two sample hypothesis test calculator

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Two Sample t-test Calculator

t = -1.608761

p-value (one-tailed) = 0.060963

p-value (two-tailed) = 0.121926

Hey there. My name is Zach Bobbitt. I have a Master of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

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t-test Calculator

When to use a t-test, which t-test, how to do a t-test, p-value from t-test, t-test critical values, how to use our t-test calculator, one-sample t-test, two-sample t-test, paired t-test, t-test vs z-test.

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.

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.

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?

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 .

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.

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! 🍺🍺🍺

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!

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.

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!

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

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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Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

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.

Two Sample T Test Calculator

Statscalculator.com.

Free statistics calculators designed for data scientists. This Two Sample t test Calculator:

• Compares the Mean of Two Data Samples
• Assesses if Difference is Significant
• Save & Recycle Data Between Projects

Using The Two Sample t test Calculator

For the details about designing your test, read the guidance below. To use the calculator, enter the data from your sample as a string of numbers, separated by commas. Adjust the calculator's settings (significance level, one or two tailed test) to match the test goals. Hit calculate. It will compute the t-statistic, p-value, and evaluate if we should accept or reject the proposed hypothesis.

For easy entry, you can copy and paste your data into the entry box from Excel. You can save your data for use with this calculator and other calculators on this site. Just hit the "save data" button. It will save the data in your browser (not on our server, it remains private). Saved data sets will appear on the list of saved datasets below the data entry panel. To retrieve it, click the "load data" button next to it.

Two Sample t test Calculator

• Statistics and Histogram Graph
• Save data sets in your browser
• Easily Share results via email

Histogram of Samples

Test results, significance level, one or two tailed, observations.

Click To Clear; enter values seperated by commas or new lines. Cut & Paste from Excel also works.

Can be comma separated or one line per data point; you can also cut and paste from Excel.

Saved Data - Click to Restore

Saved in your browser; you can retrieve these and use them in other calculators on this site.

Sharing Results of The Two Sample t-test Calculator

Need to pass an answer to a friend? It's easy to link and share the results of this calculator. Hit calculate - then hit the share button (below). Simply cut and paste the url after hitting calculate - it will retain the values you enter so you can share them via email or social media. Share

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Interpreting Two Sample t test Results

This calculator is designed to evaluate statements comparing the mean of two separate samples. We refer to this statement as the null hypothesis, a claim we would accept in the absence of other evidence. This occurs by accepting the alternate hypothesis, which should be a mutually exclusive claim. For example, in quality control, we may test the hypothesis that two finished items came from the same batch of raw materials, by checking a property like weight or color.

One of the parameters in the calculator asks you to select if you want to run a one sided or two-sided test. A one sided test can be used to test if the sample mean is significantly below the expected mean for the population. The example above was a one-sample test. A two sided test looks for any significant deviation (up or down) relative to the null hypothesis. The two sided test is best when screening for differences, the one side test is useful if checking for a particular defect.

Two Sample T Test Calculator: How to Easily Compare Means

A two-sample t-test calculator is an essential tool for anyone who wants to determine if there is a significant difference between the means of two independent groups. This type of test is commonly used in scientific research, clinical trials, and quality control. The calculator works by comparing the means of the two groups and calculating the probability that the difference between them occurred by chance.

Using a two-sample t-test calculator is relatively straightforward. First, the user inputs the data for the two groups they want to compare. The calculator then computes the t-value and the corresponding p-value. If the p-value is less than the significance level (usually 0.05), the user can conclude that there is a significant difference between the means of the two groups. On the other hand, if the p-value is greater than the significance level, the user cannot reject the null hypothesis, and there is no evidence of a difference between the two groups.

Overall, a two-sample t-test calculator is a powerful tool that can help researchers and analysts make informed decisions based on data. By providing a quick and easy way to perform a two-sample t-test, these calculators can save time and increase accuracy, making them an essential resource for anyone working with independent groups.

Overview of the Two-Sample T-Test

The two-sample t-test is a statistical test used to determine whether two populations have different means. This test is particularly useful when comparing the means of two independent groups. The two-sample t-test can be used to determine whether there is a significant difference between the means of two groups, such as the test scores of two different classes or the heights of two different populations.

The two-sample t-test is a hypothesis test that compares the means of two independent samples. The null hypothesis is that there is no significant difference between the means of the two populations, while the alternative hypothesis is that there is a significant difference.

Assumptions

The two-sample t-test assumes that the populations are normally distributed and have equal variances. It also assumes that the samples are independent and that the sample sizes are large enough to satisfy the central limit theorem. If these assumptions are not met, the results of the test may not be reliable.

The formula for the two-sample t-test is:

t = (x1 - x2) / (s1^2/n1 + s2^2/n2)^0.5

where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, n1 and n2 are the sample sizes, and t is the test statistic.

Calculation

To perform a two-sample t-test, the following steps should be followed:

• State the null and alternative hypotheses.
• Determine the significance level.
• Collect the data and calculate the sample means and standard deviations.
• Calculate the test statistic using the formula above.
• Determine the degrees of freedom and find the critical values.
• Calculate the p-value.
• Make a decision based on the p-value and the significance level.

The two-sample t-test is a statistical test used to compare the means of two independent samples. It is based on the t-distribution and requires that the populations be normally distributed with equal variances. The test can be used to determine whether there is a significant difference between the means of two groups, and it can be performed using a variety of statistical software or online calculators.

More About How to Use a Two-Sample T-Test Calculator

Step-by-step guide.

When conducting a two-sample t-test, it is important to have a reliable calculator to help you analyze your data. Here is a step-by-step guide on how to use a two-sample t-test calculator:

Input your data: Enter the data for both samples into the calculator. This can be done by typing in the values or by uploading a file containing the data.

Select the type of test: Choose whether you want to perform a one-tailed or two-tailed test, depending on your research question.

Set the significance level: Determine the level of significance for your test, typically set at 0.05.

Calculate the results: Click the "calculate" button to obtain the p-value, confidence interval, and other relevant statistics.

Interpreting Results

After calculating the results of your two-sample t-test, it is important to understand how to interpret them. Here are some key points to keep in mind:

P-value: The p-value represents the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true. If the p-value is less than the significance level, then the null hypothesis can be rejected.

Confidence interval: The confidence interval gives a range of values that the true population mean is likely to fall within. A 95% confidence interval means that if the experiment were repeated many times, 95% of the intervals would contain the true population mean.

Average height: In the context of a two-sample t-test, average height refers to the mean height of the two samples being compared.

Men: If the two samples being compared are from different populations (e.g. men vs. women), then the two-sample t-test can be used to determine if there is a significant difference in the means of the two populations.

Zero: The null hypothesis in a two-sample t-test states that there is no significant difference between the means of the two populations being compared. If the p-value is less than the significance level, then the null hypothesis can be rejected and it can be concluded that there is a significant difference between the means.

Advantages and Limitations of the Two-Sample T-Test

The two-sample t-test is a commonly used statistical method that allows researchers to compare the means of two independent groups. One of the main advantages of using the two-sample t-test is that it is a simple and easy-to-use method that can be applied to a wide range of research questions. It is also a powerful tool that can detect differences between groups even when the sample sizes are small.

Another advantage of the two-sample t-test is that it is a parametric test, which means that it assumes that the data is normally distributed. This assumption allows researchers to make more accurate inferences about the population based on the sample data.

Limitations

Despite its many advantages, the two-sample t-test also has some limitations that researchers should be aware of when using this method. One of the main limitations is that it assumes that the variances of the two groups are equal. If the variances are not equal, the results of the t-test may not be reliable.

Another limitation of the two-sample t-test is that it is sensitive to outliers. Outliers are extreme values that are far from the rest of the data. If the data contains outliers, the results of the t-test may be skewed and may not accurately reflect the population.

Finally, the two-sample t-test is a hypothesis testing method, which means that it can only tell researchers whether there is a statistically significant difference between the two groups. It cannot tell researchers anything about the magnitude or practical significance of the difference.

Overall, the two-sample t-test is a useful statistical method that can provide valuable insights into the differences between two independent groups. However, researchers should be aware of its limitations and use it appropriately to ensure that their results are accurate and reliable.

Two Sample T-Test Calculator

Two sample t-test is used to check whether the means of two groups are significantly different from each other. For example, if you want to see if mean weight of males and females have statistically significant difference between them.

Independent T-test assumes that the two samples have equal variances. Welch's t-test is used if you have unequal variances.

Either enter raw data or summary information to calculate two sample t-test. You can directly paste data from MS Excel.

Enter Raw Data

Enter Summary Data

• Scores are normally distributed within each of the two groups
• Each score is sampled independently and randomly.
• Data must be continuous

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T test calculator

A t test compares the means of two groups. There are several types of two sample t tests and this calculator focuses on the three most common: unpaired, welch's, and paired t tests. Directions for using the calculator are listed below, along with more information about two sample t tests and help on which is appropriate for your analysis. NOTE: This is not the same as a one sample t test; for that, you need this One sample t test calculator .

1. Choose data entry format

Caution: Changing format will erase your data.

2. Choose a test

Help me choose

3. Enter data

Help me arrange the data

4. View the results

What is a t test.

A t test is used to measure the difference between exactly two means. Its focus is on the same numeric data variable rather than counts or correlations between multiple variables. If you are taking the average of a sample of measurements, t tests are the most commonly used method to evaluate that data. It is particularly useful for small samples of less than 30 observations. For example, you might compare whether systolic blood pressure differs between a control and treated group, between men and women, or any other two groups.

This calculator uses a two-sample t test, which compares two datasets to see if their means are statistically different. That is different from a one sample t test , which compares the mean of your sample to some proposed theoretical value.

The most general formula for a t test is composed of two means (M1 and M2) and the overall standard error (SE) of the two samples:

See our video on How to Perform a Two-sample t test for an intuitive explanation of t tests and an example.

How to use the t test calculator

• Choose your data entry format . This will change how section 3 on the page looks. The first two options are for entering your data points themselves, either manually or by copy & paste. The last two are for entering the means for each group, along with the number of observations (N) and either the standard error of that mean (SEM) or standard deviation of the dataset (SD) standard error. If you have already calculated these summary statistics, the latter options will save you time.
• Choose a test from the three options: Unpaired t test, Welch's unpaired t test, or Paired t test. Use our Ultimate Guide to t tests if you are unsure which is appropriate, as it includes a section on "How do I know which t test to use?". Notice not all options are available if you enter means only.
• Enter data for the test, based on the format you chose in Step 1.
• Click Calculate Now and View the results. All options will perform a two-tailed test .

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Common t test confusion

In addition to the number of t test options, t tests are often confused with completely different techniques as well. Here's how to keep them all straight.

Correlation and regression are used to measure how much two factors move together. While t tests are part of regression analysis, they are focused on only one factor by comparing means in different samples.

ANOVA is used for comparing means across three or more total groups. In contrast, t tests compare means between exactly two groups.

Finally, contingency tables compare counts of observations within groups rather than a calculated average. Since t tests compare means of continuous variable between groups, contingency tables use methods such as chi square instead of t tests.

Assumptions of t tests

Because there are several versions of t tests, it's important to check the assumptions to figure out which is best suited for your project. Here are our analysis checklists for unpaired t tests and paired t tests , which are the two most common. These (and the ultimate guide to t tests ) go into detail on the basic assumptions underlying any t test:

• Exactly two groups
• Sample is normally distributed
• Independent observations
• Unequal or equal variance?
• Paired or unpaired data?

Interpreting results

The three different options for t tests have slightly different interpretations, but they all hinge on hypothesis testing and P values. You need to select a significance threshold for your P value (often 0.05) before doing the test.

While P values can be easy to misinterpret , they are the most commonly used method to evaluate whether there is evidence of a difference between the sample of data collected and the null hypothesis. Once you have run the correct t test, look at the resulting P value. If the test result is less than your threshold, you have enough evidence to conclude that the data are significantly different.

If the test result is larger or equal to your threshold, you cannot conclude that there is a difference. However, you cannot conclude that there was definitively no difference either. It's possible that a dataset with more observations would have resulted in a different conclusion.

Depending on the test you run, you may see other statistics that were used to calculate the P value, including the mean difference, t statistic, degrees of freedom, and standard error. The confidence interval and a review of your dataset is given as well on the results page.

Graphing t tests

This calculator does not provide a chart or graph of t tests, however, graphing is an important part of analysis because it can help explain the results of the t test and highlight any potential outliers. See our Prism guide for some graphing tips for both unpaired and paired t tests.

Prism is built for customized, publication quality graphics and charts. For t tests we recommend simply plotting the datapoints themselves and the mean, or an estimation plot . Another popular approach is to use a violin plot, like those available in Prism.

For more information

Our ultimate guide to t tests includes examples, links, and intuitive explanations on the subject. It is quite simply the best place to start if you're looking for more about t tests!

If you enjoyed this calculator, you will love using Prism for analysis. Take a free 30-day trial to do more with your data, such as:

• Clear guidance to pick the right t test and detailed results summaries
• Custom, publication quality t test graphics, violin plots, and more
• More t test options, including normality testing as well as nested and multiple t tests
• Non-parametric test alternatives such as Wilcoxon, Mann-Whitney, and Kolmogorov-Smirnov

Check out our video on how to perform a t test in Prism , for an example from start to finish!

Remember, this page is just for two sample t tests. If you only have one sample, you need to use this calculator instead.

We Recommend:

Analyze, graph and present your scientific work easily with GraphPad Prism. No coding required.

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Two-Sample t-test

Use this calculator to test whether samples from two independent populations provide evidence that the populations have different means. For example, based on blood pressures measurements taken from a sample of women and a sample of men, can we conclude that women and men have different mean blood pressures?

This test is known as an a two sample (or unpaired) t-test. It produces a “p-value”, which can be used to decide whether there is evidence of a difference between the two population means.

The p-value is the probability that the difference between the sample means is at least as large as what has been observed, under the assumption that the population means are equal. The smaller the p-value, the more surprised we would be by the observed difference in sample means if there really was no difference between the population means. Therefore, the smaller the p-value, the stronger the evidence is that the two populations have different means.

Typically a threshold (known as the significance level) is chosen, and a p-value less than the threshold is interpreted as indicating evidence of a difference between the population means. The most common choice of significance level is 0.05, but other values, such as 0.1 or 0.01 are also used.

This calculator should be used when the sampling units (e.g. the sampled individuals) in the two groups are independent. If you are comparing two measurements taken on the same sampling unit (e.g. blood pressure of an individual before and after a drug is administered) then the appropriate test is the paired t-test.

More Information

Worked example.

A study compares the average capillary density in the feet of individuals with and without ulcers. A sample of 10 patients with ulcers has mean capillary density of 29, with standard deviation 7.5. A control sample of 10 individuals without ulcers has mean capillary density of 34, with standard deviation 8.0. (All measurements are in capillaries per square mm.) Using this information, the p-value is calculated as 0.167. Since this p-value is greater than 0.05, it would conventionally be interpreted as meaning that the data do not provide strong evidence of a difference in capillary density between individuals with and without ulcers.

If both sample sizes were increased to 20, the p-value would reduce to 0.048 (assuming the sample means and standard deviations remained the same), which we would interpret as strong evidence of a difference. Note that this result is not inconsistent with the previous result: with bigger samples we are able to detect smaller differences between populations.

Assumptions

This test assumes that the two populations follow normal distributions (otherwise known as Gaussian distributions). Normality of the distributions can be tested using, for example, a Q-Q plot . An alternative test that can be used if you suspect that the data are drawn from non-normal distributions is the Mann-Whitney U test .

The version of the test used here also assumes that the two populations have different variances. If you think the populations have the same variance, an alternative version of the two sample t-test (two sample t-test with a pooled variance estimator) can be used. The advantage of the alternative version is that if the populations have the same variance then it has greater statistical power – that is, there is a higher probability of detecting a difference between the population means if such a difference exists.

Performing this test assesses the extent to which the difference between the sample means provides evidence of a difference between the population means. The test puts forward a “null” hypothesis that the population means are equal, and measures the probability of observing a difference at least as big as that seen in the data under the null hypothesis (the p-value). If the p-value is large then the observed difference between the sample means is unsurprising and is interpreted as being consistent with hypothesis of equal population means. If on the other hand the p-value is small then we would be surprised about the observed difference if the null hypothesis really was true. Therefore, a small p-value is interpreted as evidence that the null hypothesis is false and that there really is a difference between the population means. Typically a threshold (known as the significance level) is chosen, and a p-value less than the threshold is interpreted as indicating evidence of a difference between the population means. The most common choice of significance level is 0.05, but other values, such as 0.1 or 0.01 are also used.

Note that a large p-value (say, larger than 0.05) cannot in itself be interpreted as evidence that the populations have equal means. It may just mean that the sample size is not large enough to detect a difference. To find out how large your sample needs to be in order to detect a difference (if a difference exists), see our sample size calculator .

If evidence of a difference in the population means is found, you may wish to quantify that difference. The difference between the sample means is a point estimate of the difference between the population means, but it can be useful to assess how reliable this estimate is using a confidence interval . A confidence interval provides you with a set of limits in which you expect the difference between the population means to lie. The p-value and the confidence interval are related and have a consistent interpretation: if the p-value is less than α then a (1-α)*100% confidence interval will not contain zero. For example, if the p-value is less than 0.05 then a 95% confidence interval will not contain zero.

If you wish to calculate a confidence interval, our confidence interval calculator will do the work for you.

Definitions

Sample mean.

The sample mean is your ‘best guess’ for what the true population mean is given your sample of data and is calcuated as:

μ = (1/n)* ∑ n i=1 x i ,

where n is the sample size and x 1 ,…,x n are the n sample observations.

Sample standard deviation

The sample standard deviation is calcuated as s=√ σ 2 , where:

σ 2 = (1/(n-1))* ∑ n i=1 (x i -μ) 2 ,

μ is the sample mean, n is the sample size and x 1 ,…,x n are the n sample observations.

Sample size

This is the total number of samples randomly drawn from you population.  The larger the sample size, the more certain you can be that the estimate reflects the population.  Choosing a sample size is an important aspect when desiging your study or survey.  For some further information, see our blog post on The Importance and Effect of Sample Size and for guidance on how to choose your sample size, see our sample size calculator .

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Two sample t-Test

The calculator to perform t-Test for the Significance of the Difference between the Means of Two Independent Samples

The calculator below implements the most known statistical test, namely, the Independent Samples t-test or Two samples t-test. t-test, also known as Student's t-test, after William Sealy Gosset. "Student" was his pen name.

The test deals with the null hypothesis such that the means of two populations are equal. To put it in other words, the difference we find between the means of the two samples should not significantly differ from zero.

Again, the test works only if certain assumptions are met. These are:

• That the two samples are independently and randomly drawn from the source population(s).
• That the scale of measurement for both samples has the properties of an equal-interval scale.
• That the source population(s) can be reasonably supposed to have a normal distribution.
• And, for this particular implementation of the test, that the variance of each population is the same

The calculator displays a level of confidence for both directional and non-directional tests. Let's say you get the result of 96%. Essentially this means that you have 96% confidence that the obtained difference shows something more than simple luck. The chance that you can get the obtained difference and the means of the two samples are the same is only 4%. This is the level of significance you calculate. Now, depending on your chosen level of significance, you can reject or fail to reject your null hypothesis.

If you care to find more, you can read excellent explanations here , starting from Chapter 9.

Two samples t-Test

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Two sample t test

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Two Sample T Test Calculator

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How To Perform a Two Sample T Test

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T-Test calculator

The Student's t-test is used to determine if means of two data sets differ significantly. This calculator will generate a step by step explanation on how to apply t - test.

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Twelve younger adults and twelve older adults conducted a life satisfaction test. The data are presented in the table below. Compute the appropriate t-test.

Are the means between two data sets are significantly different at level $\alpha < 0.05$.

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Two Sample t-test Calculator

t = -1.608761

p-value (one-tailed) = 0.060963

p-value (two-tailed) = 0.121926

An Introduction to the Binomial Distribution

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Difference in Means Hypothesis Test Calculator

Use the calculator below to analyze the results of a difference in sample means hypothesis test. Enter your sample means, sample standard deviations, sample sizes, hypothesized difference in means, test type, and significance level to calculate your results.

You will find a description of how to conduct a two sample t-test below the calculator.

Define the Two Sample t-test

The difference between the sample means under the null distribution, conducting a hypothesis test for the difference in means.

When two populations are related, you can compare them by analyzing the difference between their means.

A hypothesis test for the difference in samples means can help you make inferences about the relationships between two population means.

Testing for a Difference in Means

For the results of a hypothesis test to be valid, you should follow these steps:

Check Your Conditions

State your hypothesis, determine your analysis plan, analyze your sample, interpret your results.

To use the testing procedure described below, you should check the following conditions:

• Independence of Samples - Your samples should be collected independently of one another.
• Simple Random Sampling - You should collect your samples with simple random sampling. This type of sampling requires that every occurrence of a value in a population has an equal chance of being selected when taking a sample.
• Normality of Sample Distributions - The sampling distributions for both samples should follow the Normal or a nearly Normal distribution. A sampling distribution will be nearly Normal when the samples are collected independently and when the population distribution is nearly Normal. Generally, the larger the sample size, the more normally distributed the sampling distribution. Additionally, outlier data points can make a distribution less Normal, so if your data contains many outliers, exercise caution when verifying this condition.

You must state a null hypothesis and an alternative hypothesis to conduct an hypothesis test of the difference in means.

The null hypothesis is a skeptical claim that you would like to test.

The alternative hypothesis represents the alternative claim to the null hypothesis.

Your null hypothesis and alternative hypothesis should be stated in one of three mutually exclusive ways listed in the table below.

D is the hypothesized difference between the populations' means that you would like to test.

Before conducting a hypothesis test, you must determine a reasonable significance level, α, or the probability of rejecting the null hypothesis assuming it is true. The lower your significance level, the more confident you can be of the conclusion of your hypothesis test. Common significance levels are 10%, 5%, and 1%.

To evaluate your hypothesis test at the significance level that you set, consider if you are conducting a one or two tail test:

• Two-tail tests divide the rejection region, or critical region, evenly above and below the null distribution, i.e. to the tails of the null sampling distribution. For example, in a two-tail test with a 5% significance level, your rejection region would be the upper and lower 2.5% of the null distribution. An alternative hypothesis of μ 1 - μ 2 ≠ D requires a two tail test.
• One-tail tests place the rejection region entirely on one side of the distribution i.e. to the right or left tail of the null distribution. For example, in a one-tail test evaluating if the actual difference in means, D, is above the null distribution with a 5% significance level, your rejection region would be the upper 5% of the null distribution. μ 1 - μ 2 > D and μ 1 - μ 2 < D alternative hypotheses require one-tail tests.

The graphical results section of the calculator above shades rejection regions blue.

After checking your conditions, stating your hypothesis, determining your significance level, and collecting your sample, you are ready to analyze your hypothesis.

Sample means follow the Normal distribution with the following parameters:

• The Difference in the Population Means, D - The true difference in the population means is unknown, but we use the hypothesized difference in the means, D, from the null hypothesis in the calculations.
• The Standard Error, SE - The standard error of the difference in the sample means can be computed as follows:      SE = (s 1 2 /n 1 + s 2 2 /n 2 ) (1/2) with s 1 being the standard deviation of sample one, n 1 being the sample size of sample one, s 2 being the standard deviation of sample one, and n 2 being the sample size of sample two. The standard error defines how differences in sample means are expected to vary around the null difference in means sampling distribution given the sample sizes and under the assumption that the null hypothesis is true.
• The Degrees of Freedom, DF - The degrees of freedom calculation can be estimated as the smaller of n 1 - 1 or n 2 - 1. For more accurate results, use the following formula for the degrees of freedom (DF):      DF = (s 1 2 /n 1 + s 2 2 /n 2 ) 2 / ((s 1 2 /n 1 ) 2 / (n 1 - 1) + (s 2 2 /n 2 ) 2 / (n 2 - 1))

In a difference in means hypothesis test, we calculate the probability that we would observe the difference in sample means (x̄ 1 - x̄ 2 ), assuming the null hypothesis is true, also known as the p-value . If the p-value is less than the significance level, then we can reject the null hypothesis.

You can determine a precise p-value using the calculator above, but we can find an estimate of the p-value manually by calculating the t-score, or t-statistic, as follows: t = (x̄ 1 - x̄ 2 - D) / SE

The t-score is a test statistic that tells you how far our observation is from the null hypothesis's difference in means under the null distribution. Using any t-score table, you can look up the probability of observing the results under the null distribution. You will need to look up the t-score for the type of test you are conducting, i.e. one or two tail. A hypothesis test for the difference in means is sometimes known as a two sample mean t-test because of the use of a t-score in analyzing results.

The conclusion of a hypothesis test for the difference in means is always either:

• Reject the null hypothesis
• Do not reject the null hypothesis

If you reject the null hypothesis, you cannot say that your sample difference in means is the true difference between the means. If you do not reject the null hypothesis, you cannot say that the hypothesized difference in means is true.

A hypothesis test is simply a way to look at evidence and conclude if it provides sufficient evidence to reject the null hypothesis.

Example: Hypothesis Test for the Difference in Two Means

Let’s say you are a manager at a company that designs batteries for smartphones. One of your engineers believes that she has developed a battery that will last more than two hours longer than your standard battery.

Before you can consider if you should replace your standard battery with the new one, you need to test the engineer’s claim. So, you decided to run a difference in means hypothesis test to see if her claim that the new battery will last two hours longer than the standard one is reasonable.

You direct your team to run a study. They will take a sample of 100 of the new batteries and compare their performance to 1,000 of the old standard batteries.

• Check the conditions - Your test consists of independent samples . Your team collects your samples using simple random sampling , and you have reason to believe that all your batteries' performances are always close to normally distributed . So, the conditions are met to conduct a two sample t-test.
• State Your Hypothesis - Your null hypothesis is that the charge of the new battery lasts at most two hours longer than your standard battery (i.e. μ 1 - μ 2 ≤ 2). Your alternative hypothesis is that the new battery lasts more than two hours longer than the standard battery (i.e. μ 1 - μ 2 > 2).
• Determine Your Analysis Plan - You believe that a 1% significance level is reasonable. As your test is a one-tail test, you will evaluate if the difference in mean charge between the samples would occur at the upper 1% of the null distribution.
• Analyze Your Sample - After collecting your samples (which you do after steps 1-3), you find the new battery sample had a mean charge of 10.4 hours, x̄ 1 , with a 0.8 hour standard deviation, s 1 . Your standard battery sample had a mean charge of 8.2 hours, x̄ 2 , with a standard deviation of 0.2 hours, s 2 . Using the calculator above, you find that a difference in sample means of 2.2 hours [2 = 10.4 – 8.2] would results in a t-score of 2.49 under the null distribution, which translates to a p-value of 0.72%.
• Interpret Your Results - Since your p-value of 0.72% is less than the significance level of 1%, you have sufficient evidence to reject the null hypothesis.

In this example, you found that you can reject your null hypothesis that the new battery design does not result in more than 2 hours of extra battery life. The test does not guarantee that your engineer’s new battery lasts two hours longer than your standard battery, but it does give you strong reason to believe her claim.

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31: Two Independent Samples With Statistics and Known Population Standard Deviations Hypothesis Test and Confidence Interval Calculator

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Two Independent Samples with Statistics, Population Standard Deviation Known Calculator

Enter in the statistics, the tail type and the confidence level and hit Calculate and the test statistic, z, the p-value, p, the confidence interval's lower bound, LB, and the upper bound, UB will be shown.  Be sure to enter the confidence level as a decimal, e.g., 95% has a CL of 0.95.

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T Test Calculator for 2 Dependent Means

The t -test for dependent means (also called a repeated-measures t -test, paired samples t -test, matched pairs t -test and matched samples t -test) is used to compare the means of two sets of scores that are directly related to each other. So, for example, it could be used to test whether subjects' galvanic skin responses are different under two conditions - first, on exposure to a photograph of a beach scene; second, on exposure to a photograph of a spider.

Requirements

• The data is normally distributed
• Scale of measurement should be interval or ratio
• The two sets of scores are paired or matched in some way

Null Hypothesis

H 0 : U D = U 1 - U 2 = 0, where U D equals the mean of the population of difference scores across the two measurements.

Two Sample T-Test Calculator (Welch's T-test)

Enter sample data, information, assumptions, required sample data.

T-test for two Means – Unknown Population Standard Deviations

Instructions : Use this T-Test Calculator for two Independent Means calculator to conduct a t-test for two population means (\(\mu_1\) and \(\mu_2\)), with unknown population standard deviations. This test apply when you have two-independent samples, and the population standard deviations \(\sigma_1\) and \(\sigma_2\) and not known. Please select the null and alternative hypotheses, type the significance level, the sample means, the sample standard deviations, the sample sizes, and the results of the t-test for two independent samples will be displayed for you:

The T-test for Two Independent Samples

More about the t-test for two means so you can better interpret the output presented above: A t-test for two means with unknown population variances and two independent samples is a hypothesis test that attempts to make a claim about the population means (\(\mu_1\) and \(\mu_2\)).

More specifically, a t-test uses sample information to assess how plausible it is for the population means \(\mu_1\) and \(\mu_2\) to be equal. The test has two non-overlapping hypotheses, the null and the alternative hypothesis.

The null hypothesis is a statement about the population means, specifically the assumption of no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis.

Properties of the two sample t-test

The main properties of a two sample t-test for two population means are:

• Depending on our knowledge about the "no effect" situation, the t-test can be two-tailed, left-tailed or right-tailed
• The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true
• The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true
• In a hypothesis tests there are two types of errors. Type I error occurs when we reject a true null hypothesis, and the Type II error occurs when we fail to reject a false null hypothesis

How do you compute the t-statistic for the t test for two independent samples?

The formula for a t-statistic for two population means (with two independent samples), with unknown population variances shows us how to calculate t-test with mean and standard deviation and it depends on whether the population variances are assumed to be equal or not. If the population variances are assumed to be unequal, then the formula is:

On the other hand, if the population variances are assumed to be equal, then the formula is:

Normally, the way of knowing whether the population variances must be assumed to be equal or unequal is by using an F-test for equality of variances.

With the above t-statistic, we can compute the corresponding p-value, which allows us to assess whether or not there is a statistically significant difference between two means.

Why is it called t-test for independent samples?

This is because the samples are not related with each other, in a way that the outcomes from one sample are unrelated from the other sample. If the samples are related (for example, you are comparing the answers of husbands and wives, or identical twins), you should use a t-test for paired samples instead .

What if the population standard deviations are known?

The main purpose of this calculator is for comparing two population mean when sigma is unknown for both populations. In case that the population standard deviations are known, then you should use instead this z-test for two means .

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