null hypothesis t 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.

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POPULAR USE CASES

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

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3. Enter data

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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.

T-Test Calculator

Compare the means of two samples using a single-sample or two-sample t-test below.

• Single Sample
• Two Sample (Unpaired)

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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.

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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.

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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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Descriptive Statistics

Hypothesis test, independent t-test calculator.

To calculate an independent t-Test online just select one metric Variable and one nominal Variable with two values.

If you want to use your own data, just copy your data into the upper table and make sure that the variable name is in the first row. The results of independent t tests are then displayed clearly.

At the beginning of the independent t-test calculator you can choose what your alternative hypothesis is. Then you can test the assumptions for the t-test and you will get the null and alternative hypotheses. Then you get the relative results.

Independent t-Test

An independent t-test is a statistical hypothesis test that is used to determine if there is a significant difference between the means of two independent groups or conditions. It is based on the t-distribution, which is a probability distribution that takes into account the sample size and the variability of the data.

The t-test works by comparing the t-value, which is a measure of the difference between the means, to a critical value based on the desired level of significance (usually denoted as α). The critical value is determined by the degrees of freedom, which depend on the sample sizes of the groups being compared.

If your data are not normally distributed and therefore do not meet the requirements for the independent t-test, you can simply calculate the Mann-Whitney U test online .

Independent T-test Hypotheses

In the independent t-test (also known as the independent samples t-test or two-sample t-test), we compare the means of two independent groups to see if there is a statistically significant difference between them.

The hypotheses for the independent t-test are:

• Null Hypothesis (H 0 ): The population means of the two groups are equal. This suggests that there is no significant difference between the two group means. [ H 0 : μ 1 = μ 2 ]
• Two-tailed test: We are interested in any difference between the group means, but we don't specify a direction. [ H a : μ 1 ≠ μ 2 ]
• One-tailed test (upper-tailed): We are specifically interested in whether the mean of group 1 is greater than the mean of group 2. [ H a : μ 1 > μ 2 ]
• One-tailed test (lower-tailed): We are specifically interested in whether the mean of group 1 is less than the mean of group 2. [ H a : μ 1 2 ]

When conducting an independent t-test, you'd choose the form of the alternative hypothesis based on your research question or specific hypothesis. After performing the test, if the test statistic (t-value) is found to be in the critical region (typically using a significance level like 0.05), you'd reject the null hypothesis in favor of the alternative hypothesis. If the t-value is not in the critical region, you'd fail to reject the null hypothesis, indicating that you didn't find enough evidence to suggest a significant difference between the two group means.

How to calulate an independent t-test

• Formulate the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis typically assumes that there is no significant difference between the means of the two groups, while the alternative hypothesis assumes that there is a significant difference.
• Collect data from the two groups being compared, ensuring that the samples are independent and representative of the populations being studied.
• Calculate the sample means (x̄1 and x̄2), the sample standard deviations (s1 and s2), and the sample sizes (n1 and n2) for each group.
• Compute the t-value using the formula: t = (x̄1 - x̄2) / sqrt((s1^2/n1) + (s2^2/n2))
• Determine the degrees of freedom for the t-distribution, which is calculated using the formula: df = n1 + n2 - 2
• Compare the calculated t-value to the critical value from the t-distribution table (or use statistical software) at the desired level of significance (α).
• If the calculated t-value exceeds the critical value (i.e., it falls within the rejection region), then the null hypothesis is rejected, indicating that there is a significant difference between the means. If the calculated t-value does not exceed the critical value (i.e., it falls within the non-rejection region), then the null hypothesis is not rejected, suggesting that there is no significant difference between the means.

The p-value is important to reported in a independent samples t-tests, representing the probability of obtaining a difference as extreme as the observed one, assuming the null hypothesis is true. If the p-value is below the chosen significance level (α), typically 0.05, then the null hypothesis is rejected.

Calculate Independent t-test online

Therefore, to calculate an independent t-test online, you only need two independent samples. The samples should be normally distributed. Then simply copy the data into the table above, making sure that the first row of variables contains the name! To calculate a paired samples t-test , simply select two metric variables.

Assumptions for Independent Samples T-test

Independence: The observations in each group are independent of each other. This means that the values in one group are not influenced by or related to the values in the other group.

Normally Distributed Data: The data in each group should be approximately normally distributed. If the sample size is large enough (typically >30 observations in each group), the t-test can be robust to moderate deviations from normality. However, for smaller sample sizes, normality assumption becomes more critical.

Homogeneity of Variance: The variances of the two groups should be approximately equal. This assumption is known as homoscedasticity. Unequal variances between groups can affect the accuracy of the t-test.

Calculate the Effect Size for the Independent t-Test

The effect size for an independent t-test is a measure of the magnitude or strength of the difference between the means of two independent groups. It provides information about the practical significance of the observed difference, which can be valuable in addition to the statistical significance determined by the t-test. There are several common effect size measures for independent t-tests, and two of the most widely used ones are Cohen's d and Hedges' g.

Cohen's d is a standardized measure of effect size that expresses the difference between the means in terms of standard deviation units.

The formula for Cohen's d is as follows:

Cohen's d = (M1 - M2) / S_pooled

• M1 is the mean of Group 1.
• M2 is the mean of Group 2.
• S_pooled is the pooled standard deviation, calculated as:

S_pooled = √((s1^2 + s2^2) / 2)

Where s1 and s2 are the standard deviations of Group 1 and Group 2, respectively.

Hedges' g is similar to Cohen's d but includes a correction factor for small sample sizes. It's useful when you have unequal group sizes or small sample sizes.

The formula for Hedges' g is as follows:

Hedges' g = (M1 - M2) / S_pooled * (1 - (3 / (4 * (N1 + N2) - 9)))

• S_pooled is the pooled standard deviation, calculated as described above.
• N1 is the sample size of Group 1.
• N2 is the sample size of Group 2.

Once you have calculated Cohen's d or Hedges' g, you can interpret the effect size using general guidelines. Cohen's original guidelines suggest that:

• Small effect size: d ≈ 0.2
• Medium effect size: d ≈ 0.5
• Large effect size: d ≈ 0.8

However, it's important to note that the interpretation of effect size can be context-dependent and may vary by field or research area. In some cases, even a small effect size can be practically significant, while in others, a large effect size may not have much practical relevance. Therefore, it's essential to consider the specific context of your study when interpreting the effect size.

Dive Deep into Data with the Independent t-test Calculator

The statistical realm can seem intimidating, with its barrage of numbers and terms. But when armed with the right tools, it becomes a fascinating world to explore. Introducing the independent t-test calculator - a beacon for those navigating the seas of independent data sets!

Decoding the Independent t-test

Before we delve into the intricacies of the calculator, let's decode the basics. The independent t-test, often referred to as the two-sample t-test, determines whether there's a significant difference between the means of two unrelated groups. For instance, comparing the exam scores of students from two separate classes is a fitting application of this test.

Why Choose the Independent t-test Calculator?

• Efficiency : Bypass time-consuming manual calculations and swiftly gain insights.
• Accuracy : Harness the power of precise algorithms to ensure error-free results.
• User-friendly : Designed with simplicity in mind, the tool is accessible for novices and pros alike.

How to Use Our Calculator Efficiently

• Enter Data : Populate the given fields with the data sets for both groups.
• Select Significance Level : Commonly set at 0.05, but can be tweaked based on your research needs.
• Compute : Hit 'Calculate' and witness the tool's prowess.

Deciphering the Output

Post-calculation, you'll be greeted with a t-value and a p-value.

• t-value : Signifies the size of the difference relative to the variance within the samples.
• p-value : Denotes the likelihood of observing your specific results (or more extreme) under the assumption that the null hypothesis is true.

A p-value smaller than your chosen significance level means you can reject the null hypothesis, pointing towards a significant difference between the two groups.

Final Thoughts

The independent t-test calculator stands as an indispensable ally for all venturing into the comparison of unrelated groups. For students, researchers, and data enthusiasts, this tool demystifies complex calculations, leading to enlightened data-driven decisions.

Embrace the convenience and clarity of the independent t-test calculator and elevate your analytical journey!

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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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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 $\alpha < 0.05$. 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) One sample t test calculator The One Sample t Test Calculator allows you to determine p-values, critical values, test statistics, and conclusions using the one-sample t test method. Related Calculators : List of all calculator P-value calculator Critical value Calculator One sample z test calculator How to Use the One Sample t Test Calculator Select Data Type: Choose whether to input summary statistics directly or provide a data set. Input Your Data: Enter the required values such as population mean, sample size, sample mean, and sample standard deviation. Set Hypotheses: Specify the null and alternative hypotheses. Calculate: Click the "Calculate" button to see the test statistic, p-value, and other relevant results. Interpreting the Results To interpret the results of a one-sample t-test, you must first grasp several crucial components: the test statistic (t-value), degrees of freedom, p-value, and confidence interval. Here's a step-by-step way to interpreting these findings: Components of One-Sample t-Test Results Test Statistic (t-value) : This result indicates the amount of standard deviations your sample mean is from the population mean under the null hypothesis. A larger absolute value of t indicates a wider divergence between the sample mean and the population average. Degrees of Freedom (df): This is normally the sample size less one (n-1). The degrees of freedom are utilized to calculate the critical value of t using the t-distribution table. This represents the likelihood of receiving a test statistic as extreme as the one observed, assuming the null hypothesis is correct. A low p-value ( less than 0.05) indicates that the observed data are unlikely to support the null hypothesis, resulting in its rejection. Confidence Interval: This defines a range of values within which the genuine population mean is expected to fall. If the confidence interval excludes the population mean stated in the null hypothesis, it supports the conclusion that the sample mean differs considerably from the population mean. Steps to Interpret the Results State the Hypotheses: Null Hypothesis (H₀): The population mean is equal to a specified value (e.g. \( \mu=\mu_0 \) ). Alternative Hypothesis (H₁): The population mean is different from the specified value (e.g. \( \mu \ne \mu_0 \) ). Check the t-value: Compare the t-value to the crucial value from the t-distribution table based on the specified significance level (α, typically 0.05) and degrees of freedom. If |t-value| > critical value, reject the null hypothesis. Examine the P-value: If the p-value is less than the chosen significance level \( \alpha \), reject the null hypothesis. A p-value less than \(0.05\) typically indicates strong evidence against the null hypothesis. Review the Confidence Interval: Check to see if the confidence interval for the sample mean includes the population mean under the null hypothesis. If the interval excludes the population mean, it indicates that the sample mean is significantly different than the population mean. Requirements and Assumptions for a one sample t test To effectively execute and interpret a one-sample t-test, you must first grasp the test's requirements and assumptions. Meeting these parameters assures that the test results are legitimate and reliable. Requirements for a One-Sample t-Test You will need a sample of the population you are studying. To compare the sample mean to the population mean (μ₀), you must have a known population mean. A suitable sample size is necessary. While the t-test performs well with small sample sizes, higher sample sizes yield more accurate results. Assumptions of a One-Sample t-Test Random Sampling: Data should be acquired from the population using random sampling. This guarantees that the sample is representative of the population and minimises bias. Scale of Measurement: The data must be continuous (interval or ratio scale). This means that the data points may be meaningfully arranged, and the differences between them are consistent and observable. The data should follow a roughly normal distribution. This assumption is particularly essential for small sample sizes (n < 30). For bigger samples, the Central Limit Theorem predicts that the sample mean distribution will be essentially normal, regardless of the distribution of the data. Independence : Observations in the sample must be independent of one another. This means that the value of one observation should not affect or predict the value of another. Unknown Population Standard Deviation: : The test assumes that the population standard deviation \(\alpha\) is unknown and needs to be determined from the sample. Applications of the One Sample t Test This test is widely used in various fields including: Education: Assessing whether the average test scores of a class differ from the national average. Healthcare: Comparing the mean blood pressure level of a group of patients to a known population mean. Business: Evaluating if the average sales of a product differ from the company's historical sales data. Frequently Asked Questions (FAQs) What is a one-sample t test. A one-sample t test is a statistical approach for determining whether a sample's mean differs significantly from a known population mean. When should I use a one-sample t test? Use this test when the population standard deviation is unknown and the sample size is small (typically n < 30). How do I interpret the p-value in a t test? The p-value indicates the probability of observing the test results under the null hypothesis. A low p-value less than the significance level equal to \( \alpha \) suggests rejecting the null hypothesis. Why Use Our One Sample t Test Calculator? Our calculator offers several advantages: Accuracy: Provides precise calculations for your hypothesis testing needs. User-Friendly Interface: Easy to navigate and input your data. Educational Value: Offers detailed explanations and step-by-step guides to help you understand the results. Time-Saving: Quickly computes results, allowing you to focus on analysis and interpretation. Versatility: Suitable for various fields such as education, healthcare, and business, making it a valuable tool for a wide range of users. Start your hypothesis testing today with our One Sample t Test Calculator and achieve accurate and reliable results effortlessly. One Sample T Test Calculator Enter sample data Reporting results in APA style One sample t-test, what is a one sample t-test, how to use the one sample t test calculator, calculators. 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. Find Us On Facebook Follow on Twitter Subscribe using RSS 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. The p-value is With sample means of The p-value would be With sample standard deviations of The p-value would be With sample sizes of The p-value would be 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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PRE-ALGEBRA MORE MATH FINANCE BUSINESS ENGINEERING FINANCE CHARTS MATH WORKSHEETS CURRENCY CONVERTER MULTIPLICATION TABLES HEALTH TIME & PRODUCTIVITY DEPRECIATION TAX CALCULATORS SPORTS CALCULATORS EVERYDAYLIFE Engineering t-Test Calculator T-test - work with steps. Input Data : Data set x = 3, 11, 17, 28, 34 Data set y = 5, 8, 13, 19, 28 Total number of elements = 5 Objective : Find the t-score by using mean and standard deviation. Solution : Mean 1 = (3 + 11 + 17 + 28 + 34)/5 = 93/5 Mean 1 = 18.6 Mean 2 = (5 + 8 + 13 + 19 + 28)/5 = 73/5 Mean 2 = 14.6 SD1 = √(1/5 - 1) x ((3 - 18.6) 2 + ( 11 - 18.6) 2 + ( 17 - 18.6) 2 + ( 28 - 18.6) 2 + ( 34 - 18.6) 2 ) = √(1/4) x ((-15.6) 2 + (-7.6) 2 + (-1.6) 2 + (9.4) 2 + (15.4) 2 ) = √(0.25) x ((243.36) + (57.76) + (2.56) + (88.36) + (237.16)) = √(0.25) x 629.2 = √157.3 SD1 = 12.5419 SD2 = √(1/5 - 1) x ((5 - 14.6) 2 + ( 8 - 14.6) 2 + ( 13 - 14.6) 2 + ( 19 - 14.6) 2 + ( 28 - 14.6) 2 ) = √(1/4) x ((-9.6) 2 + (-6.6) 2 + (-1.6) 2 + (4.4) 2 + (13.4) 2 ) = √(0.25) x ((92.16) + (43.56) + (2.56) + (19.36) + (179.56)) = √(0.25) x 337.2 = √84.3 SD2 = 9.1815 t-score = x 1 - x 2 √(SD1 2 /n1 + SD2 2 /n2) = 18.6 - 14.6 √((12.5419) 2 /5 + (9.1815) 2 /5) = 4 √((157.3)/5 + (84.3)/5) = 4 √(31.46 + 16.86) = 4 √(48.32) = 4 6.9513 t-score = 0.5754 Enter two samples (observed values) in the box. These values must be real numbers or variables and may be separated by commas. The values can be copied from a text document or a spreadsheet. Press the "GENERATE WORK" button to make the computation. t-Test calculator will give a test whether samples from two independent populations provide that the populations have different means. What is t-Test? When the samples from the two populations are independent; When the samples from the two populations are depended, i.e. when they are paired. If populations standard deviations are equal, $\sigma_1-\sigma_2$ If populations standard deviations are different How to Find t-Critical Value Simple and independent random samples; Normal populations or large samples. t-Test with Mean and Standard Deviation Company A Company B Mean 43.2 36.7 Standard Deviation 7 8.3 Number of Employess 50 66 Serbia United States Mean 43.2 5.2 Standard Deviation 1.2 8.3 Number of Employess 67 166 Hypergeometric distribution Calculator Error Function Calculator Beta Function Calculator Gamma Function Calculator Digamma Function Calculator F-Test Calculator Statistical Data Set Range Calculator Effect of Size Calculator Sampling Error Calculator Percent Error Calculator Percentile Calculator Signal to Noise Ratio Calculator Calculators Descriptive Statistics Merchandise Which Statistics Test? 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. Teach yourself statistics Bartlett's Test Calculator Bartlett's test is used to test the assumption that variances are equal (homogeneous) across groups. For help in using this calculator, read the Frequently-Asked Questions or review the Sample Problem . To learn more about Bartlett's test, read Stat Trek's tutorial on Bartlett's test . In the dropdown box, specify the number of groups. Enter a value in each of the unshaded textboxes. Click Calculate to compute outputs. Frequently-Asked Questions Instructions: To find the answer to a frequently-asked question, simply click on the question. What is Bartlett's test? Bartlett's test is used to test the assumption that variances are equal (i.e., homogeneous) across groups. The test is easy to implement and produces valid results, assuming data points within groups are randomly sampled from a normal distribution. Because Bartlett's test is sensitive to departures from normality, a normality test is prudent. Several ways to check for departures from normality are described at: How to Test for Normality: Three Simple Tests . Note: Unlike Hartley's Fmax test , which also tests for homogeneity, Bartlett's test does not assume equal sample sizes across groups. How does Bartlett's test work? Bartlett's test is an actual hypothesis test , where we examine observed data to choose between two statistical hypotheses: H 0 : σ 2 i = σ 2 j for all groups H 0 : σ 2 i ≠ σ 2 j for at least one pair of groups Like many other techniques for testing hypotheses, Bartlett's test for homogeneity involves computing a test-statistic and finding the P-value for the test statistic, given degrees of freedom and significance level . If the P-value is bigger than the significance level, we accept the null hypothesis; if it is smaller, we reject the null hypothesis. What steps (computations) are required to execute Bartlett's test? The steps required to conduct Bartlett's test for homogeneity are detailed below: Step 1. Specify the significance level ( α ). s 2 j = [ Σ ( X i, j - X j ) 2 ] / ( n j - 1 ) where X i, j is the score for observation i in Group j , X j is the mean of Group j , n j is the number of observations in Group j , and k is the number of groups. N =  Σ n  i s 2 p  =  [  Σ  ( n  j -1 ) s 2 j  ] / ( N - K ) where n j is the sample size in Group j , k is the number of groups, N is the total sample size, and s 2 j is the sample variance in Group   j . A = ( N - k ) * ln( s 2 p ) B = Σ [ ( n  j - 1 ) * ln( s 2 j ) ] C = 1 / [ 3 * ( k - 1 ) ] D = Σ [ 1 / ( n  j - 1 ) - 1 / ( N - k ) ] T = ( A - B ) / [ 1 + ( C * D ) ] where A is the first term in the numerator of the test statistic, B is the second term in the numerator, C is the first term in the denominator , D is the second term in the denominator, and ln is the natural logarithm. It turns out that the test statistic (T) is distributed much like a chi-square statistic with ( k-1 ) degrees of freedom. Knowing the value of T and the degrees of freedom associated with T, we can use Stat Trek's Chi-Square Calculator to find the P-value - the probability of seeing a test statistic more extreme than T. Step 7. Accept or reject the null hypothesis, based on P-value and significance level. If the P-value is bigger than the significance level, we accept the null hypothesis that variances are equal across groups. Otherwise, we reject the null hypothesis. What should I enter in the field for number of groups? Bartlett's is designed to test the hypothesis of homogeneity among nonoverlapping sets of data. The number of groups is the number of data sets under consideration. What should I enter for significance level? The significance level is the probability of rejecting the null hypothesis when it is true. Researchers often choose 0.05 or 0.01 for a significance level. What should I enter for sample size? Sample size refers to the number of observations in a group. For each group, enter the number of observations in the space provided. Note: Unlike some other tests for homogeneity (e.g., Hartley's Fmax test ), Bartlett's test does not require equal sample sizes across groups. What should I enter for variance? In the fields provided, enter an estimate of sample variance for each group. To compute sample variance ( s 2 j  ) for each group, use the following formula: where X i, j is the score for observation i in Group j , X j is the mean of Group j , and n j is the number of observations in Group j . What is degrees of freedom? Bartlett's test computes a test statistic (T) to test for normality. The degrees of freedom ( df ) for a chi-square test of that statistic is: where N is the total sample size across all groups, and k is the number of groups in the sample. What is the test statistic (T)? The test statistic (T) is the statistic used by Bartlett's test to make a decision about whether to accept or reject the null hypothesis of equal variances between groups. When T is very big, we reject the null hypothesis; when T is small, we accept the null hypothesis. The calculator computes a T statistic, based on user inputs. The formulas that the calculator uses to compute a T statistic are given at Bartlett's Test for Homogeneity of Variance . What is the P-value? If you assume that the null hypothesis of equal variance is true, the P-value is the probability of seeing a test statistic (T) that is more extreme (bigger) than the actual test statistic computed from sample data. How does the calculator test hypotheses? Like many other techniques for testing hypotheses, Bartlett's test for homogeneity of variance involves computing a test-statistic and finding the P-value for the test statistic, given degrees of freedom and significance level . If the P-value is bigger than the significance level, the calculator accepts the null hypothesis. Otherwise, it rejects the null hypothesis. Sample Problems The table below shows sample data and variance for five groups. How would you test the assumption that variances are equal across groups? Group 1 Group 2 Group 3 Group 4 Group 5 Sample Data 1 2 3 4 5 1 3 5 7 9 1 4 7 10 13 1 5 9 13 17 1 6 11 16 21 Variance 2.5 10 22.5 40 62.5 One option would be to use Stat Trek's Bartlett's Test Calculator . Simply, take the following steps: Enter the number of groups (5). Enter the significance level. For this problem, we'll use 0.05. For each group, enter sample size. In this example, the sample size is 5 for each group. For each group, enter a sample estimate of group variance. Then, click the Calculate button to produce the output shown below: From the calculator, we see that the test statistic (T) is 8.91505. Assuming equal variances in groups and given a significance level of 0.05, the probability of observing a test statistic (T) bigger than 8.91505 is given by the P-value. Since the P-value (0.06326) is bigger than the significance level (0.05), we cannot reject the null hypothesis of equal variances across groups. Note: To see the hand calculations required to solve this problem, go to Bartlett's Test for Homogeneity of Variance: Example 1 . Oops! Something went wrong. 140 COMMENTS t-test Calculator t-test calculator performs all kinds of t-tests: one-sample, two-sample, and paired. Board. Biology Chemistry Construction ... 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 ... 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 ... 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. T-Test Calculator. ... Null Hypothesis. H0: u1 - u2 = 0, where u1 is the mean of first population and u2 the mean of the second. 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 ... Hypothesis Test Calculator Use this Hypothesis Test Calculator for quick results in Python and R. Learn the step-by-step hypothesis test process and why hypothesis testing is important. Hypothesis Test Calculator | 365 Data Science t-Test Calculator t-test calculator is an online statistics tool to estimate the significance of observed differences between the means of two samples when there is a null hypothesis that is no significant difference between the means by using standard deviation. It is necessary to follow the next steps: Enter two samples (observed values) in the box. These values must be real numbers or variables and may be ... T-Test Calculator 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 ... 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. Independent t-Test Calculator In the independent t-test (also known as the independent samples t-test or two-sample t-test), we compare the means of two independent groups to see if there is a statistically significant difference between them. The hypotheses for the independent t-test are: Null Hypothesis (H 0): The population means of the two groups are equal. This ... 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. Treatment 1 ( X) Treatment 2 ( X) Significance ... Single Sample T-Test Calculator A single sample t-test (or one sample t-test) is used to compare the mean of a single sample of scores to a known or hypothetical population mean. So, for example, it could be used to determine whether the mean diastolic blood pressure of a particular group differs from 85, a value determined by a previous study. Requirements. T-Test Calculator 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. Two sample t-test One sample t-test. t-test One-sample t-test (go to the calculator) What is the one-sample t-test? The one-sample t-test checks if the known mean is statistically correct, based on a sample average and sample standard deviation. The null hypothesis assumes that the known mean is correct. One sample t test calculator Select Data Type: Choose whether to input summary statistics directly or provide a data set. Input Your Data: Enter the required values such as population mean, sample size, sample mean, and sample standard deviation. Set Hypotheses: Specify the null and alternative hypotheses. Calculate: Click the "Calculate" button to see the test statistic ... Two Sample t test calculator Two Samples T-Test Calculator: Quickly calculate the significance of the difference between two sample means, showing steps. Online Calculators. ... the null hypothesis is rejected; The number of degrees of freedom when equal population variances are assumed is \(df = n_1 + n_2\), where \(n_1\) and \(n_2\) are the corresponding sample sizes . ... T-test for One Population Mean This t-test calculator calculates the t-statistics for one sample, for the null and alternative hypotheses that you provide. Online Calculators. ... Decision about the null hypothesis. Since it is observed that \(t = 2.385 > t_c = 1.699\), it is then concluded that the null hypothesis is rejected. One Sample T Test Calculator The one-sample t-test determines if the mean of a single sample is significantly different from a known population mean. The one sample t-test calculator calculates the one sample t-test p-value and the effect size. When you enter the raw data, the one sample t-test calculator provides also the Shapiro-Wilk normality test result and the outliers. T-test for two Means Use this T-Test Calculator for two Independent Means calculator to conduct a t-test for two population means u1 and u2, with unknown pop standard deviations. ... 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 ... Two Sample t-test Calculator A two sample t-test is used to test whether or not the means of two populations are equal. This type of test assumes that the two samples have equal variances. 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 ... 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. t-Test Calculator t-test calculator is an online statistics tool to estimate the significance of observed differences between the means of two samples when there is a null hypothesis that is no significant difference between the means by using standard deviation. It is necessary to follow the next steps: Enter two samples (observed values) in the box. These values must be real numbers or variables and may be ... T-Test Calculator for 2 Dependent Means 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. Equation. A T-test calculator that compares 2 dependent population means for statistical significance. Bartlett's Test Calculator When T is very big, we reject the null hypothesis; when T is small, we accept the null hypothesis. The calculator computes a T statistic, based on user inputs. The formulas that the calculator ... From the calculator, we see that the test statistic (T) is 8.91505. Assuming equal variances in groups and given a significance level of 0.05, the ... essay homework paper phd project resume review speech study thesis