t distribution formula hypothesis test

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• T Test Formula

T-Test Formula

The t-test is any statistical hypothesis test in which the test statistic follows a Student’s t-distribution under the null hypothesis. It can be used to determine if two sets of data are significantly different from each other, and is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known.

T-test uses means and standard deviations of two samples to make a comparison. The formula for T-test is given below:

\begin{array}{l}\qquad t=\frac{\bar{X}_{1}-\bar{X}_{2}}{s_{\bar{\Delta}}} \\ \text { where } \\ \qquad s_{\bar{\Delta}}=\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}} \\ \end{array}

Where, \(\begin{array}{l}\overline{x}\end{array} \) = Mean of first set of values \(\begin{array}{l}\overline{x}_{2}\end{array} \)  = Mean of second set of values \(\begin{array}{l}S_{1}\end{array} \)   = Standard deviation of first set of values \(\begin{array}{l}S_{2}\end{array} \)   = Standard deviation of second set of values \(\begin{array}{l}n_{1}\end{array} \)   = Total number of values in first set \(\begin{array}{l}n_{2}\end{array} \)   = Total number of values in second set.

The formula for standard deviation is given by:

Where, x = Values given \(\begin{array}{l}\overline{x}\end{array} \) = Mean n = Total number of values.

T-Test Solved Examples

Question 1: Find the t-test value for the following two sets of values: 7, 2, 9, 8 and 1, 2, 3, 4?

Formula for standard deviation:  \(\begin{array}{l}S=\sqrt{\frac{\sum\left(x-\overline{x}\right)^{2}}{n-1}}\end{array} \)

Number of terms in first set:  \(\begin{array}{l}n_{1}\end{array} \) = 4

Mean for first set of data: \(\begin{array}{l}\overline{x}_{1}\end{array} \) = 6.5

Construct the following table for standard deviation:

Standard deviation for the first set of data: S 1 = 3.11

Number of terms in second set: n 2 = 4

Standard deviation for first set of data: \(\begin{array}{l}S_{2}\end{array} \) = 1.29

Formula for t-test value:

t = 2.3764 = 2.36 (approx)

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• T-distribution: What it is and how to use it

T-Distribution | What It Is and How To Use It (With Examples)

Published on August 28, 2020 by Rebecca Bevans . Revised on June 21, 2023.

The t -distribution, also known as Student’s t -distribution, is a way of describing data that follow a bell curve when plotted on a graph, with the greatest number of observations close to the mean and fewer observations in the tails.

It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.

In statistics, the t -distribution is most often used to:

• Find the critical values for a confidence interval when the data is approximately normally distributed.
• Find the corresponding p -value from a statistical test that uses the t -distribution ( t -tests , regression analysis ).

Table of contents

• What is a t-distribution?
• T-distribution and the standard normal distribution
• T-distribution and t-scores

Other interesting articles

Frequently asked questions about the t-distribution, what is a t -distribution.

The t -distribution is a type of normal distribution that is used for smaller sample sizes. Normally-distributed data form a bell shape when plotted on a graph, with more observations near the mean and fewer observations in the tails.

The t -distribution is used when data are approximately normally distributed, which means the data follow a bell shape but the population variance is unknown. The variance in a t -distribution is estimated based on the degrees of freedom of the data set (total number of observations minus 1).

It is a more conservative form of the standard normal distribution , also known as the z -distribution. This means that it gives a lower probability to the center and a higher probability to the tails than the standard normal distribution.

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T -distribution and the standard normal distribution

As the degrees of freedom (total number of observations minus 1) increases, the t -distribution will get closer and closer to matching the standard normal distribution, a.k.a. the z -distribution, until they are almost identical.

Above 30 degrees of freedom, the t -distribution roughly matches the z -distribution. Therefore, the z -distribution can be used in place of the t -distribution with large sample sizes.

The z -distribution is preferable over the t -distribution when it comes to making statistical estimates because it has a known variance. It can make more precise estimates than the t -distribution, whose variance is approximated using the degrees of freedom of the data.

T -distribution and t -scores

A t -score is the number of standard deviations from the mean in a t -distribution. You can typically look up a t -score in a t -table , or by using an online t -score calculator.

In statistics, t -scores are primarily used to find two things:

• The upper and lower bounds of a confidence interval when the data are approximately normally distributed.
• The p -value of the test statistic for t -tests and regression tests.

T -scores and confidence intervals

Confidence intervals use t -scores to calculate the upper and lower bounds of the prediction interval. The t -score used to generate the upper and lower bounds is also known as the critical value of t , or t *.

Using a two-tailed t -test, you generate an estimate of the difference between the two classes and a confidence interval around that estimate. From the t -test you find the difference in average score between class 1 and class 2 is 4.61, with a 95% confidence interval of 3.87 to 5.35.

Because the confidence interval does not cross zero, and is in fact quite far from zero, it is unlikely that this difference in test scores could have occurred under the null hypothesis of no difference between groups.

T -scores and  p -values

Statistical tests generate a test statistic showing how far from the null hypothesis of the statistical test your data is. They then calculate a  p -value that describes the likelihood of your data occurring if the null hypothesis were true.

The test statistic for t -tests and regression tests is the t -score. While most statistical programs will automatically calculate the corresponding p -value for the t -score, you can also look up the values in a t -table, using your degrees of freedom and t -score to find the p -value.

The t -score which generates a p -value below your threshold for statistical significance is known as the critical value of t , or t *.

The degrees of freedom is 38 (n–1 for each group). Looking this up in a t -table (or calculating it in your favorite stats program) you find a p -value < 0.001.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s t table
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

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The t -distribution is a way of describing a set of observations where most observations fall close to the mean , and the rest of the observations make up the tails on either side. It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.

The t -distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the standard deviation .

The t -distribution gives more probability to observations in the tails of the distribution than the standard normal distribution (a.k.a. the z -distribution).

In this way, the t -distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance , you will need to include a wider range of the data.

A t -score (a.k.a. a t -value) is equivalent to the number of standard deviations away from the mean of the t -distribution .

The t -score is the test statistic used in t -tests and regression tests. It can also be used to describe how far from the mean an observation is when the data follow a t -distribution.

A test statistic is a number calculated by a  statistical test . It describes how far your observed data is from the  null hypothesis  of no relationship between  variables or no difference among sample groups.

The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.

A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval , or which defines the threshold of statistical significance in a statistical test. It describes how far from the mean of the distribution you have to go to cover a certain amount of the total variation in the data (i.e. 90%, 95%, 99%).

If you are constructing a 95% confidence interval and are using a threshold of statistical significance of p = 0.05, then your critical value will be identical in both cases.

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Understanding t-Tests: t-values and t-distributions

Topics: Hypothesis Testing , Data Analysis

T-tests are handy hypothesis tests in statistics when you want to compare means. You can compare a sample mean to a hypothesized or target value using a one-sample t-test. You can compare the means of two groups with a two-sample t-test. If you have two groups with paired observations (e.g., before and after measurements), use the paired t-test.

How do t-tests work? How do t-values fit in? In this series of posts, I’ll answer these questions by focusing on concepts and graphs rather than equations and numbers. After all, a key reason to use statistical software like Minitab  is so you don’t get bogged down in the calculations and can instead focus on understanding your results.

In this post, I will explain t-values, t-distributions, and how t-tests use them to calculate probabilities and assess hypotheses.

What Are t-Values?

T-tests are called t-tests because the test results are all based on t-values. T-values are an example of what statisticians call test statistics. A test statistic is a standardized value that is calculated from sample data during a hypothesis test. The procedure that calculates the test statistic compares your data to what is expected under the null hypothesis .

Each type of t-test uses a specific procedure to boil all of your sample data down to one value, the t-value. The calculations behind t-values compare your sample mean(s) to the null hypothesis and incorporates both the sample size and the variability in the data. A t-value of 0 indicates that the sample results exactly equal the null hypothesis. As the difference between the sample data and the null hypothesis increases, the absolute value of the t-value increases.

Assume that we perform a t-test and it calculates a t-value of 2 for our sample data. What does that even mean? I might as well have told you that our data equal 2 fizbins! We don’t know if that’s common or rare when the null hypothesis is true.

By itself, a t-value of 2 doesn’t really tell us anything. T-values are not in the units of the original data, or anything else we’d be familiar with. We need a larger context in which we can place individual t-values before we can interpret them. This is where t-distributions come in.

What Are t-Distributions?

When you perform a t-test for a single study, you obtain a single t-value. However, if we drew multiple random samples of the same size from the same population and performed the same t-test, we would obtain many t-values and we could plot a distribution of all of them. This type of distribution is known as a sampling distribution .

Fortunately, the properties of t-distributions are well understood in statistics, so we can plot them without having to collect many samples! A specific t-distribution is defined by its degrees of freedom (DF) , a value closely related to sample size. Therefore, different t-distributions exist for every sample size.  You can graph t-distributions u sing Minitab’s probability distribution plots .

T-distributions assume that you draw repeated random samples from a population where the null hypothesis is true. You place the t-value from your study in the t-distribution to determine how consistent your results are with the null hypothesis.

The graph above shows a t-distribution that has 20 degrees of freedom, which corresponds to a sample size of 21 in a one-sample t-test. It is a symmetric, bell-shaped distribution that is similar to the normal distribution, but with thicker tails. This graph plots the probability density function (PDF), which describes the likelihood of each t-value.

The peak of the graph is right at zero, which indicates that obtaining a sample value close to the null hypothesis is the most likely. That makes sense because t-distributions assume that the null hypothesis is true. T-values become less likely as you get further away from zero in either direction. In other words, when the null hypothesis is true, you are less likely to obtain a sample that is very different from the null hypothesis.

Our t-value of 2 indicates a positive difference between our sample data and the null hypothesis. The graph shows that there is a reasonable probability of obtaining a t-value from -2 to +2 when the null hypothesis is true. Our t-value of 2 is an unusual value, but we don’t know exactly how unusual. Our ultimate goal is to determine whether our t-value is unusual enough to warrant rejecting the null hypothesis. To do that, we'll need to calculate the probability.

Ready for a demo of Minitab Statistical Software? Just ask! 

Using t-Values and t-Distributions to Calculate Probabilities

The foundation behind any hypothesis test is being able to take the test statistic from a specific sample and place it within the context of a known probability distribution. For t-tests, if you take a t-value and place it in the context of the correct t-distribution, you can calculate the probabilities associated with that t-value.

A probability allows us to determine how common or rare our t-value is under the assumption that the null hypothesis is true. If the probability is low enough, we can conclude that the effect observed in our sample is inconsistent with the null hypothesis. The evidence in the sample data is strong enough to reject the null hypothesis for the entire population.

Before we calculate the probability associated with our t-value of 2, there are two important details to address.

First, we’ll actually use the t-values of +2 and -2 because we’ll perform a two-tailed test. A two-tailed test is one that can test for differences in both directions. For example, a two-tailed 2-sample t-test can determine whether the difference between group 1 and group 2 is statistically significant in either the positive or negative direction. A one-tailed test can only assess one of those directions.

Second, we can only calculate a non-zero probability for a range of t-values. As you’ll see in the graph below, a range of t-values corresponds to a proportion of the total area under the distribution curve, which is the probability. The probability for any specific point value is zero because it does not produce an area under the curve.

With these points in mind, we’ll shade the area of the curve that has t-values greater than 2 and t-values less than -2.

The graph displays the probability for observing a difference from the null hypothesis that is at least as extreme as the difference present in our sample data while assuming that the null hypothesis is actually true. Each of the shaded regions has a probability of 0.02963, which sums to a total probability of 0.05926. When the null hypothesis is true, the t-value falls within these regions nearly 6% of the time.

This probability has a name that you might have heard of—it’s called the p-value!  While the probability of our t-value falling within these regions is fairly low, it’s not low enough to reject the null hypothesis using the common significance level of 0.05.

Learn how to correctly interpret the p-value.

t-Distributions and Sample Size

As mentioned above, t-distributions are defined by the DF, which are closely associated with sample size. As the DF increases, the probability density in the tails decreases and the distribution becomes more tightly clustered around the central value. The graph below depicts t-distributions with 5 and 30 degrees of freedom.

The t-distribution with fewer degrees of freedom has thicker tails. This occurs because the t-distribution is designed to reflect the added uncertainty associated with analyzing small samples. In other words, if you have a small sample, the probability that the sample statistic will be further away from the null hypothesis is greater even when the null hypothesis is true.

Small samples are more likely to be unusual. This affects the probability associated with any given t-value. For 5 and 30 degrees of freedom, a t-value of 2 in a two-tailed test has p-values of 10.2% and 5.4%, respectively. Large samples are better!

I’ve showed how t-values and t-distributions work together to produce probabilities. To see how each type of t-test works and actually calculates the t-values, read the other post in this series, Understanding t-Tests: 1-sample, 2-sample, and Paired t-Tests .

If you'd like to learn how the ANOVA F-test works, read my post, Understanding Analysis of Variance (ANOVA) and the F-test .

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T Test (Student’s T-Test): Definition and Examples

T Test: Contents :

• What is a T Test?
• The T Score
• T Values and P Values
• Calculating the T Test
• What is a Paired T Test (Paired Samples T Test)?

What is a T test?

The t test tells you how significant the differences between group means are. It lets you know if those differences in means could have happened by chance. The t test is usually used when data sets follow a normal distribution but you don’t know the population variance .

For example, you might flip a coin 1,000 times and find the number of heads follows a normal distribution for all trials. So you can calculate the sample variance from this data, but the population variance is unknown. Or, a drug company may want to test a new cancer drug to find out if it improves life expectancy. In an experiment, there’s always a control group (a group who are given a placebo, or “sugar pill”). So while the control group may show an average life expectancy of +5 years, the group taking the new drug might have a life expectancy of +6 years. It would seem that the drug might work. But it could be due to a fluke. To test this, researchers would use a Student’s t-test to find out if the results are repeatable for an entire population.

In addition, a t test uses a t-statistic and compares this to t-distribution values to determine if the results are statistically significant .

However, note that you can only uses a t test to compare two means. If you want to compare three or more means, use an ANOVA instead.

The T Score.

The t score is a ratio between the difference between two groups and the difference within the groups .

• Larger t scores = more difference between groups.
• Smaller t score = more similarity between groups.

A t score of 3 tells you that the groups are three times as different from each other as they are within each other. So when you run a t test, bigger t-values equal a greater probability that the results are repeatable.

T-Values and P-values

How big is “big enough”? Every t-value has a p-value to go with it. A p-value from a t test is the probability that the results from your sample data occurred by chance. P-values are from 0% to 100% and are usually written as a decimal (for example, a p value of 5% is 0.05). Low p-values indicate your data did not occur by chance . For example, a p-value of .01 means there is only a 1% probability that the results from an experiment happened by chance.

Calculating the Statistic / Test Types

There are three main types of t-test:

• An Independent Samples t-test compares the means for two groups.
• A Paired sample t-test compares means from the same group at different times (say, one year apart).
• A One sample t-test tests the mean of a single group against a known mean.

You can find the steps for an independent samples t test here . But you probably don’t want to calculate the test by hand (the math can get very messy. Use the following tools to calculate the t test:

• How to do a T test in Excel.
• T test in SPSS.
• T-distribution on the TI 89.
• T distribution on the TI 83.

What is a Paired T Test (Paired Samples T Test / Dependent Samples T Test)?

A paired t test (also called a correlated pairs t-test , a paired samples t test or dependent samples t test ) is where you run a t test on dependent samples. Dependent samples are essentially connected — they are tests on the same person or thing. For example:

• Knee MRI costs at two different hospitals,
• Two tests on the same person before and after training,
• Two blood pressure measurements on the same person using different equipment.

When to Choose a Paired T Test / Paired Samples T Test / Dependent Samples T Test

Choose the paired t-test if you have two measurements on the same item, person or thing. But you should also choose this test if you have two items that are being measured with a unique condition. For example, you might be measuring car safety performance in vehicle research and testing and subject the cars to a series of crash tests. Although the manufacturers are different, you might be subjecting them to the same conditions.

With a “regular” two sample t test , you’re comparing the means for two different samples . For example, you might test two different groups of customer service associates on a business-related test or testing students from two universities on their English skills. But if you take a random sample each group separately and they have different conditions, your samples are independent and you should run an independent samples t test (also called between-samples and unpaired-samples).

The null hypothesis for the independent samples t-test is μ 1 = μ 2 . So it assumes the means are equal. With the paired t test, the null hypothesis is that the pairwise difference between the two tests is equal (H 0 : µ d = 0).

Paired Samples T Test By hand

• The “ΣD” is the sum of X-Y from Step 2.
• ΣD 2 : Sum of the squared differences (from Step 4).
• (ΣD) 2 : Sum of the differences (from Step 2), squared.

If you’re unfamiliar with the Σ notation used in the t test, it basically means to “add everything up”. You may find this article useful: summation notation .

Step 6: Subtract 1 from the sample size to get the degrees of freedom. We have 11 items. So 11 – 1 = 10.

Step 7: Find the p-value in the t-table , using the degrees of freedom in Step 6. But if you don’t have a specified alpha level , use 0.05 (5%).

So for this example t test problem, with df = 10, the t-value is 2.228.

Step 8: In conclusion, compare your t-table value from Step 7 (2.228) to your calculated t-value (-2.74). The calculated t-value is greater than the table value at an alpha level of .05. In addition, note that the p-value is less than the alpha level: p <.05. So we can reject the null hypothesis that there is no difference between means.

However, note that you can ignore the minus sign when comparing the two t-values as ± indicates the direction; the p-value remains the same for both directions.

In addition, check out our YouTube channel for more stats help and tips!

Goulden, C. H. Methods of Statistical Analysis, 2nd ed. New York: Wiley, pp. 50-55, 1956.

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

T-test Formula

The t-test formula helps us to compare the average values of two data sets and determine if they belong to the same population or are they different. The t-score is compared with the critical value obtained from the t-table. The large t-score indicates that the groups are different and a small t-score indicates that the groups are similar.

What Is the T-test Formula?

The t-test formula is applied to the sample population. The t-test formula depends on the mean , variance, and  standard deviation of the data being compared. There are 3 types of t-tests that could be performed on the n number of samples collected.

• One-sample test,
• Independent sample t-test and
• Paired samples t-test

The critical value is obtained from the t-table looking for the degree of freedom(df = n-1) and the corresponding α value(usually 0.05 or 0.1). If the t-test obtained statistically > CV then the initial hypothesis is wrong and we conclude that the results are significantly different.

One-Sample T-Test Formula

For comparing the mean of a population \(\overline{x}\) from n samples, with a specified theoretical mean μ, we use a one-sample t-test.

\(t= \dfrac{\overline{x}- μ}{\dfrac{\sigma}{\sqrt{n}}}\)

where σ/√n is the standard error

Independent Sample T-Test

Students t-test is used to compare the mean of two groups of samples. It helps evaluate if the means of the two sets of data are statistically significantly different from each other.

\(t = \dfrac{\overline{x_{1}}-\overline{x_{2}}}{\sqrt{(\dfrac{s_{1}^2}{n_{1}}+\dfrac{s_{2}^2}{{n_{2}}}})}\)

t = Student's t-test

• \(x_{1}\) = mean of first group
• \(x_{2}\)= mean of second group
• \(s_{1}\) = standard deviation of group 1
• \(s_{2}\) = standard deviation of group 1
• \(n_{1}\)= number of observations in group 1
• \(n_{2}\)= number of observations in group 2

Paired Samples T-Test

Whenever two distributions of the variables are highly correlated, they could be pre and post test results from the same people. In such cases, we use the paired samples t-test.

\(t = \dfrac{Σ(x_{1}-x_{2})}{\dfrac{s}{\sqrt{n}}}\)

\(x_{1}-x_{2}\) = Difference mean of the pairs

s= standard deviation

n = sample size

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Examples Using t-test Formula

Example 1: Calculate a t-test for the following data of the number of times people prefer coffee or tea in five time intervals.

Solution: let \(x_{1}\) be the sample of data that prefers coffee and \(x_{2}\) be the sample of data that prefers tea.

let us find the mean, variance and the SD

\(\overline{x_{1}}\) = 31/ 5 = 6.2

\(\overline{x_{2}}\) = 28/5 = 5.6

Σ(x 1 -\(\overline{x_{1}}\)) 2 = 14.8

Σ(x 2 -\(\overline{x_{2}}\)) 2 = 17.2

S 1= 14.8/4 = 3.7

S 2 = 17.2/4 = 4.3

According to the t-test formula,

Applying the known values in the t-test formula, we get

\(t = \dfrac{6.2-5.6}{\sqrt{(\dfrac{3.7}{5}+\dfrac{4.3}{5})}}\)

\(=\dfrac{0.6}{\sqrt{1.6}}\)= 0.6/1.26 = 0.47

Example 2: A company wants to improve its sales. The previous sales data indicated that the average sale of 25 salesmen was $50 per transaction. After training, the recent data showed an average sale of $80 per transaction. If the standard deviation is $15, find the t-score. Has the training provided improved the sales?

\(H_{0}\)accepted hypothesis:the population mean = the claimed value⇒ μ = μ 0

\(H_{0}\)alternate hypothesis: the population mean not equal to the claimed value⇒ μ ≠ μ 0

t - test formula for independent test is \(t= \dfrac{m- μ}{\dfrac{s}{\sqrt{n}}}\)

Mean sale = 80, μ = 50, s= 15 and n= 25

substituting the values, we get t= (80-50)/(15/√25)

t = (30 ×5)/10 = 10

looking at the t-table we find 10 > 1.711 . (I.e. CV for α = 0.05). ∴ the accepted hypothesis is not true. Thus we conclude that the training boosted the sales.

Example 3: A pre-test and post-test conducted during a survey to find the study hours of Patrick on weekends. Calculate the t-score and determine (for α = 0.25) if the pre-test and post-test surveys are significantly different?

According to the t-test formula, we know that \(t = \dfrac{ΣX-Y}{\dfrac{s}{\sqrt{n}}}\)

Σ(X-Y)= -3 = 3

s= Σ(X-Y) 2 /(n-1) = 5 2 /1 = 25

t= 3/(25/2) = 6/25 = 0.24

here degree of freedom is n-1 = 2-1 =1 and the corresponidng critical value in the t-table for α= 0.25, is 1.

Therefore the scores are not significantly different.

FAQs on T-test Formula

How do you calculate the t-test.

The following steps are followed to calculate the t-test.

• Get the data. Find the mean.
• Subtract the mean score from each individual score
• Square the differences.
• Add up all the squared differences.
• Find the variance and standard deviation.
• Key-in the values in the formula: \(t = \dfrac{Σx_{1}- mean}{\dfrac{s}{\sqrt{n}}}\)

What is the Formula for Finding The Independent T-test?

Students t-test is used to compare the mean of two groups of samples.

t = Student's t-test score

\(x_{1}\) = mean of first group and \(x_{2}\)= mean of second group

\(s_{1}\) = standard deviation of group 1 and \(s_{2}\) = standard deviation of group 1

\(n_{1}\)= number of observations in group 1 and \(n_{2}\)= number of observations in group 2

What is a One-Sample t-test?

The one-sample t-test is the statistical test used to determine whether an unknown population mean is different from a specific value. For example, comparing the mean height of the students with respect to the national average height of an adult.

What is a T-test Formula Used For?

We use the T-test Formula to statistically determine if there is a significant difference between the means of two groups that are related in certain aspects. Examples: a gym center tests the weight loss from a few samples, a company hiring candidates is set to determine the skills of 2 candidates from two different universities at the interview, and so on.

Student's t-distribution calculator with graph generator

Critical value calculator - student's t-distribution.

This statistical calculator allows you to calculate the critical value corresponding to the Student's t-distribution, you can also see the result in a graph through our online graph generator and if you wish you can download the graph. Just enter the significance value (alpha), degrees of freedom, and left, right, or both tails.

Critical value result

P-value calculator - student's t distribution.

Use our online statistical calculator to calculate the p-value of the Student's t-distribution. You just need to enter the t-value and degrees of freedom and specify the tail. In addition to the p-value, you can get and download the graph created with our graph generator

p-value result

One sample t-test calculator.

The one sample t-test is a statistical hypothesis test calculator, use our calculator to check if you get a statistically significant result or not. To obtain it, fill in the corresponding fields and you will obtain the value of the t-score, p-value, critical value, and the degrees of freedom. You can also download a graph that will display your results in the form of the Student's t-distribution.

T-score result

Two sample t-test calculator.

To determine whether or not the means of two groups are equal, you can use our two-sample t-test calculator that applies the t-test. The results are displayed in a Student's t-distribution plot that you can download. To complete the form, you must include information for both groups, including the mean, standard deviation, sample size, significance level,and whether the test is left, right, or two-tailed.

Common questions related to the Student's t-distribution

In this section, we will try to address the most frequently asked questions about the Student's t-distribution. To give you a fundamental and complementary understanding, we will try to dive into the underlying ideas of the t-distribution. The approach we want to take is to answer the most common questions from students with relevant information. Let's tackle problems simply and offer short and understandable solutions.

Questions related to the student's t-distribution

The formula in relation to the probability density function (pdf) for Student's t-distribution, is given as follows:

Where: π is the pi (approximately 3.14), ν correspond to the degrees of freedom, and Γ is the Euler Gamma function.

A distribution of mean estimates derived from samples taken from a population is what is, by definition, the Student's t-distribution. The t-distribution, commonly known as the Student's t-distribution, is a type of symmetric bell-shaped distribution, it has a lower height but a wider spread than the normal distribution. It is symmetric around 0, but the t-distribution has a wider spread than the typical normal distribution curve, or put another way, the t-distribution has a high standard deviation. The variability of individual observations around their mean is measured by a standard deviation. The degrees of freedom (df) are n - 1. So, df is equal to n – 1, where n is the sample size. The degrees of freedom affect the shape of each t distribution curve.

When the sample size is less than 30 and the population standard deviation is unknown, the t-distribution is utilized in hypothesis testing. It is helpful when the sample size is relatively small or the population standard deviation is unknown. It resembles the normal distribution more closely as sample size grows.

A statistical metric known as the standard deviation is used to quantify the distances between each observation and the mean in a set of data. The standard deviation calculates the degree of dispersion or variability. In other words, it's used to calculate how much a random variable deviates from the mean.

The t-value and t-score have the same meanings. It is one of the relative position measurements. By definition, a value of t defines the location of a continuous random variable, X, in relation to the number of standard deviations from the mean.

The significance level is a point in the normal distribution that must be understood in order to either reject or fail to reject the null hypothesis and to assess whether or not the results are statistically significant. If you decide to make use of our t distribution calculator , you must enter the alpha value corresponding to the significance level. The most common alpha values are 0.1, 0.05 or 0.01. Generally, the most common confidence intervals are: 90%, 95% and 99% (1 − α is the confidence level).

The p-value is a probability with a value ranging from 0 to 1. It is used to test a hypothesis. As an example, in some experiment, we choose the significance level value as 0.05, in this case, the alternative hypothesis is more likely to be supported by stronger evidence when the p-value is less than 0.05 (p-value < 0.05), in case the p-value is high (p-value > 0.05), the probability of accepting the null hypothesis is also high.

The z and t distributions are symmetric and bell-shaped. However, what most characterizes the t distribution are its tails, since they are heavier than in the normal distribution. Furthermore, it can be seen that there are more values in the t-distribution located at the ends of the tail instead of the center of the distribution. You must have the population standard deviation to use the standard normal or z distribution. On the other hand, one of the important conditions for adopting the t distribution is that the population variance is unknown

The t-test , it is a parametric comparison test, is used if the means of two samples are compared using a hypothesis test, if they are independent, from two separate samples, or dependent, a sample evaluated at two different times. The procedure is carried out to evaluate if the differences between the means are significant, determining that they are not due to chance.

To interpret the results of a t-test, you can compare the t-score to the critical value and consider the p-value. A high t-score and low p-value indicate that there is a statistically significant difference between the two means, while a low t-score and high p-value indicate that the difference is not statistically significant. The degrees of freedom and the significance level (alpha) also play a role in determining the critical value and the p-value.

A one sample t-test is a statistical procedure used to test whether the mean of a single sample is significantly different from a hypothesized mean. It is used to determine whether the sample comes from a population with a mean that is different from the hypothesized mean. To perform a one sample t-test using a calculator, you need to input the following information: The sample data, including the mean and standard deviation. The hypothesized mean. The significance level (alpha). The type of tail (left, right, or two-tailed). The calculator will then calculate the t-score and p-value based on this information, and will also provide the critical value and degrees of freedom. To interpret the results, you can compare the t-score to the critical value and consider the p-value. If the t-score is greater than the critical value and the p-value is less than the significance level, you can reject the null hypothesis and conclude that the sample mean is significantly different from the hypothesized mean. If the t-score is less than the critical value or the p-value is greater than the significance level, you cannot reject the null hypothesis and must conclude that the sample mean is not significantly different from the hypothesized mean.

A two-sample t-test is a statistical procedure used to determine whether there is a significant difference between the means of two groups. It is often used to compare the means of two groups in order to determine whether a difference exists between them. For example, a researcher might use a two-sample t-test to determine whether there is a significant difference in the average scores on a test between males and females, or between two different treatment groups in a medical study. The t-test is based on the t-statistic , which is calculated from the sample data and represents the difference between the two groups in relation to the variation within the groups. The t-test is used to determine whether this difference is statistically significant, meaning that it is unlikely to have occurred by chance.

How to Cite

APA: Ait Nasser, F. (2022). Koshegio: Free graph generator for your day-to-day calculations . https://www.koshegio.com.

ISO 690: AIT NASSER, Farouk. Koshegio: Free graph generator for your day-to-day calculations [online]. Koshegio, 2022. [cit. ]. Available at: https://www.koshegio.com

MLA: Ait Nasser, Farouk. "Koshegio: Free graph generator for your day-to-day calculations." Koshegio, 2022, https://www.koshegio.com.

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Statistics By Jim

Making statistics intuitive

One Sample T Test: Definition, Using & Example

By Jim Frost Leave a Comment

What is a One Sample T Test?

Use a one sample t test to evaluate a population mean using a single sample. Usually, you conduct this hypothesis test to determine whether a population mean differs from a hypothesized value you specify. The hypothesized value can be theoretically important in the study area, a reference value, or a target.

For example, a beverage company claims its soda cans contain 12 ounces. A researcher randomly samples their cans and measures the amount of fluid in each one. A one-sample t-test can use the sample data to determine whether the entire population of soda cans differs from the hypothesized value of 12 ounces.

In this post, learn about the one-sample t-test, its hypotheses and assumptions, and how to interpret the results.

Related post : Difference between Descriptive and Inferential Statistics

One Sample T Test Hypotheses

A one sample t test has the following hypotheses:

• Null hypothesis (H 0 ): The population mean equals the hypothesized value (µ = H 0 ).
• Alternative hypothesis (H A ): The population mean does not equal the hypothesized value (µ ≠ H 0 ).

If the p-value is less than your significance level (e.g., 0.05), you can reject the null hypothesis. The difference between the sample mean and the hypothesized value is statistically significant. Your sample provides strong enough evidence to conclude that the population mean does not equal the hypothesized value.

Learn how this analysis compares to the Z Test .

Related posts : How to Interpret P Values and Null Hypothesis: Definition, Rejecting & Examples .

One Sample T Test Assumptions

For reliable one sample t test results, your data should satisfy the following assumptions:

Random Sample

Drawing a random sample from your target population helps ensure your data represent the population. Samples that don’t reflect that population tend to produce invalid results.

Related posts : Populations, Parameters, and Samples in Inferential Statistics and Representative Samples: Definition, Uses & Examples .

Continuous Data

One-sample t-tests require continuous data . These variables can take on any numeric value, and the scale can be split meaningfully into smaller increments. For example, temperature, height, weight, and volume are continuous data.

Read  Comparing Hypothesis Tests for Continuous, Binary, and Count Data  for more information. .

Normally distributed data or your sample has more than 20 observations

This hypothesis test assumes your data follow the normal distribution . However, your data can be mildly skewed when the distribution is unimodal and your sample size is greater than 20 because of the central limit theorem.

Be sure to check for outliers because they can throw off the results.

Related posts : Central Limit Theorem , Skewed Distributions , and 5 Ways to Find Outliers .

Independent Observations

The one-sample t-test assumes that observations are independent of each other, meaning that the value of one observation does not influence or depend on another observation’s value. Violating this assumption can lead to inaccurate results because the test relies on the premise that each data point provides unique and separate information.

Example One Sample T Test

Let’s return to the 12-ounce soda can example and perform a one-sample t-test on the data. Imagine we randomly collected 30 cans of soda and measured their contents.

We want to determine whether the difference between the sample mean and the hypothesized value (12) is statistically significant. Download the CSV file that contains the example data: OneSampleTTest .

Here is how a portion of the data appear in the worksheet.

The histogram shows the data are not skewed , and no outliers are present.

Interpreting the Results

Here’s how to read and report the results for a one sample t test.

The statistical output indicates that the sample mean (A) is 11.8013. Because the p-value (B) of 0.000 is less than our significance level of 0.05, the results are statistically significant. We reject the null hypothesis and conclude that the population mean does not equal 12 ounces. Specifically, it is less than that target value. The beverage company is underfilling the cans.

Learn more about Statistical Significance: Definition & Meaning .

The confidence interval (C) indicates the population mean for all cans is likely between 11.7358 and 11.8668 ounces. This range excludes our hypothesized value of 12 ounces, reaffirming the statistical significance. Learn more about confidence intervals .

To learn more about performing t-tests and how they work, read the following posts:

• T Test Overview
• Independent Samples T Test
• Paired T Test
• Running T Tests in Excel
• T-Values and T-Distributions

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How to Do a T Test in Excel (2 Ways with Interpretation of Results)

Download the Practice Workbook

T Test.xlsx

T Test Types

There are two types of t-tests. They are:

• One-tailed t-test
• Two-tailed t-test

Each of them has 3 subtypes. They are:

• Two sample equal variance
• Two sample unequal variance

How to Do a T-Test in Excel: 2 Effective Ways

Method 1 – using the excel t.test or ttest function for a t-test, case 1.1 – two sample equal variance t-test.

In the dataset, you will see the prices of different laptops and smartphones. Here is a formula that performs a T-Test on the prices of these products and returns the t-test result.

=T.TEST(B5:B14,C5:C14,2,2)

We set the 3rd argument of the function to 2 as we are doing a two-tailed t-test on the dataset. The 4th argument should be 2 for a two-sample equal variance t-test.

Case 1.2 – Paired T-Test

We are going to apply another formula to calculate the Paired T-Test . The following dataset shows the performance mark of some employees in two different criteria.

=T.TEST(C5:C13,D5:D13,2,1)

Note: The explanation of the results is described in the following sections.

Method 2 – Using the Analysis ToolPak

• Go to the Options window.
• Select Add-ins and click on the Go button in the Manage section.

• The Add-ins window will appear. Select Analysis ToolPak and click OK again.

Case 2.1 – Tw-Sample Equal Variance T-Test

• Click on the Data Analysis button from the ribbon of the Data tab.
• The Data Analysis features will appear. Select t-Test: Two Sample Assuming Equal Variances and click OK .

• Set up the parameters for the t-test operation. Insert the Laptop and Smartphone prices as Variable 1 Range and Variable 2 Range. Include the headings in the range and check Labels.
• Set the value of Hypothesized Mean Difference to 0 .
• Select an Output option of your preference and click OK .

• As we have chosen a New Worksheet for the outputs, we will see the results in a new sheet.

Comments on Results

The output shows that the mean values for Laptops and Smartphones are 1608.85 and 1409.164 respectively. We can see from the Variances row that they are not precisely equal, but they are close enough to be assumed to have equal variances. The most relevant metric is the p-value .

The difference between means is statistically significant if the p-value is less than your significance level. Excel calculates p-values for one- and two-tailed T Tests .

One-tailed T Tests can detect only one direction of difference between means. A one-tailed test, for example, might only evaluate whether Smartphones have higher prices than Laptops . Two-tailed tests can reveal differences that are larger or smaller than. There are some other disadvantages to utilizing one-tailed testing, so I’ll continue with the conventional two-tailed results.

For our results, we’ll utilize P(T=t) two-tail, which is the p-value for the t-test’s two-tailed version. We cannot reject the null hypothesis because our p-value ( 0.095639932 ) is greater than the conventional significance level of 0.05 . The hypothesis that the population means differ is supported by our sample data. The mean price of Laptops is greater than the mean price of Smartphones’ .

The Analysis ToolPak also returns results for a one-tailed t-test . Here, the one-tailed P value of the two-sample equal variance t-test is 1.734 .

Case 2.2 – Paired T-Test

Select the t-Test: Paired Two Samples for Mean when you open the Data Analysis window.

The result shows that the mean for the Workpace is 104 and the mean for the Efficiency is 96.56 .

The difference between means is statistically significant if the p-value is less than your significance level. For our results, we’ll utilize P(T=t) two-tail, which is the p-value for the t-test’s two-tailed version. We cannot reject the null hypothesis because our p-value ( 0.188 ) is greater than the conventional significance level of 0.05 . The hypothesis that the population means differ is supported by our sample data. In particular, the Workpace mean exceeds the Efficiency mean.

How to Interpret T-Test Results in Excel

Let’s bring out the results again.

• The mean of laptop prices = 1608.85
• The mean of smartphone prices = 1409.164

ii. Variance

• The variance of laptop prices = 77622.597
• The variance of smartphone prices = 51313.7904

iii. Observations

The number of observations for both laptops and smartphones are 10 .

iv. Pooled Variance

The samples’ average variance, calculated by pooling the variances of each sample.

The mathematical formula for this parameter is:

((No of observations of Sample 1-1)*(Variance of Sample 1) + (No of observations of Sample 2-1)*(Variance of Sample 2))/(No of observations of Sample 1 + No of observations of Sample 2 – 2)

So it becomes: ((10-1)*77622.59676+(10-1)*51313.7904)/(10+10-2) = 64468.19358

v. Hypothesized Mean Difference

We “hypothesize” that the number is the difference between the two population means. In this situation, we chose 0 because we want to see if the difference between the means of the two populations is zero.

It indicates the value of the Degrees of Freedom. Formula for this parameter is:

No of observations of Sample 1 + No of observations of Sample 2 – 2 = 10 + 10 – 2 = 18

vii. t-Stat

The test statistic value of the t-Test operation.

The formula for this parameter is given below.

(Mean of Sample 1 – Mean of Sample 2)/(Square root of (Pooling Variance* (1/No of observations of Sample 1 + 1/No of observations of Sample 2)))

So it becomes: (1608.85 – 1409.164)/Sqrt(64468.19358 * (1/10 + 1/10)) = 1.758570846

viii. P(T<=t) two-tail

A two-tailed t-test’s p-value. This value can be found by entering t = 1.758570846 with 18 degrees of freedom into any T Score to P Value Calculator.

In this situation, the value of p is 0.095639932 . Because this is greater than 0.05 , we cannot reject the null hypothesis. This suggests that we lack adequate evidence to conclude that the two population means differ.

ix. t-Critical two-tail

This is the test’s crucial value. A t-Critical value Calculator with 18 degrees of freedom and a 95% confidence level can be used to calculate this number.

In this instance, the critical value is 2.10092204 . We cannot reject the null hypothesis because our test statistic t is less than this number. Again, we lack adequate information to conclude that the two population means are distinct.

Things to Remember

• Excel demands that your data be arranged in columns, with data from each group in a separate column. The first row should have labels or headers.
• Clearly state your null hypothesis (usually that there is no significant difference between the group means) and your alternative hypothesis (the opposite of the null hypothesis).
• As a result of the t-test, Excel returns the p-value. A small p-value (usually less than the specified alpha level) indicates that the null hypothesis may be rejected and that there is a substantial difference between the group means.

Frequently Asked Questions

Can I perform a t-test on unequal sample sizes in Excel?

Yes, you can use the T.TEST function to do a t-test on unequal sample sizes. When calculating the test statistic, Excel automatically accounts for unequal sample sizes.

What is the difference between a one-tailed and a two-tailed t-test?

A one-tailed t-test determines if the means of the two groups differ substantially in a given direction (e.g., greater or smaller). A two-tailed t-test looks for any significant difference, regardless of direction.

Can I calculate the effect size in Excel for t-tests?

While there is no built-in tool in Excel to calculate effect size, you can manually compute Cohen’s d for independent t-tests and paired sample correlations for paired t-tests using Excel’s basic mathematical operations.

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T-Test in Statistics: Formula, Types and Steps

T-test in statistics | comprehensive guide.

A t-test is a statistical test used to determine whether there is a significant difference between the means of two groups. It is widely used when the sample size is small, and the population standard deviation is unknown. The t-test is essential in hypothesis testing, particularly in comparing group means and analyzing experimental results.

What is a T-Test?

The t-test helps determine if the difference between the means of two groups is statistically significant or if it occurred by chance. It is especially useful when working with small sample sizes and when the variance of the population is unknown.

Types of T-Tests

One-Sample T-Test :

• This test compares the mean of a single group against a known mean or a theoretical value. It is used to check if the sample mean significantly differs from the population mean.

Independent Two-Sample T-Test :

• This test compares the means of two independent groups. It is used when you want to determine whether the means of two unrelated groups are significantly different, such as comparing test scores of two different classes.

Paired Sample T-Test :

• This test compares the means of two related groups. It is used when the data comes from the same group at different times or under different conditions, such as before-and-after measurements in a study.

Assumptions of T-Test

Normal Distribution :

• The data should approximately follow a normal distribution, especially when the sample size is small. For larger sample sizes, the test is robust to deviations from normality.

Independent Observations :

• The data points in one group should be independent of the data points in the other group for independent t-tests.

Equal Variances :

• For a two-sample t-test, the variances of the two groups should be approximately equal. If the variances are not equal, a variation called Welch’s t-test can be used.

Hypothesis in a T-Test

In a t-test, we typically start with two hypotheses:

• Null Hypothesis (H₀) : There is no significant difference between the means of the two groups.
• Alternative Hypothesis (H₁) : There is a significant difference between the means of the two groups.

The t-test evaluates whether the observed data provides enough evidence to reject the null hypothesis in favor of the alternative hypothesis.

How to Interpret T-Test Results

• The t-value is the result of the t-test calculation, representing the difference between the group means in units of standard error. A larger t-value indicates a larger difference between the groups.
• The p-value tells us whether the difference observed is statistically significant. If the p-value is below a chosen significance level (usually 0.05), it indicates that the difference is significant, and the null hypothesis can be rejected.

Confidence Interval :

• The t-test also provides a confidence interval for the difference between the group means. If this interval does not include zero, it supports the conclusion that the group means are significantly different.

Applications of T-Test

Medical Research :

• T-tests are commonly used in clinical trials to compare the effectiveness of two treatments or drugs.

Education :

• In education, t-tests can be used to compare the performance of two groups of students, such as comparing test scores between two classes or before-and-after scores in an intervention study.

Business and Marketing :

• In business, t-tests help compare customer satisfaction scores, product performance, or sales data before and after a marketing campaign.

Social Sciences :

• In social science research, t-tests are used to compare the behavior or responses of different groups, such as comparing survey results from different demographic segments.

Why Learn About T-Tests?

The t-test is a powerful tool in statistics that helps in making data-driven decisions by comparing means. Whether you are working in research, business, or education, understanding how to use and interpret t-tests will allow you to analyze data effectively, determine significance, and support your findings with statistical evidence.

Topics Covered :

Types of T-Tests : One-sample, independent two-sample, and paired sample t-tests.

Hypothesis Testing : How to set up null and alternative hypotheses for t-tests.

Applications : Real-world uses in fields like medical research, education, business, and social sciences.

For more details and further examples, check out the full article on GeeksforGeeks: https://www.geeksforgeeks.org/t-test-in-statistics/ .