t test definition in research

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What Is a T-Test?

Understanding the t-test, using a t-test, which t-test to use.

• T-Test FAQs
• Fundamental Analysis

T-Test: What It Is With Multiple Formulas and When To Use Them

Read how this calculation can be used for hypothesis testing in statistics

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A t-test is an inferential statistic used to determine if there is a significant difference between the means of two groups and how they are related. T-tests are used when the data sets follow a normal distribution and have unknown variances, like the data set recorded from flipping a coin 100 times.

The t-test is a test used for hypothesis testing in statistics and uses the t-statistic, the t-distribution values, and the degrees of freedom to determine statistical significance.

Key Takeaways

• A t-test is an inferential statistic used to determine if there is a statistically significant difference between the means of two variables.
• The t-test is a test used for hypothesis testing in statistics.
• Calculating a t-test requires three fundamental data values including the difference between the mean values from each data set, the standard deviation of each group, and the number of data values.
• T-tests can be dependent or independent.

Investopedia / Sabrina Jiang

A t-test compares the average values of two data sets and determines if they came from the same population. In the above examples, a sample of students from class A and a sample of students from class B would not likely have the same mean and standard deviation. Similarly, samples taken from the placebo-fed control group and those taken from the drug prescribed group should have a slightly different mean and standard deviation.

Mathematically, the t-test takes a sample from each of the two sets and establishes the problem statement. It assumes a null hypothesis that the two means are equal.

Using the formulas, values are calculated and compared against the standard values. The assumed null hypothesis is accepted or rejected accordingly. If the null hypothesis qualifies to be rejected, it indicates that data readings are strong and are probably not due to chance.

The t-test is just one of many tests used for this purpose. Statisticians use additional tests other than the t-test to examine more variables and larger sample sizes. For a large sample size, statisticians use a  z-test . Other testing options include the chi-square test and the f-test.

Consider that a drug manufacturer tests a new medicine. Following standard procedure, the drug is given to one group of patients and a placebo to another group called the control group. The placebo is a substance with no therapeutic value and serves as a benchmark to measure how the other group, administered the actual drug, responds.

After the drug trial, the members of the placebo-fed control group reported an increase in average life expectancy of three years, while the members of the group who are prescribed the new drug reported an increase in average life expectancy of four years.

Initial observation indicates that the drug is working. However, it is also possible that the observation may be due to chance. A t-test can be used to determine if the results are correct and applicable to the entire population.

Four assumptions are made while using a t-test. The data collected must follow a continuous or ordinal scale, such as the scores for an IQ test, the data is collected from a randomly selected portion of the total population, the data will result in a normal distribution of a bell-shaped curve, and equal or homogenous variance exists when the standard variations are equal.

T-Test Formula

Calculating a t-test requires three fundamental data values. They include the difference between the mean values from each data set, or the mean difference, the standard deviation of each group, and the number of data values of each group.

This comparison helps to determine the effect of chance on the difference, and whether the difference is outside that chance range. The t-test questions whether the difference between the groups represents a true difference in the study or merely a random difference.

The t-test produces two values as its output: t-value and degrees of freedom . The t-value, or t-score, is a ratio of the difference between the mean of the two sample sets and the variation that exists within the sample sets.

The numerator value is the difference between the mean of the two sample sets. The denominator is the variation that exists within the sample sets and is a measurement of the dispersion or variability.

This calculated t-value is then compared against a value obtained from a critical value table called the T-distribution table. Higher values of the t-score indicate that a large difference exists between the two sample sets. The smaller the t-value, the more similarity exists between the two sample sets.

A large t-score, or t-value, indicates that the groups are different while a small t-score indicates that the groups are similar.

Degrees of freedom refer to the values in a study that has the freedom to vary and are essential for assessing the importance and the validity of the null hypothesis. Computation of these values usually depends upon the number of data records available in the sample set.

Paired Sample T-Test

The correlated t-test, or paired t-test, is a dependent type of test and is performed when the samples consist of matched pairs of similar units, or when there are cases of repeated measures. For example, there may be instances where the same patients are repeatedly tested before and after receiving a particular treatment. Each patient is being used as a control sample against themselves.

This method also applies to cases where the samples are related or have matching characteristics, like a comparative analysis involving children, parents, or siblings.

The formula for computing the t-value and degrees of freedom for a paired t-test is:

T = mean 1 − mean 2 s ( diff ) ( n ) where: mean 1  and  mean 2 = The average values of each of the sample sets s ( diff ) = The standard deviation of the differences of the paired data values n = The sample size (the number of paired differences) n − 1 = The degrees of freedom \begin{aligned}&T=\frac{\textit{mean}1 - \textit{mean}2}{\frac{s(\text{diff})}{\sqrt{(n)}}}\\&\textbf{where:}\\&\textit{mean}1\text{ and }\textit{mean}2=\text{The average values of each of the sample sets}\\&s(\text{diff})=\text{The standard deviation of the differences of the paired data values}\\&n=\text{The sample size (the number of paired differences)}\\&n-1=\text{The degrees of freedom}\end{aligned} ​ T = ( n ) ​ s ( diff ) ​ mean 1 − mean 2 ​ where: mean 1  and  mean 2 = The average values of each of the sample sets s ( diff ) = The standard deviation of the differences of the paired data values n = The sample size (the number of paired differences) n − 1 = The degrees of freedom ​

Equal Variance or Pooled T-Test

The equal variance t-test is an independent t-test and is used when the number of samples in each group is the same, or the variance of the two data sets is similar.

The formula used for calculating t-value and degrees of freedom for equal variance t-test is:

T-value = m e a n 1 − m e a n 2 ( n 1 − 1 ) × v a r 1 2 + ( n 2 − 1 ) × v a r 2 2 n 1 + n 2 − 2 × 1 n 1 + 1 n 2 where: m e a n 1  and  m e a n 2 = Average values of each of the sample sets v a r 1  and  v a r 2 = Variance of each of the sample sets n 1  and  n 2 = Number of records in each sample set \begin{aligned}&\text{T-value} = \frac{ mean1 - mean2 }{\frac {(n1 - 1) \times var1^2 + (n2 - 1) \times var2^2 }{ n1 +n2 - 2}\times \sqrt{ \frac{1}{n1} + \frac{1}{n2}} } \\&\textbf{where:}\\&mean1 \text{ and } mean2 = \text{Average values of each} \\&\text{of the sample sets}\\&var1 \text{ and } var2 = \text{Variance of each of the sample sets}\\&n1 \text{ and } n2 = \text{Number of records in each sample set} \end{aligned} ​ T-value = n 1 + n 2 − 2 ( n 1 − 1 ) × v a r 1 2 + ( n 2 − 1 ) × v a r 2 2 ​ × n 1 1 ​ + n 2 1 ​ ​ m e an 1 − m e an 2 ​ where: m e an 1  and  m e an 2 = Average values of each of the sample sets v a r 1  and  v a r 2 = Variance of each of the sample sets n 1  and  n 2 = Number of records in each sample set ​

Degrees of Freedom = n 1 + n 2 − 2 where: n 1  and  n 2 = Number of records in each sample set \begin{aligned} &\text{Degrees of Freedom} = n1 + n2 - 2 \\ &\textbf{where:}\\ &n1 \text{ and } n2 = \text{Number of records in each sample set} \\ \end{aligned} ​ Degrees of Freedom = n 1 + n 2 − 2 where: n 1  and  n 2 = Number of records in each sample set ​

Unequal Variance T-Test

The unequal variance t-test is an independent t-test and is used when the number of samples in each group is different, and the variance of the two data sets is also different. This test is also called Welch's t-test.

The formula used for calculating t-value and degrees of freedom for an unequal variance t-test is:

T-value = m e a n 1 − m e a n 2 ( v a r 1 n 1 + v a r 2 n 2 ) where: m e a n 1  and  m e a n 2 = Average values of each of the sample sets v a r 1  and  v a r 2 = Variance of each of the sample sets n 1  and  n 2 = Number of records in each sample set \begin{aligned}&\text{T-value}=\frac{mean1-mean2}{\sqrt{\bigg(\frac{var1}{n1}{+\frac{var2}{n2}\bigg)}}}\\&\textbf{where:}\\&mean1 \text{ and } mean2 = \text{Average values of each} \\&\text{of the sample sets} \\&var1 \text{ and } var2 = \text{Variance of each of the sample sets} \\&n1 \text{ and } n2 = \text{Number of records in each sample set} \end{aligned} ​ T-value = ( n 1 v a r 1 ​ + n 2 v a r 2 ​ ) ​ m e an 1 − m e an 2 ​ where: m e an 1  and  m e an 2 = Average values of each of the sample sets v a r 1  and  v a r 2 = Variance of each of the sample sets n 1  and  n 2 = Number of records in each sample set ​

Degrees of Freedom = ( v a r 1 2 n 1 + v a r 2 2 n 2 ) 2 ( v a r 1 2 n 1 ) 2 n 1 − 1 + ( v a r 2 2 n 2 ) 2 n 2 − 1 where: v a r 1  and  v a r 2 = Variance of each of the sample sets n 1  and  n 2 = Number of records in each sample set \begin{aligned} &\text{Degrees of Freedom} = \frac{ \left ( \frac{ var1^2 }{ n1 } + \frac{ var2^2 }{ n2 } \right )^2 }{ \frac{ \left ( \frac{ var1^2 }{ n1 } \right )^2 }{ n1 - 1 } + \frac{ \left ( \frac{ var2^2 }{ n2 } \right )^2 }{ n2 - 1}} \\ &\textbf{where:}\\ &var1 \text{ and } var2 = \text{Variance of each of the sample sets} \\ &n1 \text{ and } n2 = \text{Number of records in each sample set} \\ \end{aligned} ​ Degrees of Freedom = n 1 − 1 ( n 1 v a r 1 2 ​ ) 2 ​ + n 2 − 1 ( n 2 v a r 2 2 ​ ) 2 ​ ( n 1 v a r 1 2 ​ + n 2 v a r 2 2 ​ ) 2 ​ where: v a r 1  and  v a r 2 = Variance of each of the sample sets n 1  and  n 2 = Number of records in each sample set ​

The following flowchart can be used to determine which t-test to use based on the characteristics of the sample sets. The key items to consider include the similarity of the sample records, the number of data records in each sample set, and the variance of each sample set.

Image by Julie Bang © Investopedia 2019

Example of an Unequal Variance T-Test

Assume that the diagonal measurement of paintings received in an art gallery is taken. One group of samples includes 10 paintings, while the other includes 20 paintings. The data sets, with the corresponding mean and variance values, are as follows:

Though the mean of Set 2 is higher than that of Set 1, we cannot conclude that the population corresponding to Set 2 has a higher mean than the population corresponding to Set 1.

Is the difference from 19.4 to 21.6 due to chance alone, or do differences exist in the overall populations of all the paintings received in the art gallery? We establish the problem by assuming the null hypothesis that the mean is the same between the two sample sets and conduct a t-test to test if the hypothesis is plausible.

Since the number of data records is different (n1 = 10 and n2 = 20) and the variance is also different, the t-value and degrees of freedom are computed for the above data set using the formula mentioned in the Unequal Variance T-Test section.

The t-value is -2.24787. Since the minus sign can be ignored when comparing the two t-values, the computed value is 2.24787.

The degrees of freedom value is 24.38 and is reduced to 24, owing to the formula definition requiring rounding down of the value to the least possible integer value.

One can specify a level of probability (alpha level, level of significance,  p ) as a criterion for acceptance. In most cases, a 5% value can be assumed.

Using the degree of freedom value as 24 and a 5% level of significance, a look at the t-value distribution table gives a value of 2.064. Comparing this value against the computed value of 2.247 indicates that the calculated t-value is greater than the table value at a significance level of 5%. Therefore, it is safe to reject the null hypothesis that there is no difference between means. The population set has intrinsic differences, and they are not by chance.

How Is the T-Distribution Table Used?

The T-Distribution Table is available in one-tail and two-tails formats. The former is used for assessing cases that have a fixed value or range with a clear direction, either positive or negative. For instance, what is the probability of the output value remaining below -3, or getting more than seven when rolling a pair of dice? The latter is used for range-bound analysis, such as asking if the coordinates fall between -2 and +2.

What Is an Independent T-Test?

The samples of independent t-tests are selected independent of each other where the data sets in the two groups don’t refer to the same values. They may include a group of 100 randomly unrelated patients split into two groups of 50 patients each. One of the groups becomes the control group and is administered a placebo, while the other group receives a prescribed treatment. This constitutes two independent sample groups that are unpaired and unrelated to each other.

What Does a T-Test Explain and How Are They Used?

A t-test is a statistical test that is used to compare the means of two groups. It is often used in hypothesis testing to determine whether a process or treatment has an effect on the population of interest, or whether two groups are different from one another.

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The Ultimate Guide to T Tests

Get all of your t test questions answered here

The ultimate guide to t tests

The t test is one of the simplest statistical techniques that is used to evaluate whether there is a statistical difference between the means from up to two different samples. The t test is especially useful when you have a small number of sample observations (under 30 or so), and you want to make conclusions about the larger population.

The characteristics of the data dictate the appropriate type of t test to run. All t tests are used as standalone analyses for very simple experiments and research questions as well as to perform individual tests within more complicated statistical models such as linear regression. In this guide, we’ll lay out everything you need to know about t tests, including providing a simple workflow to determine what t test is appropriate for your particular data or if you’d be better suited using a different model.

What is a t test?

A t test is a statistical technique used to quantify the difference between the mean (average value) of a variable from up to two samples (datasets). The variable must be numeric. Some examples are height, gross income, and amount of weight lost on a particular diet.

A t test tells you if the difference you observe is “surprising” based on the expected difference. They use t-distributions to evaluate the expected variability. When you have a reasonable-sized sample (over 30 or so observations), the t test can still be used, but other tests that use the normal distribution (the z test) can be used in its place.

Sometimes t tests are called “Student’s” t tests, which is simply a reference to their unusual history.

It got its name because a brewer from the Guinness Brewery, William Gosset , published about the method under the pseudonym "Student". He wanted to get information out of very small sample sizes (often 3-5) because it took so much effort to brew each keg for his samples.

When should I use a t test?

A t test is appropriate to use when you’ve collected a small, random sample from some statistical “population” and want to compare the mean from your sample to another value. The value for comparison could be a fixed value (e.g., 10) or the mean of a second sample.

For example, if your variable of interest is the average height of sixth graders in your region, then you might measure the height of 25 or 30 randomly-selected sixth graders. A t test could be used to answer questions such as, “Is the average height greater than four feet?”

How does a t test work?

Based on your experiment, t tests make enough assumptions about your experiment to calculate an expected variability, and then they use that to determine if the observed data is statistically significant. To do this, t tests rely on an assumed “null hypothesis.” With the above example, the null hypothesis is that the average height is less than or equal to four feet.

Say that we measure the height of 5 randomly selected sixth graders and the average height is five feet. Does that mean that the “true” average height of all sixth graders is greater than four feet or did we randomly happen to measure taller than average students?

To evaluate this, we need a distribution that shows every possible average value resulting from a sample of five individuals in a population where the true mean is four. That may seem impossible to do, which is why there are particular assumptions that need to be made to perform a t test.

With those assumptions, then all that’s needed to determine the “sampling distribution of the mean” is the sample size (5 students in this case) and standard deviation of the data (let’s say it’s 1 foot).

That’s enough to create a graphic of the distribution of the mean, which is:

Notice the vertical line at x = 5, which was our sample mean. We (use software to) calculate the area to the right of the vertical line, which gives us the P value (0.09 in this case). Note that because our research question was asking if the average student is greater than four feet, the distribution is centered at four. Since we’re only interested in knowing if the average is greater than four feet, we use a one-tailed test in this case.

Using the standard confidence level of 0.05 with this example, we don’t have evidence that the true average height of sixth graders is taller than 4 feet.

What are the assumptions for t tests?

• One variable of interest : This is not correlation or regression, where you are interested in the relationship between multiple variables. With a t test, you can have different samples, but they are all measuring the same variable (e.g., height).
• Numeric data: You are dealing with a list of measurements that can be averaged. This means you aren’t just counting occurrences in various categories (e.g., eye color or political affiliation).
• Two groups or less: If you have more than two samples of data, a t test is the wrong technique. You most likely need to try ANOVA.
• Random sample : You need a random sample from your statistical “population of interest” in order to draw valid conclusions about the larger population. If your population is so small that you can measure everything, then you have a “census” and don’t need statistics. This is because you don’t need to estimate the truth, since you have measured the truth without variability.
• Normally Distributed : The smaller your sample size, the more important it is that your data come from a normal, Gaussian distribution bell curve. If you have reason to believe that your data are not normally distributed, consider nonparametric t test alternatives . This isn’t necessary for larger samples (usually 25 or 30 unless the data is heavily skewed). The reason is that the Central Limit Theorem applies in this case, which says that even if the distribution of your data is not normal, the distribution of the mean of your data is, so you can use a z-test rather than a t test.

How do I know which t test to use?

There are many types of t tests to choose from, but you don’t necessarily have to understand every detail behind each option.

You just need to be able to answer a few questions, which will lead you to pick the right t test. To that end, we put together this workflow for you to figure out which test is appropriate for your data.

Do you have one or two samples?

Are you comparing the means of two different samples, or comparing the mean from one sample to a fixed value? An example research question is, “Is the average height of my sample of sixth grade students greater than four feet?”

If you only have one sample of data, you can click here to skip to a one-sample t test example, otherwise your next step is to ask: 

Are observations in the two samples matched up or related in some way?

This could be as before-and-after measurements of the same exact subjects, or perhaps your study split up “pairs” of subjects (who are technically different but share certain characteristics of interest) into the two samples. The same variable is measured in both cases.

If so, you are looking at some kind of paired samples t test . The linked section will help you dial in exactly which one in that family is best for you, either difference (most common) or ratio.

If you aren’t sure paired is right, ask yourself another question:

Are you comparing different observations in each of the two samples?

If the answer is yes, then you have an unpaired or independent samples t test. The two samples should measure the same variable (e.g., height), but are samples from two distinct groups (e.g., team A and team B). 

The goal is to compare the means to see if the groups are significantly different. For example, “Is the average height of team A greater than team B?” Unlike paired, the only relationship between the groups in this case is that we measured the same variable for both. There are two versions of unpaired samples t tests (pooled and unpooled) depending on whether you assume the same variance for each sample.

Have you run the same experiment multiple times on the same subject/observational unit?

If so, then you have a nested t test (unless you have more than two sample groups). This is a trickier concept to understand. One example is if you are measuring how well Fertilizer A works against Fertilizer B. Let’s say you have 12 pots to grow plants in (6 pots for each fertilizer), and you grow 3 plants in each pot.

In this case you have 6 observational units for each fertilizer, with 3 subsamples from each pot. You would want to analyze this with a nested t test . The “nested” factor in this case is the pots. It’s important to note that we aren’t interested in estimating the variability within each pot, we just want to take it into account.

You might be tempted to run an unpaired samples t test here, but that assumes you have 6*3 = 18 replicates for each fertilizer. However, the three replicates within each pot are related, and an unpaired samples t test wouldn’t take that into account.

What if none of these sound like my experiment?

If you’re not seeing your research question above, note that t tests are very basic statistical tools. Many experiments require more sophisticated techniques to evaluate differences. If the variable of interest is a proportion (e.g., 10 of 100 manufactured products were defective), then you’d use z-tests. If you take before and after measurements and have more than one treatment (e.g., control vs a treatment diet), then you need ANOVA.

How do I perform a t test using software?

If you’re wondering how to do a t test, the easiest way is with statistical software such as Prism or an online t test calculator .

If you’re using software, then all you need to know is which t test is appropriate ( use the workflow here ) and understand how to interpret the output. To do that, you’ll also need to:

• Determine whether your test is one or two-tailed
• Choose the level of significance

Is my test one or two-tailed? 

Whether or not you have a one- or two-tailed test depends on your research hypothesis. Choosing the appropriately tailed test is very important and requires integrity from the researcher. This is because you have more “power” with one-tailed tests, meaning that you can detect a statistically significant difference more easily. Unless you have written out your research hypothesis as one directional before you run your experiment, you should use a two-tailed test.

Two-tailed tests

Two-tailed tests are the most common, and they are applicable when your research question is simply asking, “is there a difference?”

One-tailed tests

Contrast that with one-tailed tests, where the research questions are directional, meaning that either the question is, “is it greater than ” or the question is, “is it less than ”. These tests can only detect a difference in one direction.

Choosing the level of significance

All t tests estimate whether a mean of a population is different than some other value, and with all estimates come some variability, or what statisticians call “error.” Before analyzing your data, you want to choose a level of significance, usually denoted by the Greek letter alpha, 𝛼. The scientific standard is setting alpha to be 0.05.

An alpha of 0.05 results in 95% confidence intervals, and determines the cutoff for when P values are considered statistically significant.

One sample t test

If you only have one sample of a list of numbers, you are doing a one-sample t test. All you are interested in doing is comparing the mean from this group with some known value to test if there is evidence, that it is significantly different from that standard. Use our free one-sample t test calculator for this.

A one sample t test example research question is, “Is the average fifth grader taller than four feet?”

It is the simplest version of a t test, and has all sorts of applications within hypothesis testing. Sometimes the “known value” is called the “null value”. While the null value in t tests is often 0, it could be any value. The name comes from being the value which exactly represents the null hypothesis, where no significant difference exists. 

Any time you know the exact number you are trying to compare your sample of data against, this could work well. And of course: it can be either one or two-tailed.

One sample t test formula

Statistical software handles this for you, but if you want the details, the formula for a one sample t test is:

• M: Calculated mean of your sample
• μ: Hypothetical mean you are testing against
• s: The standard deviation of your sample
• n: The number of observations in your sample.

In a one-sample t test, calculating degrees of freedom is simple: one less than the number of objects in your dataset (you’ll see it written as n-1 ).

Example of a one sample t test

For our example within Prism, we have a dataset of 12 values from an experiment labeled “% of control”. Perhaps these are heights of a sample of plants that have been treated with a new fertilizer. A value of 100 represents the industry-standard control height. Likewise, 123 represents a plant with a height 123% that of the control (that is, 23% larger).

We’ll perform a two-tailed, one-sample t test to see if plants are shorter or taller on average with the fertilizer. We will use a significance threshold of 0.05. Here is the output:

You can see in the output that the actual sample mean was 111. Is that different enough from the industry standard (100) to conclude that there is a statistical difference?

The quick answer is yes, there’s strong evidence that the height of the plants with the fertilizer is greater than the industry standard (p=0.015). The nice thing about using software is that it handles some of the trickier steps for you. In this case, it calculates your test statistic (t=2.88), determines the appropriate degrees of freedom (11), and outputs a P value.

More informative than the P value is the confidence interval of the difference, which is 2.49 to 18.7. The confidence interval tells us that, based on our data, we are confident that the true difference between our sample and the baseline value of 100 is somewhere between 2.49 and 18.7. As long as the difference is statistically significant, the interval will not contain zero.

You can follow these tips for interpreting your own one-sample test.

Graphing a one-sample t test

For some techniques (like regression), graphing the data is a very helpful part of the analysis. For t tests, making a chart of your data is still useful to spot any strange patterns or outliers, but the small sample size means you may already be familiar with any strange things in your data.

Here we have a simple plot of the data points, perhaps with a mark for the average. We’ve made this as an example, but the truth is that graphing is usually more visually telling for two-sample t tests than for just one sample.

Two sample t tests

There are several kinds of two sample t tests, with the two main categories being paired and unpaired (independent) samples.

Paired samples t test

In a paired samples t test, also called dependent samples t test, there are two samples of data, and each observation in one sample is “paired” with an observation in the second sample. The most common example is when measurements are taken on each subject before and after a treatment. A paired t test example research question is, “Is there a statistical difference between the average red blood cell counts before and after a treatment?”

Having two samples that are closely related simplifies the analysis. Statistical software, such as this paired t test calculator , will simply take a difference between the two values, and then compare that difference to 0.

In some (rare) situations, taking a difference between the pairs violates the assumptions of a t test, because the average difference changes based on the size of the before value (e.g., there’s a larger difference between before and after when there were more to start with). In this case, instead of using a difference test, use a ratio of the before and after values, which is referred to as ratio t tests .

Paired t test formula

The formula for paired samples t test is:

• Md: Mean difference between the samples
• sd: The standard deviation of the differences
• n: The number of differences

Degrees of freedom are the same as before. If you’re studying for an exam, you can remember that the degrees of freedom are still n-1 (not n-2) because we are converting the data into a single column of differences rather than considering the two groups independently.

Also note that the null value here is simply 0. There is no real reason to include “minus 0” in an equation other than to illustrate that we are still doing a hypothesis test. After you take the difference between the two means, you are comparing that difference to 0.

For our example data, we have five test subjects and have taken two measurements from each: before (“control”) and after a treatment (“treated”). If we set alpha = 0.05 and perform a two-tailed test, we observe a statistically significant difference between the treated and control group (p=0.0160, t=4.01, df = 4). We are 95% confident that the true mean difference between the treated and control group is between 0.449 and 2.47.

Graphing a paired t test

The significant result of the P value suggests evidence that the treatment had some effect, and we can also look at this graphically. The lines that connect the observations can help us spot a pattern, if it exists. In this case the lines show that all observations increased after treatment. While not all graphics are this straightforward, here it is very consistent with the outcome of the t test. 

Prism’s estimation plot is even more helpful because it shows both the data (like above) and the confidence interval for the difference between means. You can easily see the evidence of significance since the confidence interval on the right does not contain zero.

Here are some more graphing tips for paired t tests .

Unpaired samples t test

Unpaired samples t test, also called independent samples t test, is appropriate when you have two sample groups that aren’t correlated with one another. A pharma example is testing a treatment group against a control group of different subjects. Compare that with a paired sample, which might be recording the same subjects before and after a treatment.

With unpaired t tests, in addition to choosing your level of significance and a one or two tailed test, you need to determine whether or not to assume that the variances between the groups are the same or not. If you assume equal variances, then you can “pool” the calculation of the standard error between the two samples. Otherwise, the standard choice is Welch’s t test which corrects for unequal variances. This choice affects the calculation of the test statistic and the power of the test, which is the test’s sensitivity to detect statistical significance. 

It’s best to choose whether or not you’ll use a pooled or unpooled (Welch’s) standard error before running your experiment, because the standard statistical test is notoriously problematic. See more details about unequal variances here .

As long as you’re using statistical software, such as this two-sample t test calculator , it’s just as easy to calculate a test statistic whether or not you assume that the variances of your two samples are the same. If you’re doing it by hand, however, the calculations get more complicated with unequal variances.

Unpaired (independent) samples t test formula

The general two-sample t test formula is:

• M1 and M2: Two means you are comparing, one from each dataset
• SE : The combined standard error of the two samples (calculated using pooled or unpooled standard error)

The denominator (standard error) calculation can be complicated, as can the degrees of freedom. If the groups are not balanced (the same number of observations in each), you will need to account for both when determining n for the test as a whole.

As an example for this family, we conduct a paired samples t test assuming equal variances (pooled). Based on our research hypothesis, we’ll conduct a two-tailed test, and use alpha=0.05 for our level of significance. Our samples were unbalanced, with two samples of 6 and 5 observations respectively. 

The P value (p=0.261, t = 1.20, df = 9) is higher than our threshold of 0.05. We have not found sufficient evidence to suggest a significant difference. You can see the confidence interval of the difference of the means is -9.58 to 31.2.

Note that the F-test result shows that the variances of the two groups are not significantly different from each other.

Graphing an unpaired samples t test

For an unpaired samples t test, graphing the data can quickly help you get a handle on the two groups and how similar or different they are. Like the paired example, this helps confirm the evidence (or lack thereof) that is found by doing the t test itself.

Below you can see that the observed mean for females is higher than that for males. But because of the variability in the data, we can’t tell if the means are actually different or if the difference is just by chance. 

Nonparametric alternatives for t tests

If your data comes from a normal distribution (or something close enough to a normal distribution), then a t test is valid. If that assumption is violated, you can use nonparametric alternatives.

T tests evaluate whether the mean is different from another value, whereas nonparametric alternatives compare either the median or the rank. Medians are well-known to be much more robust to outliers than the mean.

The downside to nonparametric tests is that they don’t have as much statistical power, meaning a larger difference is required in order to determine that it’s statistically significant.

Wilcoxon signed-rank test

The Wilcoxon signed-rank test is the nonparametric cousin to the one-sample t test. This compares a sample median to a hypothetical median value. It is sometimes erroneously even called the Wilcoxon t test (even though it calculates a “W” statistic).

And if you have two related samples, you should use the Wilcoxon matched pairs test instead. The two versions of Wilcoxon are different, and the matched pairs version is specifically for comparing the median difference for paired samples. 

Mann-Whitney and Kolmogorov-Smirnov tests

For unpaired (independent) samples, there are multiple options for nonparametric testing. Mann-Whitney is more popular and compares the mean ranks (the ordering of values from smallest to largest) of the two samples. Mann-Whitney is often misrepresented as a comparison of medians, but that’s not always the case. Kolmogorov-Smirnov tests if the overall distributions differ between the two samples. 

More t test FAQs

What is the formula for a t test.

The exact formula depends on which type of t test you are running, although there is a basic structure that all t tests have in common. All t test statistics will have the form:

• t : The t test statistic you calculate for your test
• Mean1 and Mean2: Two means you are comparing, at least 1 from your own dataset
• Standard Error of the Mean : The standard error of the mean , also called the standard deviation of the mean, which takes into account the variance and size of your dataset

The exact formula for any t test can be slightly different, particularly the calculation of the standard error. Not only does it matter whether one or two samples are being compared, the relationship between the samples can make a difference too.

What is a t-distribution?

A t-distribution is similar to a normal distribution. It’s a bell-shaped curve, but compared to a normal it has fatter tails, which means that it’s more common to observe extremes. T-distributions are identified by the number of degrees of freedom. The higher the number, the closer the t-distribution gets to a normal distribution. After about 30 degrees of freedom, a t and a standard normal are practically the same.

What are degrees of freedom?

Degrees of freedom are a measure of how large your dataset is. They aren’t exactly the number of observations, because they also take into account the number of parameters (e.g., mean, variance) that you have estimated.

What is the difference between paired vs unpaired t tests?

Both paired and unpaired t tests involve two sample groups of data. With a paired t test, the values in each group are related (usually they are before and after values measured on the same test subject). In contrast, with unpaired t tests, the observed values aren’t related between groups. An unpaired, or independent t test, example is comparing the average height of children at school A vs school B. 

When do I use a z-test versus a t test?

Z-tests, which compare data using a normal distribution rather than a t-distribution, are primarily used for two situations. The first is when you’re evaluating proportions (number of failures on an assembly line). The second is when your sample size is large enough (usually around 30) that you can use a normal approximation to evaluate the means.

When should I use ANOVA instead of a t test?

Use ANOVA if you have more than two group means to compare.

What are the differences between t test vs chi square?

Chi square tests are used to evaluate contingency tables , which record a count of the number of subjects that fall into particular categories (e.g., truck, SUV, car). t tests compare the mean(s) of a variable of interest (e.g., height, weight).

What are P values?

P values are the probability that you would get data as or more extreme than the observed data given that the null hypothesis is true. It’s a mouthful, and there are a lot of issues to be aware of with P values.

What are t test critical values?

Critical values are a classical form (they aren’t used directly with modern computing) of determining if a statistical test is significant or not. Historically you could calculate your test statistic from your data, and then use a t-table to look up the cutoff value (critical value) that represented a “significant” result. You would then compare your observed statistic against the critical value.

How do I calculate degrees of freedom for my t test?

In most practical usage, degrees of freedom are the number of observations you have minus the number of parameters you are trying to estimate. The calculation isn’t always straightforward and is approximated for some t tests.

Statistical software calculates degrees of freedom automatically as part of the analysis, so understanding them in more detail isn’t needed beyond assuaging any curiosity.

Perform your own t test

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Neag School of Education

Educational Research Basics by Del Siegle

An introduction to statistics usually covers t tests, anovas, and chi-square. for this course we will concentrate on t tests, although background information will be provided on anovas and chi-square., a powerpoint presentation on t tests has been created for your use..

The t test is one type of inferential statistics. It is used to determine whether there is a significant difference between the means of two groups. With all inferential statistics, we assume the dependent variable fits a normal distribution . When we assume a normal distribution exists, we can identify the probability of a particular outcome. We specify the level of probability (alpha level, level of significance, p ) we are willing to accept before we collect data ( p < .05 is a common value that is used). After we collect data we calculate a test statistic with a formula. We compare our test statistic with a critical value found on a table to see if our results fall within the acceptable level of probability. Modern computer programs calculate the test statistic for us and also provide the exact probability of obtaining that test statistic with the number of subjects we have.

Student’s test ( t test) Notes

When the difference between two population averages is being investigated, a t test is used. In other words, a t test is used when we wish to compare two means (the scores must be measured on an interval or ratio measurement scale ). We would use a t test if we wished to compare the reading achievement of boys and girls. With a t test, we have one independent variable and one dependent variable. The independent variable (gender in this case) can only have two levels (male and female). The dependent variable would be reading achievement. If the independent had more than two levels, then we would use a one-way analysis of variance (ANOVA).

The test statistic that a t test produces is a t -value. Conceptually, t -values are an extension of z -scores. In a way, the t -value represents how many standard units the means of the two groups are apart.

With a t tes t, the researcher wants to state with some degree of confidence that the obtained difference between the means of the sample groups is too great to be a chance event and that some difference also exists in the population from which the sample was drawn. In other words, the difference that we might find between the boys’ and girls’ reading achievement in our sample might have occurred by chance, or it might exist in the population. If our t test produces a t -value that results in a probability of .01, we say that the likelihood of getting the difference we found by chance would be 1 in a 100 times. We could say that it is unlikely that our results occurred by chance and the difference we found in the sample probably exists in the populations from which it was drawn.

Five factors contribute to whether the difference between two groups’ means can be considered significant:

• How large is the difference between the means of the two groups? Other factors being equal, the greater the difference between the two means, the greater the likelihood that a statistically significant mean difference exists. If the means of the two groups are far apart, we can be fairly confident that there is a real difference between them.
• How much overlap is there between the groups? This is a function of the variation within the groups. Other factors being equal, the smaller the variances of the two groups under consideration, the greater the likelihood that a statistically significant mean difference exists. We can be more confident that two groups differ when the scores within each group are close together.
• How many subjects are in the two samples? The size of the sample is extremely important in determining the significance of the difference between means. With increased sample size, means tend to become more stable representations of group performance. If the difference we find remains constant as we collect more and more data, we become more confident that we can trust the difference we are finding.
• What alpha level is being used to test the mean difference (how confident do you want to be about your statement that there is a mean difference). A larger alpha level requires less difference between the means. It is much harder to find differences between groups when you are only willing to have your results occur by chance 1 out of a 100 times ( p < .01) as compared to 5 out of 100 times ( p < .05).
• Is a directional (one-tailed) or non-directional (two-tailed) hypothesis being tested? Other factors being equal, smaller mean differences result in statistical significance with a directional hypothesis. For our purposes we will use non-directional (two-tailed) hypotheses.

I have created an Excel spreadsheet that performs t-tests (with a PowerPoint presentation that explains how enter data and read it) and a PowerPoint presentation on t tests   (you will probably find this useful).

Assumptions underlying the t test.

• The samples have been randomly drawn from their respective populations
• The scores in the population are normally distributed
• The scores in the populations have the same variance (s1=s2) Note: We use a different calculation for the standard error if they are not.

Three Types of t tests

• Pair-difference t test (a.k.a. t-test for dependent groups, correlated t test) df = n (number of pairs) -1

This is concerned with the difference between the average scores of a single sample of individuals who are assessed at two different times (such as before treatment and after treatment). It can also compare average scores of samples of individuals who are paired in some way (such as siblings, mothers, daughters, persons who are matched in terms of a particular characteristics).

• Equal Variance (Pooled-variance t-test) df=n (total of both groups) -2      Note: Used when both samples have the same number of subject or when s1=s2 (Levene or F-max tests have p > .05).
• Unequal Variance (Separate-variance t test) df dependents on a formula, but a rough estimate is one less than the smallest group    Note: Used when the samples have different numbers of subjects and they have different variances —  s1<>s2 (Levene or F-max tests have p < .05).

How do I decide which type of t test to use?

Note: The F-Max test can be substituted for the Levene test. The   t test Excel spreadsheet that I created for our class uses the F -Max.

Type I and II errors

• Type I error — reject a null hypothesis that is really true (with tests of difference this means that you say there was a difference between the groups when there really was not a difference). The probability of making a Type I error is the alpha level you choose. If you set your probability (alpha level) at p   < 05, then there is a 5% chance that you will make a Type I error. You can reduce the chance of making a Type I error by setting a smaller alpha level ( p < .01). The problem with this is that as you lower the chance of making a Type I error, you increase the chance of making a Type II error.
• Type II error — fail to reject a null hypothesis that is false (with tests of differences this means that you say there was no difference between the groups when there really was one)

Hypotheses (some ideas…)

• Non directional (two-tailed) Research Question: Is there a (statistically) significant difference between males and females with respect to math achievement? H0: There is no (statistically) significant difference between males and females with respect to math achievement. HA: There is a (statistically) significant difference between males and females with respect to math achievement.
• Directional (one-tailed) Research Question: Do males score significantly higher than females with respect to math achievement? H0: Males do not score significantly higher than females with respect to math achievement. HA: Males score significantly higher than females with respect to math achievement. The basic idea for calculating a t-test is to find the difference between the means of the two groups and divide it by the STANDARD ERROR (OF THE DIFFERENCE) — which is the standard deviation of the distribution of differences. Just for your information: A CONFIDENCE INTERVAL for a two-tailed t-test is calculated by multiplying the CRITICAL VALUE times the STANDARD ERROR and adding and subtracting that to and from the difference of the two means. EFFECT SIZE is used to calculate practical difference.   If you have several thousand subjects, it is very easy to find a statistically significant difference. Whether that difference is practical or meaningful is another questions. This is where effect size becomes important. With studies involving group differences, effect size is the difference of the two means divided by the standard deviation of the control group (or the average standard deviation of both groups if you do not have a control group). Generally, effect size is only important if you have statistical significance. An effect size of .2 is considered small, .5 is considered medium, and .8 is considered large.

A bit of history… William Sealy Gosset (1905) first published a t-test. He worked at the Guiness Brewery in Dublin and published under the name Student. The test was called Studen t Test (later shortened to t test).

t tests can be easily computed with the Excel or SPSS computer  application. I have created an Excel Spreadsheet that does a very nice job of calculating t values and other pertinent information.

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The t-test assesses whether the means of two groups are statistically different from each other. This analysis is appropriate whenever you want to compare the means of two groups, and especially appropriate as the analysis for the posttest-only two-group randomized experimental design .

Figure 1 shows the distributions for the treated (blue) and control (green) groups in a study. Actually, the figure shows the idealized distribution – the actual distribution would usually be depicted with a histogram or bar graph . The figure indicates where the control and treatment group means are located. The question the t-test addresses is whether the means are statistically different.

What does it mean to say that the averages for two groups are statistically different? Consider the three situations shown in Figure 2. The first thing to notice about the three situations is that the difference between the means is the same in all three . But, you should also notice that the three situations don’t look the same – they tell very different stories. The top example shows a case with moderate variability of scores within each group. The second situation shows the high variability case. the third shows the case with low variability. Clearly, we would conclude that the two groups appear most different or distinct in the bottom or low-variability case. Why? Because there is relatively little overlap between the two bell-shaped curves. In the high variability case, the group difference appears least striking because the two bell-shaped distributions overlap so much.

This leads us to a very important conclusion: when we are looking at the differences between scores for two groups, we have to judge the difference between their means relative to the spread or variability of their scores. The t-test does just this.

Statistical Analysis of the t-test

The formula for the t-test is a ratio. The top part of the ratio is just the difference between the two means or averages. The bottom part is a measure of the variability or dispersion of the scores. This formula is essentially another example of the signal-to-noise metaphor in research: the difference between the means is the signal that, in this case, we think our program or treatment introduced into the data; the bottom part of the formula is a measure of variability that is essentially noise that may make it harder to see the group difference. Figure 3 shows the formula for the t-test and how the numerator and denominator are related to the distributions.

The top part of the formula is easy to compute – just find the difference between the means. The bottom part is called the standard error of the difference . To compute it, we take the variance for each group and divide it by the number of people in that group. We add these two values and then take their square root. The specific formula for the standard error of the difference between the means is:

Remember, that the variance is simply the square of the standard deviation .

The final formula for the t-test is:

The t -value will be positive if the first mean is larger than the second and negative if it is smaller. Once you compute the t -value you have to look it up in a table of significance to test whether the ratio is large enough to say that the difference between the groups is not likely to have been a chance finding. To test the significance, you need to set a risk level (called the alpha level ). In most social research, the “rule of thumb” is to set the alpha level at .05 . This means that five times out of a hundred you would find a statistically significant difference between the means even if there was none (i.e. by “chance”). You also need to determine the degrees of freedom (df) for the test. In the t-test , the degrees of freedom is the sum of the persons in both groups minus 2 . Given the alpha level, the df, and the t -value, you can look the t -value up in a standard table of significance (available as an appendix in the back of most statistics texts) to determine whether the t -value is large enough to be significant. If it is, you can conclude that the difference between the means for the two groups is different (even given the variability). Fortunately, statistical computer programs routinely print the significance test results and save you the trouble of looking them up in a table.

The t-test, one-way Analysis of Variance (ANOVA) and a form of regression analysis are mathematically equivalent (see the statistical analysis of the posttest-only randomized experimental design ) and would yield identical results.

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Dependent t-test for paired samples

What does this test do.

The dependent t-test (also called the paired t-test or paired-samples t-test) compares the means of two related groups to determine whether there is a statistically significant difference between these means.

What variables do you need for a dependent t-test?

You need one dependent variable that is measured on an interval or ratio scale (see our Types of Variable guide if you need clarification). You also need one categorical variable that has only two related groups.

What is meant by "related groups"?

A dependent t-test is an example of a "within-subjects" or "repeated-measures" statistical test. This indicates that the same participants are tested more than once. Thus, in the dependent t-test, "related groups" indicates that the same participants are present in both groups. The reason that it is possible to have the same participants in each group is because each participant has been measured on two occasions on the same dependent variable. For example, you might have measured the performance of 10 participants in a spelling test (the dependent variable) before and after they underwent a new form of computerised teaching method to improve spelling. You would like to know if the computer training improved their spelling performance. Here, we can use a dependent t-test because we have two related groups. The first related group consists of the participants at the beginning (prior to) the computerised spell training and the second related group consists of the same participants, but now at the end of the computerised training.

Does the dependent t-test test for "changes" or "differences" between related groups?

The dependent t-test can be used to test either a "change" or a "difference" in means between two related groups, but not both at the same time. Whether you are measuring a "change" or "difference" between the means of the two related groups depends on your study design. The two types of study design are indicated in the following diagrams.

How do you detect differences between experimental conditions using the dependent t-test?

The dependent t-test can look for "differences" between means when participants are measured on the same dependent variable under two different conditions. For example, you might have tested participants' eyesight (dependent variable) when wearing two different types of spectacle (independent variable). See the diagram below for a general schematic of this design approach (click the image to enlarge):

Find out more about the dependent t-test on the next page .

Statistics Made Easy

Paired Samples t-test: Definition, Formula, and Example

A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.

This tutorial explains the following:

• The motivation for performing a paired samples t-test.
• The formula to perform a paired samples t-test.
• The assumptions that should be met to perform a paired samples t-test.
• An example of how to perform a paired samples t-test.

Paired Samples t-test: Motivation

A paired samples t-test is commonly used in two scenarios:

1. A measurement is taken on a subject before and after some treatment – e.g. the max vertical jump of college basketball players is measured before and after participating in a training program.

2. A measurement is taken under two different conditions  – e.g. the response time of a patient is measured on two different drugs.

In both cases we are interested in comparing the mean measurement between two groups in which each observation in one sample can be paired with an observation in the other sample.

Paired Samples t-test: Formula

A paired samples 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:

• H 1 (two-tailed): μ 1  ≠ μ 2 (the two population means are not equal)
• H 1 (left-tailed): μ 1  < μ 2  (population 1 mean is less than population 2 mean)
• H 1 (right-tailed):  μ 1 > μ 2  (population 1 mean is greater than population 2 mean)

We use the following formula to calculate the test statistic t:

t = x diff  / (s diff /√n)

• x diff :  sample mean of the differences
• s:  sample standard deviation of the differences
• n:  sample size (i.e. number of pairs)

If the p-value that corresponds to the test statistic t with (n-1) degrees of freedom is less than your chosen significance level (common choices are 0.10, 0.05, and 0.01) then you can reject the null hypothesis.

Paired Samples t-test: Assumptions

For the results of a paired samples t-test to be valid, the following assumptions should be met:

• The participants should be selected randomly from the population.
• The differences between the pairs should be approximately normally distributed.
• There should be no extreme outliers in the differences.

Paired Samples t-test : Example

Suppose we want to know whether or not a certain training program is able to increase the max vertical jump (in inches) of college basketball players.

To test this, we may recruit a simple random sample of 20 college basketball players and measure each of their max vertical jumps. Then, we may have each player use the training program for one month and then measure their max vertical jump again at the end of the month.

To determine whether or not the training program actually had an effect on max vertical jump, we will perform a paired samples t-test at significance level α = 0.05 using the following steps:

Step 1: Calculate the summary data for the differences.

• x diff :  sample mean of the differences =  -0.95
• s:  sample standard deviation of the differences =  1.317
• n:  sample size (i.e. number of pairs) =  20

Step 2: Define the hypotheses.

We will perform the paired samples t-test with the following hypotheses:

• H 0 :  μ 1  = μ 2 (the two population means are equal)
• H 1 :  μ 1  ≠ μ 2 (the two population means are not equal)

Step 3: Calculate the test statistic  t .

t = x diff  / (s diff /√n)  = -0.95 / (1.317/ √ 20) =  -3.226

Step 4: Calculate the p-value of the test statistic  t .

According to the T Score to P Value Calculator , the p-value associated with t = -3.226 and degrees of freedom = n-1 = 20-1 = 19 is  0.00445 .

Step 5: Draw a conclusion.

Since this p-value is less than our significance level α = 0.05, we reject the null hypothesis. We have sufficient evidence to say that the mean max vertical jump of players is different before and after participating in the training program.

Note:  You can also perform this entire paired samples t-test by simply using the Paired Samples t-test Calculator .

Additional Resources

The following tutorials explain how to perform a paired samples t-test using different statistical programs:

How to Perform a Paired Samples t-Test in Excel How to Perform a Paired Samples t-test in SPSS How to Perform a Paired Samples t-test in Stata How to Perform a Paired Samples t-test on a TI-84 Calculator How to Perform a Paired Samples t-test in R How to Perform a Paired Samples t-Test in Python How to Perform a Paired Samples t-Test by Hand

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

5 Replies to “Paired Samples t-test: Definition, Formula, and Example”

What will happen if the p-value is more than the significance level of 0.05? Does it mean it is not significant or effective?

Thank you for this. It helped me a lot.

Excellent it really helps me

Hey Zach , thanks for statology! I love this site. It helps me a lot to remember and learn new things. I have a question. What if rather then test if the difference between tests is zero or greater/lower than zero, I’d like to test if the difference is at least x? Eg: I’d like to check if the time to change tire was reduced by at least 5 seconds after a training. Not only if the time is different or less than before. Could you include this example, please? Which changes in calculation we should do? Thanks a lot. Marcelo

in paired t test are we supposed to deduct the before from the after ?

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