrong, but that’s hardly the only thing we’re interested in. If the null hypothesis claimed that θ=.5, and we show that it’s wrong, we’ve only really told half of the story. Rejecting the null hypothesis implies that we believe that θ≠.5, but there’s a big difference between θ=.51 and θ=.8. If we find that θ=.8, then not only have we found that the null hypothesis is wrong, it appears to be very wrong. On the other hand, suppose we’ve successfully rejected the null hypothesis, but it looks like the true value of θ is only .51 (this would only be possible with a large study). Sure, the null hypothesis is wrong, but it’s not at all clear that we actually care , because the effect size is so small. In the context of my ESP study we might still care, since any demonstration of real psychic powers would actually be pretty cool 166 , but in other contexts a 1% difference isn’t very interesting, even if it is a real difference. For instance, suppose we’re looking at differences in high school exam scores between males and females, and it turns out that the female scores are 1% higher on average than the males. If I’ve got data from thousands of students, then this difference will almost certainly be statistically significant , but regardless of how small the p value is it’s just not very interesting. You’d hardly want to go around proclaiming a crisis in boys education on the basis of such a tiny difference would you? It’s for this reason that it is becoming more standard (slowly, but surely) to report some kind of standard measure of effect size along with the the results of the hypothesis test. The hypothesis test itself tells you whether you should believe that the effect you have observed is real (i.e., not just due to chance); the effect size tells you whether or not you should care.

Increasing the power of your study

Not surprisingly, scientists are fairly obsessed with maximising the power of their experiments. We want our experiments to work, and so we want to maximise the chance of rejecting the null hypothesis if it is false (and of course we usually want to believe that it is false!) As we’ve seen, one factor that influences power is the effect size. So the first thing you can do to increase your power is to increase the effect size. In practice, what this means is that you want to design your study in such a way that the effect size gets magnified. For instance, in my ESP study I might believe that psychic powers work best in a quiet, darkened room; with fewer distractions to cloud the mind. Therefore I would try to conduct my experiments in just such an environment: if I can strengthen people’s ESP abilities somehow, then the true value of θ will go up 167 and therefore my effect size will be larger. In short, clever experimental design is one way to boost power; because it can alter the effect size.

Unfortunately, it’s often the case that even with the best of experimental designs you may have only a small effect. Perhaps, for example, ESP really does exist, but even under the best of conditions it’s very very weak. Under those circumstances, your best bet for increasing power is to increase the sample size. In general, the more observations that you have available, the more likely it is that you can discriminate between two hypotheses. If I ran my ESP experiment with 10 participants, and 7 of them correctly guessed the colour of the hidden card, you wouldn’t be terribly impressed. But if I ran it with 10,000 participants and 7,000 of them got the answer right, you would be much more likely to think I had discovered something. In other words, power increases with the sample size. This is illustrated in Figure 11.7, which shows the power of the test for a true parameter of θ=0.7, for all sample sizes N from 1 to 100, where I’m assuming that the null hypothesis predicts that θ 0 =0.5.

Because power is important, whenever you’re contemplating running an experiment it would be pretty useful to know how much power you’re likely to have. It’s never possible to know for sure, since you can’t possibly know what your effect size is. However, it’s often (well, sometimes) possible to guess how big it should be. If so, you can guess what sample size you need! This idea is called power analysis , and if it’s feasible to do it, then it’s very helpful, since it can tell you something about whether you have enough time or money to be able to run the experiment successfully. It’s increasingly common to see people arguing that power analysis should be a required part of experimental design, so it’s worth knowing about. I don’t discuss power analysis in this book, however. This is partly for a boring reason and partly for a substantive one. The boring reason is that I haven’t had time to write about power analysis yet. The substantive one is that I’m still a little suspicious of power analysis. Speaking as a researcher, I have very rarely found myself in a position to be able to do one – it’s either the case that (a) my experiment is a bit non-standard and I don’t know how to define effect size properly, (b) I literally have so little idea about what the effect size will be that I wouldn’t know how to interpret the answers. Not only that, after extensive conversations with someone who does stats consulting for a living (my wife, as it happens), I can’t help but notice that in practice the only time anyone ever asks her for a power analysis is when she’s helping someone write a grant application. In other words, the only time any scientist ever seems to want a power analysis in real life is when they’re being forced to do it by bureaucratic process. It’s not part of anyone’s day to day work. In short, I’ve always been of the view that while power is an important concept, power analysis is not as useful as people make it sound, except in the rare cases where (a) someone has figured out how to calculate power for your actual experimental design and (b) you have a pretty good idea what the effect size is likely to be. Maybe other people have had better experiences than me, but I’ve personally never been in a situation where both (a) and (b) were true. Maybe I’ll be convinced otherwise in the future, and probably a future version of this book would include a more detailed discussion of power analysis, but for now this is about as much as I’m comfortable saying about the topic.

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Sample Size Calculation for a Hypothesis Test

Lynne Stokes

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Introduction, use of the method.

• CAVEATS TO CONSIDER WHEN LOOKING AT RESULTS BASED ON POWER ANALYSIS
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This JAMA Guide to Statistics and Methods explains the importance of considering sample size when interpreting study results, how the power analysis can help calculate the appropriate sample size, and the potential pitfalls of this approach.

Koegelenberg et al 1 reported the results of a randomized clinical trial (RCT) that investigated whether treatment with a nicotine patch in addition to varenicline produced higher rates of smoking abstinence than varenicline alone. The primary results were positive; that is, patients receiving the combination therapy were more likely to achieve continuous abstinence at 12 weeks than patients receiving varenicline alone. The absolute difference in the abstinence rate was estimated to be approximately 14%, which was statistically significant at level α = .05.

These findings differed from the results reported in 2 previous studies 2,3 of the same question, which detected no difference in treatments. What explains this difference? One explanation offered by the authors is that the previous studies “…may have been inadequately powered,” which means the sample size in those studies may have been too small to identify a difference between the treatments tested.

Why Is Power Analysis Used?

The sample size in a research investigation should be large enough that differences occurring by chance are rare but should not be larger than necessary, to avoid waste of resources and to prevent exposure of research participants to risk associated with the interventions. With any study, but especially if the study sample size is very small, any difference in observed rates can happen by chance and thus cannot be considered statistically significant.

In developing the methods for a study, investigators conduct a power analysis to calculate sample size. The power of a hypothesis test is the probability of obtaining a statistically significant result when there is a true difference in treatments. For example, suppose, as Koegelenberg et al 1 did, that the smoking abstinence rate were 45% for varenicline alone and 14% larger, or 59%, for the combination regimen. Power is the probability that, under these conditions, the trial would detect a difference in rates large enough to be statistically significant at a certain level α (ie, α is the probability of a type I error, which occurs by rejecting a null hypothesis that is actually true).

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Power and Sample Size Determination

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Issues in Estimating Sample Size for Hypothesis Testing

Ensuring that a test has high power.

All Modules

In the module on hypothesis testing for means and proportions, we introduced techniques for means, proportions, differences in means, and differences in proportions. While each test involved details that were specific to the outcome of interest (e.g., continuous or dichotomous) and to the number of comparison groups (one, two, more than two), there were common elements to each test. For example, in each test of hypothesis, there are two errors that can be committed. The first is called a Type I error and refers to the situation where we incorrectly reject H 0 when in fact it is true.   In the first step of any test of hypothesis, we select a level of significance, α , and α = P(Type I error) = P(Reject H 0 | H 0 is true). Because we purposely select a small value for α , we control the probability of committing a Type I error. The second type of error is called a Type II error and it is defined as the probability we do not reject H 0 when it is false. The probability of a Type II error is denoted β , and β =P(Type II error) = P(Do not Reject H 0 | H 0 is false). In hypothesis testing, we usually focus on power, which is defined as the probability that we reject H 0 when it is false, i.e., power = 1- β = P(Reject H 0 | H 0 is false). Power is the probability that a test correctly rejects a false null hypothesis. A good test is one with low probability of committing a Type I error (i.e., small α ) and high power (i.e., small β, high power).  

Here we present formulas to determine the sample size required to ensure that a test has high power. The sample size computations depend on the level of significance, aα, the desired power of the test (equivalent to 1-β), the variability of the outcome, and the effect size. The effect size is the difference in the parameter of interest that represents a clinically meaningful difference. Similar to the margin of error in confidence interval applications, the effect size is determined based on clinical or practical criteria and not statistical criteria.  

The concept of statistical power can be difficult to grasp. Before presenting the formulas to determine the sample sizes required to ensure high power in a test, we will first discuss power from a conceptual point of view.  

Suppose we want to test the following hypotheses at aα=0.05:  H 0 : μ = 90 versus H 1 : μ ≠ 90. To test the hypotheses, suppose we select a sample of size n=100. For this example, assume that the standard deviation of the outcome is σ=20. We compute the sample mean and then must decide whether the sample mean provides evidence to support the alternative hypothesis or not. This is done by computing a test statistic and comparing the test statistic to an appropriate critical value. If the null hypothesis is true (μ=90), then we are likely to select a sample whose mean is close in value to 90. However, it is also possible to select a sample whose mean is much larger or much smaller than 90. Recall from the Central Limit Theorem (see page 11 in the module on Probability ), that for large n (here n=100 is sufficiently large), the distribution of the sample means is approximately normal with a mean of

If the null hypothesis is true, it is possible to observe any sample mean shown in the figure below; all are possible under H 0 : μ = 90.  

Rejection Region for Test H 0 : μ = 90 versus H 1 : μ ≠ 90 at α =0.05

The areas in the two tails of the curve represent the probability of a Type I Error, α= 0.05. This concept was discussed in the module on Hypothesis Testing .  

Now, suppose that the alternative hypothesis, H 1 , is true (i.e., μ ≠ 90) and that the true mean is actually 94. The figure below shows the distributions of the sample mean under the null and alternative hypotheses.The values of the sample mean are shown along the horizontal axis.  

If the true mean is 94, then the alternative hypothesis is true. In our test, we selected α = 0.05 and reject H 0 if the observed sample mean exceeds 93.92 (focusing on the upper tail of the rejection region for now). The critical value (93.92) is indicated by the vertical line. The probability of a Type II error is denoted β, and β = P(Do not Reject H 0 | H 0 is false), i.e., the probability of not rejecting the null hypothesis if the null hypothesis were true. β is shown in the figure above as the area under the rightmost curve (H 1 ) to the left of the vertical line (where we do not reject H 0 ). Power is defined as 1- β = P(Reject H 0 | H 0 is false) and is shown in the figure as the area under the rightmost curve (H 1 ) to the right of the vertical line (where we reject H 0 ).  

Note that β and power are related to α, the variability of the outcome and the effect size. From the figure above we can see what happens to β and power if we increase α. Suppose, for example, we increase α to α=0.10.The upper critical value would be 92.56 instead of 93.92. The vertical line would shift to the left, increasing α, decreasing β and increasing power. While a better test is one with higher power, it is not advisable to increase α as a means to increase power. Nonetheless, there is a direct relationship between α and power (as α increases, so does power).

β and power are also related to the variability of the outcome and to the effect size. The effect size is the difference in the parameter of interest (e.g., μ) that represents a clinically meaningful difference. The figure above graphically displays α, β, and power when the difference in the mean under the null as compared to the alternative hypothesis is 4 units (i.e., 90 versus 94). The figure below shows the same components for the situation where the mean under the alternative hypothesis is 98.

Notice that there is much higher power when there is a larger difference between the mean under H 0 as compared to H 1 (i.e., 90 versus 98). A statistical test is much more likely to reject the null hypothesis in favor of the alternative if the true mean is 98 than if the true mean is 94. Notice also in this case that there is little overlap in the distributions under the null and alternative hypotheses. If a sample mean of 97 or higher is observed it is very unlikely that it came from a distribution whose mean is 90. In the previous figure for H 0 : μ = 90 and H 1 : μ = 94, if we observed a sample mean of 93, for example, it would not be as clear as to whether it came from a distribution whose mean is 90 or one whose mean is 94.

In designing studies most people consider power of 80% or 90% (just as we generally use 95% as the confidence level for confidence interval estimates). The inputs for the sample size formulas include the desired power, the level of significance and the effect size. The effect size is selected to represent a clinically meaningful or practically important difference in the parameter of interest, as we will illustrate.  

The formulas we present below produce the minimum sample size to ensure that the test of hypothesis will have a specified probability of rejecting the null hypothesis when it is false (i.e., a specified power). In planning studies, investigators again must account for attrition or loss to follow-up. The formulas shown below produce the number of participants needed with complete data, and we will illustrate how attrition is addressed in planning studies.

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Sample Size and its Importance in Research

Chittaranjan andrade.

Clinical Psychopharmacology Unit, Department of Clinical Psychopharmacology and Neurotoxicology, National Institute of Mental Health and Neurosciences, Bengaluru, Karnataka, India

The sample size for a study needs to be estimated at the time the study is proposed; too large a sample is unnecessary and unethical, and too small a sample is unscientific and also unethical. The necessary sample size can be calculated, using statistical software, based on certain assumptions. If no assumptions can be made, then an arbitrary sample size is set for a pilot study. This article discusses sample size and how it relates to matters such as ethics, statistical power, the primary and secondary hypotheses in a study, and findings from larger vs. smaller samples.

Studies are conducted on samples because it is usually impossible to study the entire population. Conclusions drawn from samples are intended to be generalized to the population, and sometimes to the future as well. The sample must therefore be representative of the population. This is best ensured by the use of proper methods of sampling. The sample must also be adequate in size – in fact, no more and no less.

SAMPLE SIZE AND ETHICS

A sample that is larger than necessary will be better representative of the population and will hence provide more accurate results. However, beyond a certain point, the increase in accuracy will be small and hence not worth the effort and expense involved in recruiting the extra patients. Furthermore, an overly large sample would inconvenience more patients than might be necessary for the study objectives; this is unethical. In contrast, a sample that is smaller than necessary would have insufficient statistical power to answer the primary research question, and a statistically nonsignificant result could merely be because of inadequate sample size (Type 2 or false negative error). Thus, a small sample could result in the patients in the study being inconvenienced with no benefit to future patients or to science. This is also unethical.

In this regard, inconvenience to patients refers to the time that they spend in clinical assessments and to the psychological and physical discomfort that they experience in assessments such as interviews, blood sampling, and other procedures.

ESTIMATING SAMPLE SIZE

So how large should a sample be? In hypothesis testing studies, this is mathematically calculated, conventionally, as the sample size necessary to be 80% certain of identifying a statistically significant outcome should the hypothesis be true for the population, with P for statistical significance set at 0.05. Some investigators power their studies for 90% instead of 80%, and some set the threshold for significance at 0.01 rather than 0.05. Both choices are uncommon because the necessary sample size becomes large, and the study becomes more expensive and more difficult to conduct. Many investigators increase the sample size by 10%, or by whatever proportion they can justify, to compensate for expected dropout, incomplete records, biological specimens that do not meet laboratory requirements for testing, and other study-related problems.

Sample size calculations require assumptions about expected means and standard deviations, or event risks, in different groups; or, upon expected effect sizes. For example, a study may be powered to detect an effect size of 0.5; or a response rate of 60% with drug vs. 40% with placebo.[ 1 ] When no guesstimates or expectations are possible, pilot studies are conducted on a sample that is arbitrary in size but what might be considered reasonable for the field.

The sample size may need to be larger in multicenter studies because of statistical noise (due to variations in patient characteristics, nonspecific treatment characteristics, rating practices, environments, etc. between study centers).[ 2 ] Sample size calculations can be performed manually or using statistical software; online calculators that provide free service can easily be identified by search engines. G*Power is an example of a free, downloadable program for sample size estimation. The manual and tutorial for G*Power can also be downloaded.

PRIMARY AND SECONDARY ANALYSES

The sample size is calculated for the primary hypothesis of the study. What is the difference between the primary hypothesis, primary outcome and primary outcome measure? As an example, the primary outcome may be a reduction in the severity of depression, the primary outcome measure may be the Montgomery-Asberg Depression Rating Scale (MADRS) and the primary hypothesis may be that reduction in MADRS scores is greater with the drug than with placebo. The primary hypothesis is tested in the primary analysis.

Studies almost always have many hypotheses; for example, that the study drug will outperform placebo on measures of depression, suicidality, anxiety, disability and quality of life. The sample size necessary for adequate statistical power to test each of these hypotheses will be different. Because a study can have only one sample size, it can be powered for only one outcome, the primary outcome. Therefore, the study would be either overpowered or underpowered for the other outcomes. These outcomes are therefore called secondary outcomes, and are associated with secondary hypotheses, and are tested in secondary analyses. Secondary analyses are generally considered exploratory because when many hypotheses in a study are each tested at a P < 0.05 level for significance, some may emerge statistically significant by chance (Type 1 or false positive errors).[ 3 ]

INTERPRETING RESULTS

Here is an interesting question. A test of the primary hypothesis yielded a P value of 0.07. Might we conclude that our sample was underpowered for the study and that, had our sample been larger, we would have identified a significant result? No! The reason is that larger samples will more accurately represent the population value, whereas smaller samples could be off the mark in either direction – towards or away from the population value. In this context, readers should also note that no matter how small the P value for an estimate is, the population value of that estimate remains the same.[ 4 ]

On a parting note, it is unlikely that population values will be null. That is, for example, that the response rate to the drug will be exactly the same as that to placebo, or that the correlation between height and age at onset of schizophrenia will be zero. If the sample size is large enough, even such small differences between groups, or trivial correlations, would be detected as being statistically significant. This does not mean that the findings are clinically significant.

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Power & Sample Size Calculator

Use this advanced sample size calculator to calculate the sample size required for a one-sample statistic, or for differences between two proportions or means (two independent samples). More than two groups supported for binomial data. Calculate power given sample size, alpha, and the minimum detectable effect (MDE, minimum effect of interest).

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Post-hoc power (Observed power)

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• Absolute versus relative difference and why it matters for sample size determination

    Using the power & sample size calculator

This calculator allows the evaluation of different statistical designs when planning an experiment (trial, test) which utilizes a Null-Hypothesis Statistical Test to make inferences. It can be used both as a sample size calculator and as a statistical power calculator . Usually one would determine the sample size required given a particular power requirement, but in cases where there is a predetermined sample size one can instead calculate the power for a given effect size of interest.

1. Number of test groups. The sample size calculator supports experiments in which one is gathering data on a single sample in order to compare it to a general population or known reference value (one-sample), as well as ones where a control group is compared to one or more treatment groups ( two-sample, k-sample ) in order to detect differences between them. For comparing more than one treatment group to a control group the sample size adjustments based on the Dunnett's correction are applied. These are only approximately accurate and subject to the assumption of about equal effect size in all k groups, and can only support equal sample sizes in all groups and the control. Power calculations are not currently supported for more than one treatment group due to their complexity.

2. Type of outcome . The outcome of interest can be the absolute difference of two proportions (binomial data, e.g. conversion rate or event rate), the absolute difference of two means (continuous data, e.g. height, weight, speed, time, revenue, etc.), or the relative difference between two proportions or two means (percent difference, percent change, etc.). See Absolute versus relative difference for additional information. One can also calculate power and sample size for the mean of just a single group. The sample size and power calculator uses the Z-distribution (normal distribution) .

3. Baseline The baseline mean (mean under H 0 ) is the number one would expect to see if all experiment participants were assigned to the control group. It is the mean one expects to observe if the treatment has no effect whatsoever.

4. Minimum Detectable Effect . The minimum effect of interest, which is often called the minimum detectable effect ( MDE , but more accurately: MRDE, minimum reliably detectable effect) should be a difference one would not like to miss , if it existed. It can be entered as a proportion (e.g. 0.10) or as percentage (e.g. 10%). It is always relative to the mean/proportion under H 0 ± the superiority/non-inferiority or equivalence margin. For example, if the baseline mean is 10 and there is a superiority alternative hypothesis with a superiority margin of 1 and the minimum effect of interest relative to the baseline is 3, then enter an MDE of 2 , since the MDE plus the superiority margin will equal exactly 3. In this case the MDE (MRDE) is calculated relative to the baseline plus the superiority margin, as it is usually more intuitive to be interested in that value.

If entering means data, one needs to specify the mean under the null hypothesis (worst-case scenario for a composite null) and the standard deviation of the data (for a known population or estimated from a sample).

5. Type of alternative hypothesis . The calculator supports superiority , non-inferiority and equivalence alternative hypotheses. When the superiority or non-inferiority margin is zero, it becomes a classical left or right sided hypothesis, if it is larger than zero then it becomes a true superiority / non-inferiority design. The equivalence margin cannot be zero. See Types of null and alternative hypothesis below for an in-depth explanation.

6. Acceptable error rates . The type I error rate, α , should always be provided. Power, calculated as 1 - β , where β is the type II error rate, is only required when determining sample size. For an in-depth explanation of power see What is statistical power below. The type I error rate is equivalent to the significance threshold if one is doing p-value calculations and to the confidence level if using confidence intervals.

The sample size calculator will output the sample size of the single group or of all groups, as well as the total sample size required. If used to solve for power it will output the power as a proportion and as a percentage.

    Why is sample size determination important?

While this online software provides the means to determine the sample size of a test, it is of great importance to understand the context of the question, the "why" of it all.

Estimating the required sample size before running an experiment that will be judged by a statistical test (a test of significance, confidence interval, etc.) allows one to:

• determine the sample size needed to detect an effect of a given size with a given probability
• be aware of the magnitude of the effect that can be detected with a certain sample size and power
• calculate the power for a given sample size and effect size of interest

This is crucial information with regards to making the test cost-efficient. Having a proper sample size can even mean the difference between conducting the experiment or postponing it for when one can afford a sample of size that is large enough to ensure a high probability to detect an effect of practical significance.

For example, if a medical trial has low power, say less than 80% (β = 0.2) for a given minimum effect of interest, then it might be unethical to conduct it due to its low probability of rejecting the null hypothesis and establishing the effectiveness of the treatment. Similarly, for experiments in physics, psychology, economics, marketing, conversion rate optimization, etc. Balancing the risks and rewards and assuring the cost-effectiveness of an experiment is a task that requires juggling with the interests of many stakeholders which is well beyond the scope of this text.

    What is statistical power?

Statistical power is the probability of rejecting a false null hypothesis with a given level of statistical significance , against a particular alternative hypothesis. Alternatively, it can be said to be the probability to detect with a given level of significance a true effect of a certain magnitude. This is what one gets when using the tool in "power calculator" mode. Power is closely related with the type II error rate: β, and it is always equal to (1 - β). In a probability notation the type two error for a given point alternative can be expressed as [1] :

β(T α ; μ 1 ) = P(d(X) ≤ c α ; μ = μ 1 )

It should be understood that the type II error rate is calculated at a given point, signified by the presence of a parameter for the function of beta. Similarly, such a parameter is present in the expression for power since POW = 1 - β [1] :

POW(T α ; μ 1 ) = P(d(X) > c α ; μ = μ 1 )

In the equations above c α represents the critical value for rejecting the null (significance threshold), d(X) is a statistical function of the parameter of interest - usually a transformation to a standardized score, and μ 1 is a specific value from the space of the alternative hypothesis.

One can also calculate and plot the whole power function, getting an estimate of the power for many different alternative hypotheses. Due to the S-shape of the function, power quickly rises to nearly 100% for larger effect sizes, while it decreases more gradually to zero for smaller effect sizes. Such a power function plot is not yet supported by our statistical software, but one can calculate the power at a few key points (e.g. 10%, 20% ... 90%, 100%) and connect them for a rough approximation.

Statistical power is directly and inversely related to the significance threshold. At the zero effect point for a simple superiority alternative hypothesis power is exactly 1 - α as can be easily demonstrated with our power calculator. At the same time power is positively related to the number of observations, so increasing the sample size will increase the power for a given effect size, assuming all other parameters remain the same.

Power calculations can be useful even after a test has been completed since failing to reject the null can be used as an argument for the null and against particular alternative hypotheses to the extent to which the test had power to reject them. This is more explicitly defined in the severe testing concept proposed by Mayo & Spanos (2006).

Computing observed power is only useful if there was no rejection of the null hypothesis and one is interested in estimating how probative the test was towards the null . It is absolutely useless to compute post-hoc power for a test which resulted in a statistically significant effect being found [5] . If the effect is significant, then the test had enough power to detect it. In fact, there is a 1 to 1 inverse relationship between observed power and statistical significance, so one gains nothing from calculating post-hoc power, e.g. a test planned for α = 0.05 that passed with a p-value of just 0.0499 will have exactly 50% observed power (observed β = 0.5).

I strongly encourage using this power and sample size calculator to compute observed power in the former case, and strongly discourage it in the latter.

    Sample size formula

The formula for calculating the sample size of a test group in a one-sided test of absolute difference is:

where Z 1-α is the Z-score corresponding to the selected statistical significance threshold α , Z 1-β is the Z-score corresponding to the selected statistical power 1-β , σ is the known or estimated standard deviation, and δ is the minimum effect size of interest. The standard deviation is estimated analytically in calculations for proportions, and empirically from the raw data for other types of means.

The formula applies to single sample tests as well as to tests of absolute difference between two samples. A proprietary modification is employed when calculating the required sample size in a test of relative difference . This modification has been extensively tested under a variety of scenarios through simulations.

    Types of null and alternative hypotheses in significance tests

When doing sample size calculations, it is important that the null hypothesis (H 0 , the hypothesis being tested) and the alternative hypothesis is (H 1 ) are well thought out. The test can reject the null or it can fail to reject it. Strictly logically speaking it cannot lead to acceptance of the null or to acceptance of the alternative hypothesis. A null hypothesis can be a point one - hypothesizing that the true value is an exact point from the possible values, or a composite one: covering many possible values, usually from -∞ to some value or from some value to +∞. The alternative hypothesis can also be a point one or a composite one.

In a Neyman-Pearson framework of NHST (Null-Hypothesis Statistical Test) the alternative should exhaust all values that do not belong to the null, so it is usually composite. Below is an illustration of some possible combinations of null and alternative statistical hypotheses: superiority, non-inferiority, strong superiority (margin > 0), equivalence.

All of these are supported in our power and sample size calculator.

Careful consideration has to be made when deciding on a non-inferiority margin, superiority margin or an equivalence margin . Equivalence trials are sometimes used in clinical trials where a drug can be performing equally (within some bounds) to an existing drug but can still be preferred due to less or less severe side effects, cheaper manufacturing, or other benefits, however, non-inferiority designs are more common. Similar cases exist in disciplines such as conversion rate optimization [2] and other business applications where benefits not measured by the primary outcome of interest can influence the adoption of a given solution. For equivalence tests it is assumed that they will be evaluated using a two one-sided t-tests (TOST) or z-tests, or confidence intervals.

Note that our calculator does not support the schoolbook case of a point null and a point alternative, nor a point null and an alternative that covers all the remaining values. This is since such cases are non-existent in experimental practice [3][4] . The only two-sided calculation is for the equivalence alternative hypothesis, all other calculations are one-sided (one-tailed) .

    Absolute versus relative difference and why it matters for sample size determination

When using a sample size calculator it is important to know what kind of inference one is looking to make: about the absolute or about the relative difference, often called percent effect, percentage effect, relative change, percent lift, etc. Where the fist is μ 1 - μ the second is μ 1 -μ / μ or μ 1 -μ / μ x 100 (%). The division by μ is what adds more variance to such an estimate, since μ is just another variable with random error, therefore a test for relative difference will require larger sample size than a test for absolute difference. Consequently, if sample size is fixed, there will be less power for the relative change equivalent to any given absolute change.

For the above reason it is important to know and state beforehand if one is going to be interested in percentage change or if absolute change is of primary interest. Then it is just a matter of fliping a radio button.

    References

1 Mayo D.G., Spanos A. (2010) – "Error Statistics", in P. S. Bandyopadhyay & M. R. Forster (Eds.), Philosophy of Statistics, (7, 152–198). Handbook of the Philosophy of Science . The Netherlands: Elsevier.

2 Georgiev G.Z. (2017) "The Case for Non-Inferiority A/B Tests", [online] https://blog.analytics-toolkit.com/2017/case-non-inferiority-designs-ab-testing/ (accessed May 7, 2018)

3 Georgiev G.Z. (2017) "One-tailed vs Two-tailed Tests of Significance in A/B Testing", [online] https://blog.analytics-toolkit.com/2017/one-tailed-two-tailed-tests-significance-ab-testing/ (accessed May 7, 2018)

4 Hyun-Chul Cho Shuzo Abe (2013) "Is two-tailed testing for directional research hypotheses tests legitimate?", Journal of Business Research 66:1261-1266

5 Lakens D. (2014) "Observed power, and what to do if your editor asks for post-hoc power analyses" [online] http://daniellakens.blogspot.bg/2014/12/observed-power-and-what-to-do-if-your.html (accessed May 7, 2018)

Cite this calculator & page

If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Sample Size Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/power-sample-size-calculator.php URL [Accessed Date: 29 Apr, 2024].

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• How To Determine Sample Size

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How to determine sample size.

12 min read Sample size can make or break your research project. Here’s how to master the delicate art of choosing the right sample size.

Author:  Will Webster

Sample size is the beating heart of any research project. It’s the invisible force that gives life to your data, making your findings robust, reliable and believable.

Sample size is what determines if you see a broad view or a focus on minute details; the art and science of correctly determining it involves a careful balancing act. Finding an appropriate sample size demands a clear understanding of the level of detail you wish to see in your data and the constraints you might encounter along the way.

Remember, whether you’re studying a small group or an entire population, your findings are only ever as good as the sample you choose.

Free eBook: Empower your market research efforts today

Let’s delve into the world of sampling and uncover the best practices for determining sample size for your research.

“How much sample do we need?” is one of the most commonly-asked questions and stumbling points in the early stages of research design . Finding the right answer to it requires first understanding and answering two other questions:

How important is statistical significance to you and your stakeholders?

What are your real-world constraints.

At the heart of this question is the goal to confidently differentiate between groups, by describing meaningful differences as statistically significant. Statistical significance isn’t a difficult concept, but it needs to be considered within the unique context of your research and your measures.

First, you should consider when you deem a difference to be meaningful in your area of research. While the standards for statistical significance are universal, the standards for “meaningful difference” are highly contextual.

For example, a 10% difference between groups might not be enough to merit a change in a marketing campaign for a breakfast cereal, but a 10% difference in efficacy of breast cancer treatments might quite literally be the difference between life and death for hundreds of patients. The exact same magnitude of difference has very little meaning in one context, but has extraordinary meaning in another. You ultimately need to determine the level of precision that will help you make your decision.

Within sampling, the lowest amount of magnification – or smallest sample size – could make the most sense, given the level of precision needed, as well as timeline and budgetary constraints.

If you’re able to detect statistical significance at a difference of 10%, and 10% is a meaningful difference, there is no need for a larger sample size, or higher magnification. However, if the study will only be useful if a significant difference is detected for smaller differences – say, a difference of 5% — the sample size must be larger to accommodate this needed precision. Similarly, if 5% is enough, and 3% is unnecessary, there is no need for a larger statistically significant sample size.

You should also consider how much you expect your responses to vary. When there isn’t a lot of variability in response, it takes a lot more sample to be confident that there are statistically significant differences between groups.

For instance, it will take a lot more sample to find statistically significant differences between groups if you are asking, “What month do you think Christmas is in?” than if you are asking, “How many miles are there between the Earth and the moon?”. In the former, nearly everybody is going to give the exact same answer, while the latter will give a lot of variation in responses. Simply put, when your variables do not have a lot of variance, larger sample sizes make sense.

Statistical significance

The likelihood that the results of a study or experiment did not occur randomly or by chance, but are meaningful and indicate a genuine effect or relationship between variables.

Magnitude of difference

The size or extent of the difference between two or more groups or variables, providing a measure of the effect size or practical significance of the results.

Actionable insights

Valuable findings or conclusions drawn from data analysis that can be directly applied or implemented in decision-making processes or strategies to achieve a particular goal or outcome.

It’s crucial to understand the differences between the concepts of “statistical significance”, “magnitude of difference” and “actionable insights” – and how they can influence each other:

• Even if there is a statistically significant difference, it doesn’t mean the magnitude of the difference is large: with a large enough sample, a 3% difference could be statistically significant
• Even if the magnitude of the difference is large, it doesn’t guarantee that this difference is statistically significant: with a small enough sample, an 18% difference might not be statistically significant
• Even if there is a large, statistically significant difference, it doesn’t mean there is a story, or that there are actionable insights

There is no way to guarantee statistically significant differences at the outset of a study – and that is a good thing.

Even with a sample size of a million, there simply may not be any differences – at least, any that could be described as statistically significant. And there are times when a lack of significance is positive.

Imagine if your main competitor ran a multi-million dollar ad campaign in a major city and a huge pre-post study to detect campaign effects, only to discover that there were no statistically significant differences in brand awareness . This may be terrible news for your competitor, but it would be great news for you.

With Stats iQ™ you can analyze your research results and conduct significance testing

As you determine your sample size, you should consider the real-world constraints to your research.

Factors revolving around timings, budget and target population are among the most common constraints, impacting virtually every study. But by understanding and acknowledging them, you can definitely navigate the practical constraints of your research when pulling together your sample.

Timeline constraints

Gathering a larger sample size naturally requires more time. This is particularly true for elusive audiences, those hard-to-reach groups that require special effort to engage. Your timeline could become an obstacle if it is particularly tight, causing you to rethink your sample size to meet your deadline.

Budgetary constraints

Every sample, whether large or small, inexpensive or costly, signifies a portion of your budget. Samples could be like an open market; some are inexpensive, others are pricey, but all have a price tag attached to them.

Population constraints

Sometimes the individuals or groups you’re interested in are difficult to reach; other times, they’re a part of an extremely small population. These factors can limit your sample size even further.

What’s a good sample size?

A good sample size really depends on the context and goals of the research. In general, a good sample size is one that accurately represents the population and allows for reliable statistical analysis.

Larger sample sizes are typically better because they reduce the likelihood of sampling errors and provide a more accurate representation of the population. However, larger sample sizes often increase the impact of practical considerations, like time, budget and the availability of your audience. Ultimately, you should be aiming for a sample size that provides a balance between statistical validity and practical feasibility.

4 tips for choosing the right sample size

Choosing the right sample size is an intricate balancing act, but following these four tips can take away a lot of the complexity.

1) Start with your goal

The foundation of your research is a clearly defined goal. You need to determine what you’re trying to understand or discover, and use your goal to guide your research methods – including your sample size.

If your aim is to get a broad overview of a topic, a larger, more diverse sample may be appropriate. However, if your goal is to explore a niche aspect of your subject, a smaller, more targeted sample might serve you better. You should always align your sample size with the objectives of your research.

2) Know that you can’t predict everything

Research is a journey into the unknown. While you may have hypotheses and predictions, it’s important to remember that you can’t foresee every outcome – and this uncertainty should be considered when choosing your sample size.

A larger sample size can help to mitigate some of the risks of unpredictability, providing a more diverse range of data and potentially more accurate results. However, you shouldn’t let the fear of the unknown push you into choosing an impractically large sample size.

3) Plan for a sample that meets your needs and considers your real-life constraints

Every research project operates within certain boundaries – commonly budget, timeline and the nature of the sample itself. When deciding on your sample size, these factors need to be taken into consideration.

Be realistic about what you can achieve with your available resources and time, and always tailor your sample size to fit your constraints – not the other way around.

4) Use best practice guidelines to calculate sample size

There are many established guidelines and formulas that can help you in determining the right sample size.

The easiest way to define your sample size is using a sample size calculator , or you can use a manual sample size calculation if you want to test your math skills. Cochran’s formula is perhaps the most well known equation for calculating sample size, and widely used when the population is large or unknown.

Beyond the formula, it’s vital to consider the confidence interval, which plays a significant role in determining the appropriate sample size – especially when working with a random sample – and the sample proportion. This represents the expected ratio of the target population that has the characteristic or response you’re interested in, and therefore has a big impact on your correct sample size.

If your population is small, or its variance is unknown, there are steps you can still take to determine the right sample size. Common approaches here include conducting a small pilot study to gain initial estimates of the population variance, and taking a conservative approach by assuming a larger variance to ensure a more representative sample size.

Empower your market research

Conducting meaningful research and extracting actionable intelligence are priceless skills in today’s ultra competitive business landscape. It’s never been more crucial to stay ahead of the curve by leveraging the power of market research to identify opportunities, mitigate risks and make informed decisions.

Equip yourself with the tools for success with our essential eBook, “The ultimate guide to conducting market research” .

With this front-to-back guide, you’ll discover the latest strategies and best practices that are defining effective market research. Learn about practical insights and real-world applications that are demonstrating the value of research in driving business growth and innovation.

Related resources

Selection bias 11 min read, systematic random sampling 15 min read, convenience sampling 18 min read, probability sampling 8 min read, non-probability sampling 17 min read, stratified random sampling 12 min read, simple random sampling 9 min read, request demo.

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t-test Calculator

Table of contents

Welcome to our t-test calculator! Here you can not only easily perform one-sample t-tests , but also two-sample t-tests , as well as paired t-tests .

Do you prefer to find the p-value from t-test, or would you rather find the t-test critical values? Well, this t-test calculator can do both! 😊

What does a t-test tell you? Take a look at the text below, where we explain what actually gets tested when various types of t-tests are performed. Also, we explain when to use t-tests (in particular, whether to use the z-test vs. t-test) and what assumptions your data should satisfy for the results of a t-test to be valid. If you've ever wanted to know how to do a t-test by hand, we provide the necessary t-test formula, as well as tell you how to determine the number of degrees of freedom in a t-test.

When to use a t-test?

A t-test is one of the most popular statistical tests for location , i.e., it deals with the population(s) mean value(s).

There are different types of t-tests that you can perform:

• A one-sample t-test;
• A two-sample t-test; and
• A paired t-test.

In the next section , we explain when to use which. Remember that a t-test can only be used for one or two groups . If you need to compare three (or more) means, use the analysis of variance ( ANOVA ) method.

The t-test is a parametric test, meaning that your data has to fulfill some assumptions :

• The data points are independent; AND
• The data, at least approximately, follow a normal distribution .

If your sample doesn't fit these assumptions, you can resort to nonparametric alternatives. Visit our Mann–Whitney U test calculator or the Wilcoxon rank-sum test calculator to learn more. Other possibilities include the Wilcoxon signed-rank test or the sign test.

Which t-test?

Your choice of t-test depends on whether you are studying one group or two groups:

One sample t-test

Choose the one-sample t-test to check if the mean of a population is equal to some pre-set hypothesized value .

The average volume of a drink sold in 0.33 l cans — is it really equal to 330 ml?

The average weight of people from a specific city — is it different from the national average?

Two-sample t-test

Choose the two-sample t-test to check if the difference between the means of two populations is equal to some pre-determined value when the two samples have been chosen independently of each other.

In particular, you can use this test to check whether the two groups are different from one another .

The average difference in weight gain in two groups of people: one group was on a high-carb diet and the other on a high-fat diet.

The average difference in the results of a math test from students at two different universities.

This test is sometimes referred to as an independent samples t-test , or an unpaired samples t-test .

Paired t-test

A paired t-test is used to investigate the change in the mean of a population before and after some experimental intervention , based on a paired sample, i.e., when each subject has been measured twice: before and after treatment.

In particular, you can use this test to check whether, on average, the treatment has had any effect on the population .

The change in student test performance before and after taking a course.

The change in blood pressure in patients before and after administering some drug.

How to do a t-test?

So, you've decided which t-test to perform. These next steps will tell you how to calculate the p-value from t-test or its critical values, and then which decision to make about the null hypothesis.

Decide on the alternative hypothesis :

Use a two-tailed t-test if you only care whether the population's mean (or, in the case of two populations, the difference between the populations' means) agrees or disagrees with the pre-set value.

Use a one-tailed t-test if you want to test whether this mean (or difference in means) is greater/less than the pre-set value.

Compute your T-score value :

Formulas for the test statistic in t-tests include the sample size , as well as its mean and standard deviation . The exact formula depends on the t-test type — check the sections dedicated to each particular test for more details.

Determine the degrees of freedom for the t-test:

The degrees of freedom are the number of observations in a sample that are free to vary as we estimate statistical parameters. In the simplest case, the number of degrees of freedom equals your sample size minus the number of parameters you need to estimate . Again, the exact formula depends on the t-test you want to perform — check the sections below for details.

The degrees of freedom are essential, as they determine the distribution followed by your T-score (under the null hypothesis). If there are d degrees of freedom, then the distribution of the test statistics is the t-Student distribution with d degrees of freedom . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from N(0,1).

💡 The t-Student distribution owes its name to William Sealy Gosset, who, in 1908, published his paper on the t-test under the pseudonym "Student". Gosset worked at the famous Guinness Brewery in Dublin, Ireland, and devised the t-test as an economical way to monitor the quality of beer. Cheers! 🍺🍺🍺

p-value from t-test

Recall that the p-value is the probability (calculated under the assumption that the null hypothesis is true) that the test statistic will produce values at least as extreme as the T-score produced for your sample . As probabilities correspond to areas under the density function, p-value from t-test can be nicely illustrated with the help of the following pictures:

The following formulae say how to calculate p-value from t-test. By cdf t,d we denote the cumulative distribution function of the t-Student distribution with d degrees of freedom:

p-value from left-tailed t-test:

p-value = cdf t,d (t score )

p-value from right-tailed t-test:

p-value = 1 − cdf t,d (t score )

p-value from two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

or, equivalently: p-value = 2 − 2 × cdf t,d (|t score |)

However, the cdf of the t-distribution is given by a somewhat complicated formula. To find the p-value by hand, you would need to resort to statistical tables, where approximate cdf values are collected, or to specialized statistical software. Fortunately, our t-test calculator determines the p-value from t-test for you in the blink of an eye!

t-test critical values

Recall, that in the critical values approach to hypothesis testing, you need to set a significance level, α, before computing the critical values , which in turn give rise to critical regions (a.k.a. rejection regions).

Formulas for critical values employ the quantile function of t-distribution, i.e., the inverse of the cdf :

Critical value for left-tailed t-test: cdf t,d -1 (α)

critical region:

(-∞, cdf t,d -1 (α)]

Critical value for right-tailed t-test: cdf t,d -1 (1-α)

[cdf t,d -1 (1-α), ∞)

Critical values for two-tailed t-test: ±cdf t,d -1 (1-α/2)

(-∞, -cdf t,d -1 (1-α/2)] ∪ [cdf t,d -1 (1-α/2), ∞)

To decide the fate of the null hypothesis, just check if your T-score lies within the critical region:

If your T-score belongs to the critical region , reject the null hypothesis and accept the alternative hypothesis.

If your T-score is outside the critical region , then you don't have enough evidence to reject the null hypothesis.

How to use our t-test calculator

Choose the type of t-test you wish to perform:

A one-sample t-test (to test the mean of a single group against a hypothesized mean);

A two-sample t-test (to compare the means for two groups); or

A paired t-test (to check how the mean from the same group changes after some intervention).

Two-tailed;

Left-tailed; or

Right-tailed.

This t-test calculator allows you to use either the p-value approach or the critical regions approach to hypothesis testing!

Enter your T-score and the number of degrees of freedom . If you don't know them, provide some data about your sample(s): sample size, mean, and standard deviation, and our t-test calculator will compute the T-score and degrees of freedom for you .

Once all the parameters are present, the p-value, or critical region, will immediately appear underneath the t-test calculator, along with an interpretation!

One-sample t-test

The null hypothesis is that the population mean is equal to some value μ 0 \mu_0 μ 0 ​ .

The alternative hypothesis is that the population mean is:

• different from μ 0 \mu_0 μ 0 ​ ;
• smaller than μ 0 \mu_0 μ 0 ​ ; or
• greater than μ 0 \mu_0 μ 0 ​ .

One-sample t-test formula :

• μ 0 \mu_0 μ 0 ​ — Mean postulated in the null hypothesis;
• n n n — Sample size;
• x ˉ \bar{x} x ˉ — Sample mean; and
• s s s — Sample standard deviation.

Number of degrees of freedom in t-test (one-sample) = n − 1 n-1 n − 1 .

The null hypothesis is that the actual difference between these groups' means, μ 1 \mu_1 μ 1 ​ , and μ 2 \mu_2 μ 2 ​ , is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the difference μ 1 − μ 2 \mu_1 - \mu_2 μ 1 ​ − μ 2 ​ is:

• Different from Δ \Delta Δ ;
• Smaller than Δ \Delta Δ ; or
• Greater than Δ \Delta Δ .

In particular, if this pre-determined difference is zero ( Δ = 0 \Delta = 0 Δ = 0 ):

The null hypothesis is that the population means are equal.

The alternate hypothesis is that the population means are:

• μ 1 \mu_1 μ 1 ​ and μ 2 \mu_2 μ 2 ​ are different from one another;
• μ 1 \mu_1 μ 1 ​ is smaller than μ 2 \mu_2 μ 2 ​ ; and
• μ 1 \mu_1 μ 1 ​ is greater than μ 2 \mu_2 μ 2 ​ .

Formally, to perform a t-test, we should additionally assume that the variances of the two populations are equal (this assumption is called the homogeneity of variance ).

There is a version of a t-test that can be applied without the assumption of homogeneity of variance: it is called a Welch's t-test . For your convenience, we describe both versions.

Two-sample t-test if variances are equal

Use this test if you know that the two populations' variances are the same (or very similar).

Two-sample t-test formula (with equal variances) :

where s p s_p s p ​ is the so-called pooled standard deviation , which we compute as:

• Δ \Delta Δ — Mean difference postulated in the null hypothesis;
• n 1 n_1 n 1 ​ — First sample size;
• x ˉ 1 \bar{x}_1 x ˉ 1 ​ — Mean for the first sample;
• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• n 2 n_2 n 2 ​ — Second sample size;
• x ˉ 2 \bar{x}_2 x ˉ 2 ​ — Mean for the second sample; and
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

Number of degrees of freedom in t-test (two samples, equal variances) = n 1 + n 2 − 2 n_1 + n_2 - 2 n 1 ​ + n 2 ​ − 2 .

Two-sample t-test if variances are unequal (Welch's t-test)

Use this test if the variances of your populations are different.

Two-sample Welch's t-test formula if variances are unequal:

• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

The number of degrees of freedom in a Welch's t-test (two-sample t-test with unequal variances) is very difficult to count. We can approximate it with the help of the following Satterthwaite formula :

Alternatively, you can take the smaller of n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 as a conservative estimate for the number of degrees of freedom.

🔎 The Satterthwaite formula for the degrees of freedom can be rewritten as a scaled weighted harmonic mean of the degrees of freedom of the respective samples: n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 , and the weights are proportional to the standard deviations of the corresponding samples.

As we commonly perform a paired t-test when we have data about the same subjects measured twice (before and after some treatment), let us adopt the convention of referring to the samples as the pre-group and post-group.

The null hypothesis is that the true difference between the means of pre- and post-populations is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the actual difference between these means is:

Typically, this pre-determined difference is zero. We can then reformulate the hypotheses as follows:

The null hypothesis is that the pre- and post-means are the same, i.e., the treatment has no impact on the population .

The alternative hypothesis:

• The pre- and post-means are different from one another (treatment has some effect);
• The pre-mean is smaller than the post-mean (treatment increases the result); or
• The pre-mean is greater than the post-mean (treatment decreases the result).

Paired t-test formula

In fact, a paired t-test is technically the same as a one-sample t-test! Let us see why it is so. Let x 1 , . . . , x n x_1, ... , x_n x 1 ​ , ... , x n ​ be the pre observations and y 1 , . . . , y n y_1, ... , y_n y 1 ​ , ... , y n ​ the respective post observations. That is, x i , y i x_i, y_i x i ​ , y i ​ are the before and after measurements of the i -th subject.

For each subject, compute the difference, d i : = x i − y i d_i := x_i - y_i d i ​ := x i ​ − y i ​ . All that happens next is just a one-sample t-test performed on the sample of differences d 1 , . . . , d n d_1, ... , d_n d 1 ​ , ... , d n ​ . Take a look at the formula for the T-score :

Δ \Delta Δ — Mean difference postulated in the null hypothesis;

n n n — Size of the sample of differences, i.e., the number of pairs;

x ˉ \bar{x} x ˉ — Mean of the sample of differences; and

s s s  — Standard deviation of the sample of differences.

Number of degrees of freedom in t-test (paired): n − 1 n - 1 n − 1

t-test vs Z-test

We use a Z-test when we want to test the population mean of a normally distributed dataset, which has a known population variance . If the number of degrees of freedom is large, then the t-Student distribution is very close to N(0,1).

Hence, if there are many data points (at least 30), you may swap a t-test for a Z-test, and the results will be almost identical. However, for small samples with unknown variance, remember to use the t-test because, in such cases, the t-Student distribution differs significantly from the N(0,1)!

🙋 Have you concluded you need to perform the z-test? Head straight to our z-test calculator !

What is a t-test?

A t-test is a widely used statistical test that analyzes the means of one or two groups of data. For instance, a t-test is performed on medical data to determine whether a new drug really helps.

What are different types of t-tests?

Different types of t-tests are:

• One-sample t-test;
• Two-sample t-test; and
• Paired t-test.

How to find the t value in a one sample t-test?

To find the t-value:

• Subtract the null hypothesis mean from the sample mean value.
• Divide the difference by the standard deviation of the sample.
• Multiply the resultant with the square root of the sample size.

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t-test for the population mean, μ, based on one independent sample . Null hypothesis H 0 : μ = μ 0  

Alternative hypothesis H 1

Test details

Significance level α

The probability that we reject a true H 0 (type I error).

Degrees of freedom

Calculated as sample size minus one.

Test results

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S.3.2 hypothesis testing (p-value approach).

The P -value approach involves determining "likely" or "unlikely" by determining the probability — assuming the null hypothesis was true — of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed. If the P -value is small, say less than (or equal to) \(\alpha\), then it is "unlikely." And, if the P -value is large, say more than \(\alpha\), then it is "likely."

If the P -value is less than (or equal to) \(\alpha\), then the null hypothesis is rejected in favor of the alternative hypothesis. And, if the P -value is greater than \(\alpha\), then the null hypothesis is not rejected.

Specifically, the four steps involved in using the P -value approach to conducting any hypothesis test are:

• Specify the null and alternative hypotheses.
• Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. Again, to conduct the hypothesis test for the population mean μ , we use the t -statistic \(t^*=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) which follows a t -distribution with n - 1 degrees of freedom.
• Using the known distribution of the test statistic, calculate the P -value : "If the null hypothesis is true, what is the probability that we'd observe a more extreme test statistic in the direction of the alternative hypothesis than we did?" (Note how this question is equivalent to the question answered in criminal trials: "If the defendant is innocent, what is the chance that we'd observe such extreme criminal evidence?")
• Set the significance level, \(\alpha\), the probability of making a Type I error to be small — 0.01, 0.05, or 0.10. Compare the P -value to \(\alpha\). If the P -value is less than (or equal to) \(\alpha\), reject the null hypothesis in favor of the alternative hypothesis. If the P -value is greater than \(\alpha\), do not reject the null hypothesis.

Example S.3.2.1

Mean gpa section  .

In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t * equaling 2.5. Since n = 15, our test statistic t * has n - 1 = 14 degrees of freedom. Also, suppose we set our significance level α at 0.05 so that we have only a 5% chance of making a Type I error.

Right Tailed

The P -value for conducting the right-tailed test H 0 : μ = 3 versus H A : μ > 3 is the probability that we would observe a test statistic greater than t * = 2.5 if the population mean \(\mu\) really were 3. Recall that probability equals the area under the probability curve. The P -value is therefore the area under a t n - 1 = t 14 curve and to the right of the test statistic t * = 2.5. It can be shown using statistical software that the P -value is 0.0127. The graph depicts this visually.

The P -value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0127, is less than \(\alpha\) = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ > 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ > 3 if we lowered our willingness to make a Type I error to \(\alpha\) = 0.01 instead, as the P -value, 0.0127, is then greater than \(\alpha\) = 0.01.

Left Tailed

In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t * instead of equaling -2.5. The P -value for conducting the left-tailed test H 0 : μ = 3 versus H A : μ < 3 is the probability that we would observe a test statistic less than t * = -2.5 if the population mean μ really were 3. The P -value is therefore the area under a t n - 1 = t 14 curve and to the left of the test statistic t* = -2.5. It can be shown using statistical software that the P -value is 0.0127. The graph depicts this visually.

The P -value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0127, is less than α = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ < 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ < 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the P -value, 0.0127, is then greater than \(\alpha\) = 0.01.

In our example concerning the mean grade point average, suppose again that our random sample of n = 15 students majoring in mathematics yields a test statistic t * instead of equaling -2.5. The P -value for conducting the two-tailed test H 0 : μ = 3 versus H A : μ ≠ 3 is the probability that we would observe a test statistic less than -2.5 or greater than 2.5 if the population mean μ really was 3. That is, the two-tailed test requires taking into account the possibility that the test statistic could fall into either tail (hence the name "two-tailed" test). The P -value is, therefore, the area under a t n - 1 = t 14 curve to the left of -2.5 and to the right of 2.5. It can be shown using statistical software that the P -value is 0.0127 + 0.0127, or 0.0254. The graph depicts this visually.

Note that the P -value for a two-tailed test is always two times the P -value for either of the one-tailed tests. The P -value, 0.0254, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0254, is less than α = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ ≠ 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ ≠ 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the P -value, 0.0254, is then greater than \(\alpha\) = 0.01.

Now that we have reviewed the critical value and P -value approach procedures for each of the three possible hypotheses, let's look at three new examples — one of a right-tailed test, one of a left-tailed test, and one of a two-tailed test.

The good news is that, whenever possible, we will take advantage of the test statistics and P -values reported in statistical software, such as Minitab, to conduct our hypothesis tests in this course.

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.