two sample mean hypothesis test calculator

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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ol{padding-top:0;}.css-63uqft ul:not(:first-child),.css-63uqft ol:not(:first-child){padding-top:4px;} Test setup

Choose test type

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

Hypothesis Testing 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.

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.

Two sample t test calculator

Welcome to our Two Sample T Test Calculator, the ideal tool for comparing mean values from two independent samples. This calculator calculates test statistics, p-values, critical values, judgments, and conclusions using both equal and unequal variance approaches. This tool is intended to help students, researchers, and data analysts simplify their statistical analyses.

Enter below values for sample 1 :

Enter below values for sample 2:

Related Calculators :

• List of all calculator
• P-value calculator
• Critical value Calculator

What is a Two Sample T Test?

A Two Sample T Test is used to see if there is a significant difference in the means of two independent groups. This test is frequently used in experiments and research to compare two groups and draw conclusions about the population mean.

Features of Our Two Sample T-Test Calculator

• Direct Data Entry: Enter each sample's raw data values directly into the calculator.
• Summary Statistics: If you have summary statistics rather than raw data, please provide the sample size, mean, and standard deviation.
• Hypothesis Testing: Determine whether your null and alternative hypotheses are two-tailed, right-tailed, or left-tailed.
• Significance Level: Enter the significance level (alpha) for the test.
• Variance Type: For more accurate results, select either equal or unequal variance.
• Detailed Results: Receive complete results, including test statistics, p-values, critical values, and decision-making conclusions.

How To Use The Calculator

Select Data Type: Determine whether you have raw data values or summary statistics.

Enter Data: Fill in the data values or summary statistics for both samples.

Hypothesis Selection: Determine the relevant null and alternative hypotheses for your test.

Set the Significance Level: The alpha level is used to set the threshold for statistical significance.

Variance Type: Determine whether the variances of the two samples are equal or unequal.

Calculate: To view the results, simply click the "Calculate" button.

Example Use Cases

Our Two Sample T-Test Calculator can be applied in various fields, including:

• Medical Research: Determine the efficacy of two different drugs.
• Education: Compare test scores from two distinct teaching approaches.
• Marketing: Evaluate the effectiveness of two marketing efforts.

Why Use Our Calculator?

• Accuracy: Our calculator can perform precise estimates for both equal and unequal variance cases.
• Ease of Use: A user-friendly interface with clear directions and inputs.
• Comprehensive Results: Detailed output, including statistical computations and decision-making advice.

Frequently Asked Questions

Q: What is the difference between equal and unequal variances? A: Equal variance assumes that the two populations have equal variance, whereas unequal variance does not make this assumption. Selecting the proper option guarantees accurate results.

Q: How do I determine whether to conduct a two-tailed, right-tailed, or left-tailed test? A: It depends on your research hypothesis. If you want to find a significant difference, conduct a two-tailed test. If you predict the first sample's mean to be greater than the second, perform a right-tailed test. If you predict the first sample's mean to be less than the second, perform a left-tailed test.

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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 .

Performing t tests? We can help.

Sign up for more information on how to perform t tests and other common statistical analyses.

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 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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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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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.

Groups Have Equal Variance (default) Groups Have Unequal Variance (Welch t-test) Two Tailed Test (default) One Tailed Test 0.05 (default) 0.01 0.001 Unpaired T Test (default) Paired (Dependent) T Test Get Widget Code Long Division Evaluate Expressions Fraction Calculator Greatest Common Divisor GCD Least Common Multiple LCM Prime Factorization Scientific Notation Percentage Calculator Dec / Bin / Hex Factoring Polynomials Polynomial Roots Synthetic Division Polynomial Operations Graphing Polynomials Simplify Polynomials Generate From Roots Simplify Expression Multiplication / Division Addition / Subtraction Rationalize Denominator Simplifying Quadratic Equations Solver Polynomial Equations Solving Equations - With Steps Solving (with steps) Quadratic Plotter Factoring Trinomials Equilateral Triangle Right Triangle Oblique Triangle Square Calculator Rectangle Calculator Circle Calculator Hexagon Calculator Rhombus Calculator Trapezoid Calculator Triangular Prism Distance calculator Midpoint Calculator Triangle Calculator Graphing Lines Lines Intersection Two Point Form Line-Point Distance Parallel/Perpendicular Circle Equation Circle From 3 Points Circle-line Intersection Modulus, inverse, polar form Vectors (2D & 3D) Add, Subtract, Multiply Determinant Calculator Matrix Inverse Characteristic Polynomial Eigenvalues Eigenvectors Matrix Decomposition Limit Calculator Derivative Calculator Integral Calculator Arithmetic Sequences Geometric Sequences Find n th Term Degrees to Radians Trig. Equations Probability Calculator Probability Distributions Descriptive Statistics Standard Deviation Z - score Calculator Normal Distribution T-Test Calculator Correlation & Regression Simple Interest Compound Interest Amortization Calculator Annuity Calculator Work Problems Hire MATHPORTAL experts to do math homework for you. Prices start at $3 per problem. 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 $\alpha < 0.05$. Standard deviation calculator – Widget Code Feel free to copy the widget code below and paste it into your website or blog. The widget will look like the example below. Widget preview: Standard deviation calculator Copy code to clipboard Welcome to MathPortal. This website's owner is mathematician Miloš Petrović. I designed this website and wrote all the calculators, lessons, and formulas . If you want to contact me, probably have some questions, write me using the contact form or email me on [email protected] Email (optional) 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 . Related Calculators log in to your account Reset password. Two-Sample $t$ Test Calculator The two-sample $t$ procedures assume that both our samples come from an SRS and will give trustworthy conclusions only if this condition is met. If the sum of the sample size is less than 15, the $t$ procedures yield trustworthy conclusions only if you can reasonably assume that your data from both samples come from a normal distribution, that is, if the distribution appears to be symmetric with one peak and no outliers. If your data are obviously skewed or if there are any outliers, it is not advisable to use the $t$ procedures. Non-parametric methods may be more advisable: try the Wilcoxon Rank Sum Test . If the sum of the sample sizes is 15 or larger, the $t$ procedures can be trusted if there are no outliers and the distribution is not obviously skewed. If the sum of the sample sizes is 40 or larger, you may use $t$ procedures even if your distributions appear to be skewed. The two-sample $t$ procedures are more robust against non-normal data when the sample sizes of both samples are equal. Therefore, when planning a two-sample study, try to choose equal sample sizes. Sample 1 Sample 2 Sample data goes here (enter numbers in columns): Sample Means: $\bar{x}_1=$ $\bar{x}_2=$ Sample Standard Deviations: $s_1=$ $s_2=$ Sample Sizes: $n_1=$ $n_2=$ Null Hypothesis: $H_0: \mu_1=\mu_2$ Alternative Hypothesis: $H_a:\mu_1$ $\mu_2$ Level of Significance: $\alpha=$ Use Summary Statistics: % Confidence Interval: Sample Sizes: $n_1=$ $n_2=$ Sample Means: $\overline{x}_1=$ $\overline{x}_2=$ Sample Standard Deviations: $s_1=$ $s_2=$ Degrees of Freedom: $df=$ Critical $t$ Value: $t^{*}=$ $t$ statistic: $t=$ $p\mbox{-value}$: $p\mbox{-value}=$ T-Test Calculator Compare the means of two samples using a single-sample or two-sample t-test below. Single Sample Two Sample (Unpaired) Have a Question or Feedback? Two-tailed P: Left-tailed P: Right-tailed P: Test Statistic (t): Degrees of Freedom (df): Two-tailed P: Left-tailed P: Right-tailed P: Test Statistic (t): Degrees of Freedom (df): Pooled Standard Deviation: Difference of Means: Standard Error of Difference: Chevron Down Icon scroll down On this page: How to do a t-test, types of t-tests, how to calculate t using a one-sample t-test, how to calculate t using a student’s t-test, how to calculate t using welch’s t-test, find the p-value. Joe is the creator of Inch Calculator and has over 20 years of experience in engineering and construction. He holds several degrees and certifications. Full bio Chevron Right Icon Brian specializes in political science and statistics with several advanced degrees from Harvard. A t-test calculates how significant the difference between the means of two groups are. The results let you know if those differences could have occurred by chance, or rather, whether the difference is statistically significant. A t-test uses the test statistic, sometimes called a t-value or t-score, the t-distribution values, and the degrees of freedom to calculate the statistical significance of the difference. Since a t-test is a parametric test, it relies on assumptions about the process that generated the underlying data. In particular, the likelihood or unlikelihood that the t-test provides for a difference being due to chance depends on the assumption that the data are normally distributed and each data point’s values are independent of one another. Depending on how plausible those assumptions are, the analysis that follows will be more or less useful. If your data is continuous and comes from a relatively large random sample from some population, the central limit theorem implies that these assumptions will likely be approximately satisfied. The first part of doing a t-test is determining which type of t-test you need to do. There are three different types of t-tests: one-sample t-test: used to compare the mean of a sample to the known mean of a population two-sample t-test: used to compare the mean of two different independent samples paired t-test: used to compare the mean of two different samples after an intervention or change A one-sample t-test, or single-sample test, is used to compare a sample mean to a population mean when the null hypothesis is that the sample mean is equal to the population mean. Those who first encounter this test often wonder why they would use it, since the population mean is often not known (and the data is often collected to determine the population mean in the first place). It often does make sense to use a one-sample t-test if you have a particular interest in whether a sample’s mean is different from some reference value that is determined to be substantively important for other reasons. For example, let’s suppose that 5 micrograms of lead per liter of blood is the maximum safe amount, according to most medical references. Then, you may well consider doing a one-sample t-test to examine whether the average blood lead level of a sample of individuals was above that medically acceptable limit. One-Sample T-Test Formula To calculate the t value using a one-sample t-test, use the following formula: Where: x̄ = sample mean μ = population mean s = sample standard deviation n = sample size Thus, the test statistic t is equal to the difference between the sample mean x̄ and the population mean μ , divided by the standard error s / √n . A Student’s t-test is used for test statistics that follow a Student’s t-distribution under the null hypothesis that two populations have equal means. The name “Student” refers to the pseudonym of the author who first proposed the test in an academic journal, and does not refer to the fact it is one of the most commonly taught tests in statistics courses (although the latter is also true). The Student’s t-test assumes that the variances of two populations are equal and asks whether their means differ significantly. This is a type of two-sample test used to compare two sample means, where a large t-value suggests that the samples are very different, and a small t-value suggests that they are similar. Similar to the one-sample t-test, individuals who first encounter this test may wonder about the plausibility of its assumptions. In particular, you might question how the variances in two samples could possibly be equal if the means are different. In some contexts (for example, the industrial experiments that motivated Student’s efforts), there might be substantive reasons to assume equal variances. More informally, if you calculate the standard deviations in each sample and sees that they are close, you might proceed to calculate Student’s t-test. More formally, some analysts would recommend that you initially conduct an F-test to determine whether variances are different, and then proceed to consider the means. But many analysts would also simply not make the equal variances assumption and proceed directly to Welch’s t-test. Student’s T-Test Formula The formula for a Student’s t-test is: Given the formula to calculate the pooled standard deviation s p : Where: x̄ 1 = first sample mean x̄ 2 = second sample mean n 1 = first sample size n 2 = second sample size s 1 = first sample standard deviation s 2 = second sample standard deviation n 1 + n 2 – 2 = degrees of freedom ν In a Student’s t-test, the test statistic t is equal to the difference between sample means x̄ 1 and x̄ 2 , divided by the pooled standard deviation s p times the square root of 1 divided by the first sample size n 1 plus 1 divided by the second sample size n 2 . The pooled standard deviation s p is equal to the first sample size n 1 minus 1 times the first sample standard deviation s 1 plus the second sample size n 2 minus 1 times the second sample standard deviation s 2 , divided by the degrees of freedom, in this case the sum of the sample sizes minus two. It is called the “pooled” standard deviation because it combines or “pools” the data between both samples to determine the overall variability of the data. This formula can be broken down into a few simple steps. Step One: Calculate the Degrees of Freedom Step two: calculate the pooled standard deviation, step three: calculate the test statistic. Recall that the Student’s t-test assumes that the variances of two populations are equal. As was mentioned above, this is often a questionable assumption, and ultimately unverifiable. In this case, you can use Welch’s t-test, which is sometimes also called an unequal variances t-test or an “unpooled” t-test. Like before, the null hypothesis with this test is that two populations have equal means. Welch’s T-Test Formula The formula for Welch’s t-test is: Degrees of Freedom Formula To find the degrees of freedom when using Welch’s t-test, use the Satterthwaite formula: The next step is to find the p-value for the test statistic. The p-value is a measure of how “surprising” or “unlikely” some statistic would be given the particular assumptions that the analyst makes. In the case of these t-tests for differences in means, the p-value is the probability of calculating a t-statistic that is as large or larger than what was actually calculated from the observed data if, in fact, the population means were identical. More generally, a p-value is used to determine whether to reject the null hypothesis. In formal hypothesis testing, you would specify beforehand the p-value that would lead you to conclude that the two samples came from different populations. What is the Right P-Value? These standards differ by field and disciplines a lot, for example, in social and biological sciences, a p-value of 0.05 or smaller (implying 5% or lower chance of observing the data under the null hypothesis) is common, although in some cases 0.1 or 0.01 might be the standard. In the physical sciences, it is not uncommon to pre-specify a “6 sigma” standard for certain kinds of evidence, which requires an astronomically small p-value. How to Calculate the P-Value To calculate the p-value from a t-statistic, use a t-table and locate the degrees of freedom in the leftmost column. Then, locate the desired p-value in the heading row, 0.05 is most commonly used for a 95% confidence level. Then, find the intersection of the row and column to find the critical value. Drawing Conclusions Using the P-Value If the calculated t-value is larger than the critical value, then you can reject the null hypothesis. If it is less than the critical value, then you fail to reject the null hypothesis. The t-distribution is related to the normal distribution; indeed, it can be thought of as the normal distribution’s “heavy-tailed” cousin. The degrees of freedom in the t-distribution determines how heavy the tails are, with fewer degrees of freedom resulting in greater departures from normality. As the degrees of freedom increase, it becomes harder and harder to tell the differences between the associated t-distribution and the normal distribution. Because of this fact, experienced statistical analysts are often able to approximately estimate the p-value of a particular t-statistic through their familiarity with the normal distribution. A t-statistic of 2 or greater is typically enough to confirm statistical significance in the social and biological contexts. Recommended Statistics Resources Winning Percentage Calculator Right Arrow Icon Standard Error Calculator Right Arrow Icon Average Calculator Right Arrow Icon Central Limit Theorem Calculator Right Arrow Icon T Table Right Arrow Icon Calculators Descriptive Statistics Merchandise Which Statistics Test? T -Test Calculator for 2 Independent Means Enter the values for your two treatment conditions into the text boxes below, either one score per line or as a comma delimited list. Select your significance level and whether your hypothesis is one or two-tailed. Then give your data a final check, and press the "Calculate T and P Values" button. No calculation has yet been performed. For optimal use, please visit DATAtab on your desktop PC! Metric Variables: Ordinal variables:, nominal variables:, t-test calculator. You want to calculate a t-test? It's easy with DATAtab, just copy your data into the table above and select your variables for which you want to calculate the t-test! DATAtab will automatically use the appropriate test and interpret your results. If you want to calculate a t-test with your own data online, empty the upper table (click on Empty Table), copy your own data into it and make sure that the variable name is in the first row. Afterwards the variables are displayed below the table. Now click on the variables you want to evaluate. After selecting your variables, the t-test calculator will suggest which t-test you should use. You can choose from the following options: Simple t-Test t-Test for paired samples t-Test for independent Samples Calculate t-Test In the results section of the online t-test calculator you will find the mean and standard deviation of the samples and of course the calculated t-value and p-value. Which t-test you have to use is determined by the type of your sample or samples and how they are related to each other. Simple t-Test Calculator You want to test whether the mean of a sample is equal to that of the population? Then select a metric variable and specify the test value. Independent t-Test Calculator You want to compare the means of two independent groups? Then select two metric variables or one metric variable and one nominal variable with two values. Dependent t-Test Calculator You want to compare two groups where the measured values belong together in pairs? Then select two metric variables. p-value Calculator Of course you also get the p-value calculated and displayed in a table. You can specify the significance level right at the beginning of the calculation. If you want to calculate a one-sided t-test, you can either specify this as well or you simply divide your p-value by two at the end. More information about the theory behind the t-test and detailed examples can be found here: One sample t-test Paired t-test Independent sample t-test t-Value Calculator In order to calculate the p-value, the t-value must first be calculated. The p-value is then calculated from the t-value and the degrees of freedom. A t-test is a type of inferential statistical test that determines if there is a significant difference between the means of two groups. It can be used when the populations are normally distributed and samples are independent and randomly selected. The t-test compares the means of two groups and determines if they are statistically different from each other. Cite DATAtab: DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net Want to understand hypothesis testing better? Get our beautiful new e-Book here! T-test: two independent samples, when should i use this calculator. A t-test is any hypothesis test where the test statistic follows a Student's t-distribution . In this version of a t-test, we are testing the probability that two independent samples were drawn from the same population based on the means (and variances) of those samples. More specifically, this version of a t-test is used when: You have two independent samples. E.g. a treatment group and a control group, not a before and after treatment comparison ("paired samples"). You want to assess if a treatment led to some measureable difference in the groups (e.g. giving the treatment group a low fat diet led to a reduction in average weight). You do not know the population standard deviation(s). If you do, you can instead use a z -test. What can I calculate? You can use this calculator to estimate: The sample size a planned study will need to detect an effect size at a given power level The statistical power a planned study will have based on the expected sample and effect sizes 1 The smallest effect size a planned study can detect for a given power level and sample size The t-statistic and/or p-value for a completed study [1] It is a common mistake to try to calculate the power of a completed study based on the observed effect size. You need to know (or estimate) the true effect size to calculate the power of a study. Assumptions When conducting a t-test with two independent samples, the following assumptions are made about your data: Your data consists of two independent and identically distributed samples, one from each of the two populations being compared (although they may turn out to be the same population). The sample means ( X 1 and X 2 ) are normally distributed. 1 The sample variances ( s 2 1 and s 2 2 ) are χ 2 distributed. 2 The sample means and sample variances are statistically independent. This calculator does not require the groups to have equal variance as it uses the Welch's unequal variances t-test formulation by default 3 [1] This does not require your underlying data to be normally distributed. With larger samples, the Central Limit Theorem typically means the sample means will be normally distributed. [2] This assumption holds if the underlying data are normally distributed, but not neccessarily if you are relying on the Central Limit Theorem for normally distributed sample means. [3] You probably should as well . Key Concepts Significance Level (α) : The probability of incorrectly rejecting the null hypothesis (H 0 : θ = 0; where θ = μ 1 - μ 2 ), also known as the false positive rate or the Type I error rate. An α of 0.05 (5%) means that if we repeated an experiment where we drew samples from the same population many times, we would expect to incorrectly reject the null hypothesis in 5% of cases. α can also be thought of as a measure of how extreme the observed difference in sample means has to be before we reject the null hypothesis. With an α of 0.05, we would reject the null hypothesis when observing a difference that we would expect to see 5% (or less) of the time when drawing two samples from the same population. Statistical Power (1 - β) : β is the probability that we will fail to reject the null hypothesis when the samples are drawn from different populations. This is also known as the false negative rate or the Type II error rate. Statistical power or 1 - β is therefore the probablity that we will correctly reject the null hypothesis. In the same way that we can draw samples with different means from the same population, there is also a risk that we draw samples with very similar means from two different populations. Effect Size (Cohen's d) : A standardized measure of the difference in the means (can be sample or population means depending on the context). The difference in means is divided by the pooled standard deviation of the two samples/populations to provide a metric, in units of standard deviations, that can be compared across studies. It can also be used directly in some calculations instead of the means and standard deviations of the samples. What do you want to do? Calculations Chart description. Paired T Test Calculator (Dependent T test) Enter sample data Reporting results in APA style Paired t-test online, what is a paired t-test, how to use the paired t-test calculator, calculators. 398 IMAGES Two Sample Z Hypothesis Test Hypothesis Testing Example Two Sample t-Test Two Sample t Test (Independent Samples) Hypothesis Testing with Two Samples Two sample t hypothesis test calculator t test Two mean hypothesis test calculator VIDEO One Sample Mean Hypothesis Test (updated) Mean Hypothesis Test with Sample Data by hand One-mean hypothesis test Calculate the test statistic for a two-mean hypothesis test for population SDs assumed unequal #2 Hypothesis Testing for Population Mean (Large sample, Z test) (Hindi/Urdu) Hypothesis Test for a Mean (Rejected) with a TI 84 COMMENTS Two Sample t-test Calculator If this is not the case, you should instead use the Welch's t-test calculator. To perform a two sample t-test, simply fill in the information below and then click the "Calculate" button. Enter raw data Enter summary data. Sample 1. 301, 298, 295, 297, 304, 305, 309, 298, 291, 299, 293, 304. Sample 2. Two Sample T-Test Calculator (Pooled-Variance) The two sample t test calculator provides the p-value, effect size, test power, outliers, distribution chart, and a histogram. Statistics Kingdom ... Target: To check if the difference between the average (mean) of two groups (populations) is significant, using sample data Example1: A man of average is expected to be 10cm taller than a woman of ... t-test Calculator 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). Decide on the alternative hypothesis: Two-tailed; Left-tailed; or. Right-tailed. Hypothesis Testing Calculator with Steps Hypothesis Testing Calculator. 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 ... Difference in Means Hypothesis Test Calculator Calculate the results of your two sample t-test. 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 ... Two sample t test calculator Two sample t test calculator can be used to find test statistic, p-value, critical value, decision and conclusion by using equal and unequal variance. ... please provide the sample size, mean, and standard deviation. Hypothesis Testing: Determine whether your null and alternative hypotheses are two-tailed, right-tailed, or left -tailed ... 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 ... T-Test Calculator for 2 Independent Means This simple t -test calculator, provides full details of the t-test calculation, including sample mean, sum of squares and standard deviation. A t -test is used when you're looking at a numerical variable - for example, height - and then comparing the averages of two separate populations or groups (e.g., males and females). Two Sample t test calculator In that case, the sample data provided is usually the sample means, sample standard deviations and sample sizes. Other type of t-test calculators include the t-test for one sample . For different types of statistics, you can try this ANOVA calculator , which is similar to the t-test only that with ANOVA you can compare more than 2 groups. Two-Sample t-test 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. Two Sample T-Test Calculator 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. Online calculator: Two sample t-Test 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. Free Online Two Sample T Test Calculator Two Sample T Test Calculator. Upload your data set below to get started. Upload File. Or input your data as csv. column_one,column_two,column_three 1,2,3 4,5,6 7,8,9. Submit CSV. Sharing helps us build more free tools. T-Test calculator Two sample t-test calculator. Use this calculator to test whether population means are significantly different from each other. Input numbers separated by comma (,) , colon (:), semicolon (;) or blank space. 1. Group description: 2. Number of tails: 3. Significance Level: T-test for two Means Instructions : Use this T-Test Calculator for two Independent Means calculator to conduct a t-test for two population means (\mu_1 μ1 and \mu_2 μ2), with unknown population standard deviations. This test apply when you have two-independent samples, and the population standard deviations \sigma_1 σ1 and \sigma_2 σ2 and not known. Two-Sample $t$ Test Calculator Two-Sample t t Test Calculator. procedures assume that both our samples come from an SRS and will give trustworthy conclusions only if this condition is met. procedures yield trustworthy conclusions only if you can reasonably assume that your data from both samples come from a normal distribution, that is, if the distribution appears to be ... T-Test Calculator Where: x̄ 1 = first sample mean x̄ 2 = second sample mean n 1 = first sample size n 2 = second sample size s 1 = first sample standard deviation s 2 = second sample standard deviation n 1 + n 2 - 2 = degrees of freedom ν In a Student's t-test, the test statistic t is equal to the difference between sample means x̄ 1 and x̄ 2, divided by the pooled standard deviation s p times the ... T -Test Calculator for 2 Independent Means T -Test Calculator for 2 Independent Means. Note: You can find further information about this calculator, here. Enter the values for your two treatment conditions into the text boxes below, either one score per line or as a comma delimited list. Select your significance level and whether your hypothesis is one or two-tailed. t-Test calculator online: One sample, paired sample & unpaired sample t Calculate t-Test. In the results section of the online t-test calculator you will find the mean and standard deviation of the samples and of course the calculated t-value and p-value. Which t-test you have to use is determined by the type of your sample or samples and how they are related to each other. Two Sample t-test: Definition, Formula, and Example Fortunately, a two sample t-test allows us to answer this question. Two Sample t-test: Formula. A two-sample t-test always uses the following null hypothesis: H 0: μ 1 = μ 2 (the two population means are equal) The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed: The Power Calculator When conducting a t-test with two independent samples, the following assumptions are made about your data: Your data consists of two independent and identically distributed samples, one from each of the two populations being compared (although they may turn out to be the same population).; The sample means (X 1 and X 2) are normally distributed. 1The sample variances (s 2 1 and s 2 2) are χ 2 ... Two Sample T-Test Calculator (Welch's T-test) 1. Two tailed test example: A factory uses two identical machines to produce plastic plates. You would expect both machines to produce the same number of plates per minute. Let μ1 = average number of plates produced by machine1 per minute. Let μ2 = average number of plates produced by machine2 per minute. We would expect μ1 to be equal to μ2. Paired T Test Calculator (Dependent T test) The paired t-test calculator also called the dependent t-test calculator compares the means of the same items in two different conditions or any others connection between the two samples when there is a one to one connection between the samples - each value in one group is connected to one value in the other group. The test uses the t distribution. assignment essay homework paper phd report resume review speech thesis