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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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t-distribution table

If a hypothesis is to be tested with the t-test, the t value from the calculated test must be compared with the critical t value. The critical t-value can be read from the table below for a selected significance level alpha. Usually the significance level alpha is 0.05. If the calculated chi-squared value is smaller than the critical value, the null hypothesis can not be rejected.

Probability of error

Critical t-value, table t-value.

Calculate t-value

The t-distribution results from a combination of a random variable X with chi-squared distribution and a random variable Y with standard normal distribution to

where Y and X are independent and n is the number of degrees of freedom

t-distribution table: The desired critical t-values ​​can be read from the above table of the t-distribution. In the case of a directional hypothesis, the area is read off at the 1-alpha point; in the case of a non-directional hypothesis, the area is read off at 1-alpha / 2.

This means that the critical t-value can be read from the t-test table above and can thus determine whether the null hypothesis is rejected or not rejected. This means that a hypothesis test can be calculated with the t-test table without statistical software.

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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

• Math Article

T-test Table

In Statistics, a t-test, can be expressed as a statistical hypothesis test where the test statistic maintains a student’s t-distribution, if the null hypothesis is set. Hence, we use the t-test table here. In Paired T-Test , they analyse the means of two groups of observations. The observations need to be randomly allocated to each of the two groups. Hence, the difference in response observed is due to the procedure and not because of any other factors.

If two samples are provided, then we can pair the observation of one sample with the observation of another sample. This test can be applied in making observations on the identical sample before and after an event.

T-test Table (One-tail & Two-tail)

The t-test table is used to evaluate proportions combined with z-scores. This table is used to find the ratio for t-statistics. The t-distribution table displays the probability of t-values from a given value. The acquired probability is the t-curve area between the t-distribution ordinates, i.e., the given value and infinity.

In the t-test table, the significant values are determined for degrees of freedom(df) to the probabilities of t-distribution, α.

Also, read:

• Z Score Table
• T Distribution

T-Test Formula

The t-test is any statistical theory test in which the analysis statistic supports a student’s t-distribution under the null hypothesis . It could be used to conclude if two sets of data are significantly distinct from each other, and is most usually used when the test statistic would match a normal distribution, if the value of a scaling session in the test statistic were known.

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

S 1 = Standard deviation of first set of values

S 2 = Standard deviation of second set of values

n 1 = Total number of values in first set

n 2 = Total number of values in the second set.

We can also represent the formula as:

X̄ = Mean of first sample

μ = Mean of second sample

s/√N = Estimate of standard error difference between the means.

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Statistics By Jim

Making statistics intuitive

Independent Samples T Test: Definition, Using & Interpreting

By Jim Frost 3 Comments

What is an Independent Samples T Test?

Use an independent samples t test when you want to compare the means of precisely two groups—no more and no less! Typically, you perform this test to determine whether two population means are different. This procedure is an inferential statistical hypothesis test, meaning it uses samples to draw conclusions about populations. The independent samples t test is also known as the two sample t test.

For an example of an independent t test, do students who learn using Method A have a different mean score than those who learn using Method B?

In this post, you’ll learn about the hypotheses, assumptions, and how to interpret the results for independent samples t tests.

Related post : Difference between Descriptive and Inferential Statistics

Independent Samples T Tests Hypotheses

Independent samples t tests have the following hypotheses:

• Null hypothesis: The means for the two populations are equal.
• Alternative hypothesis : The means for the two populations are not equal.

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

Notice how the hypotheses for the two sample t test relate to independent populations. They do not contain the same subjects.

Learn how this analysis compares to the Z Test .

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

Independent Samples T Test Assumptions

For reliable independent samples t test results, your data should satisfy the following assumptions:

You have a random sample

Drawing a random sample from the population you are studying helps ensure that your data represent the population. Representative samples are vital when you want to make inferences about the population. If your data do not represent the population, your analysis results will not be valid for that population.

You must draw a random sample from your population of interest. Each item or person in the population must have an equal probability of being selected.

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

Your data must be continuous

T tests require continuous data . Continuous variables can take on any numeric value, and the scale can be meaningfully divided into smaller increments, including fractional and decimal values. There are an infinite number of possible values between any two values. Typically, you measure continuous variables on a scale. For example, when you measure temperature, weight, and height, you have continuous data.

Other hypothesis tests can handle different types of data. For more information, read Comparing Hypothesis Tests for Continuous, Binary, and Count Data .

Your sample data should follow a normal distribution or each group has more than 15 observations

All t-tests assume that your data follow the normal distribution . However, your group distributions can be skewed if your sample size is large enough thanks to the central limit theorem.

For the independent samples t test, when each group is larger than 15, your data can be mildly skewed and the test results will still be valid. However, if your sample size is less than 15 per group, graph your data and determine whether the two distributions are skewed. In this case, you might need to use a nonparametric test . The Mann Whitney U test is the nonparametric test that corresponds to the independent samples t-test.

Fortunately, if you have more than 15 observations in each group for a two sample t test, you don’t have to worry about the normality assumption too much.

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

Related post : Central Limit Theorem and Skewed Distributions

The groups are independent

Independent samples contain different sets of items in each sample. Independent samples t tests compare two distinct samples. Hence, it’s a two sample t test. If you have the same people or items in both groups, you can use the paired t-test .

Related post : Independent and Dependent Samples

Groups can have equal or unequal variances but use the correct form of the test

Variance, and the closely related standard deviation, are measures of variability. Because the two sample t test uses two independent samples, each sample has its own variance. Consequently, the independent samples t test has two methods. One method assumes that the two groups have equal variances while the other does not assume they are equal. The form that does not assume equal variances is known as Welch’s t-test.

When the sample sizes for both groups are roughly equal, and you have a moderate sample size, t-tests are robust to unequal variances. If one group has twice the standard deviation of another group, it’s time to use Welch’s t-test! However, you don’t need to worry about smaller differences.

If you have unequal variances and unequal sample sizes, it’s vital to use the unequal variances version of the two sample t test!

Related post : Standard Deviations

Example Independent Samples T Test

Let’s run an example independent sample t test! Our hypothetical scenario is that we are comparing scores from two teaching methods. We drew two random samples of students. Students in one group learned using Method A while the other group used Method B. These samples contain entirely separate students.

Now, we want to determine whether the two means are different. Download the CSV file that contains the independent samples t test example data: t-TestExamples .

Here is what the data look like in the datasheet.

Let’s assume that the variances are equal and use the Assuming Equal Variances version.

Interpreting the Results

Here’s how to read and report the results for an independent samples t test.

The output indicates that the mean for Method A is 71.50 and for Method B it is 84.74. Looking in the Standard Deviation column, we can see that they are not exactly equal, but they are close enough to assume equal variances.

Because the p-value (0.000) for our independent samples t test is less than the standard significance level of 0.05, we can reject the null hypothesis. If the p-value is low, the null must go! Our sample data support the claim that the population means are different. Specifically, Method B’s mean is greater than Method A’s mean. If high scores are better, then Method B is significantly better than Method A.

Learn more about Statistical Significance: Definition & Meaning .

The two sample t test estimates that the mean difference is -13.24. However, that estimate is based on 30 observations split between the two groups and it is unlikely to equal the population difference. The confidence interval indicates that the mean difference between these two methods for the entire population is likely between -19.89 and -6.59. Learn more about confidence intervals .

The negative values reflect the fact that Method A has a lower mean than Method B (i.e., Method A – Method B < 0). Because the confidence interval excludes zero (no difference), we can conclude that the population means are different.

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

• T Test Overview
• One-Sample T-Test
• Running T Tests in Excel
• T-Values and T-Distributions

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Reader Interactions

June 15, 2022 at 12:30 am

Hi Jim. Just to say thank you. All I needed to learn was how to interpret “independent t test” results. and after reading this article, I am looking no further. Many thanks.

December 1, 2021 at 11:08 am

Lily, I don’t know if Jim will reply as he posted this in Oct. I am just now reading it too. From my work in education, I would look at combining the three tests (average score or total points) so that each student in each group has one test.

November 28, 2021 at 8:23 am

Hi, thanks for your articles about statistics and I would like to ask you some questions. How many test variables can a T-test analyse? I’ve selected 2 groups of students to test two different teaching methods and collected the results from three exams (Is it means I have 3 dependent variables?) Then I used an independent sample T-test to analyse the data. My research purpose is to find out which teaching method is more effective. Did I use the wrong statistical method? Look forward to your reply.

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The t Table In Statistics — Tutorial With Examples

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In the realm of statistics , a t table is a critical tool used extensively in hypothesis testing. The t distribution appears when the mean of a normally distributed population is estimated in scenarios with a small sample size and an unknown population standard deviation . In such cases, the t table provides critical values for t tests, which are used to determine whether the difference between sample means and a population mean is statistically significant. Find an in-depth insight in this article.

Inhaltsverzeichnis

• 1 t Table — In a Nutshell
• 2 Definition: t table
• 3 Using the t table

t Table — In a Nutshell

• The t table in statistics lists the critical values of the t distribution.
• Its columns contain all the t distribution probabilities denoted by the letter “p.”
• The rows of the t table contain the degrees of freedom denoted by the symbol “df.”
• There are two types of t tables in statistics: the one-tail t table and the two-tail t table .

Definition: t table

A t table is a reference statistical table that contains critical values of the t distribution, also known as the t score or t value. The t value explains the significance threshold for specific tests in statistics and the upper or lower confines of confidence intervals for explicit estimates.

The t table is used in statistics when the sample size is small, or when you don’t know the population’s standard deviation. You can also use the t table during a t -test . A t -test is a statistical tes t used to liken the means of two sets or groups of data.

It is also used in hypothesis testing . You can also use the t table to test the difference between two means, if two variable quantities are significantly correlated, and to calculate the confidence intervals of statistical means or lapse/regression coefficients.

Using the t table

You can use a t table to determine a critical value of t to execute statistical tests or find a confidence interval.

Assume you are testing a new pharmaceutical product for treating acne. You group the participants randomly into a treatment group and a control group. The treatment group receives the acne medication, while the control group receives a placebo cream (unmedicated). You must compare the mean of the number of pimples the partakers in the treatment and placebo sample group have in the treatment to discover the effectiveness of the acne treatment. You use an autonomous samples t -test.

Null hypothesis ( H 0 ):

The treatment and placebo sample group have the same mean value of pimples.

Alternative hypothesis ( H 1 ):

The treatment and placebo sample groups do not have the same mean quantity of pimples.

After calculating the t value of the sample, you can compare it to the critical value of t to determine if you can discard the null hypothesis or not.

Step 1: One-tailed and two-tailed tests

Firstly, you must determine if you want to use a two or one-tailed test. Below are guidelines for determining when to use one-tailed and two-tailed tests.

Use a one-tailed test when you have a directional alternative hypothesis . A directional premise emphasizes that the population parameter (like mean or reversion constant) is more or a smaller amount than a specific value, like zero.

A directional alternative hypothesis features words like g reater than, less than, increases, or decreases. So, a hypothesis that does not feature these words is usually non-directional.

Use a two-tailed test when the alternate premise is non-directional. This type of hypothesis states that the mean or regression coefficient (population parameter) is unequal to a certain value, like zero.

You can also use two-tailed t-tests when calculating a confidence interval. Many studies use two-tailed tests.

Alternative hypothesis ( H 1 ): The treatment and control groups differentiate in the mean number of pimples.

This alternative hypothesis is non-directional because it does not state whether the treatment group’s mean is more or less than the control group’s.

In conclusion, since the alternate (premise) hypothesis is non-directional, you should use a two-tailed test.

Step 2: Calculating the degrees of freedom

Once you have decided to use a two-tailed test , the next step is calculating the degrees of freedom .

You can calculate the (df) degree of freedom from the general trial size (n). Also, the type of equation you need will depend on the test you decide to perform.

The df equation for autonomous t -tests is:

So, if you steered a tentative treatment trial with 15 partakers in the control group (placebo) and 19 participants in the treatment group, then the degree of freedom is:

Step 3: Choosing a significance level

Traditionally, the level of significance, denoted as, is 0.05. However, in certain situations, you can decrease the α to reduce the chances of Type I errors or increase the α to decrease the risk of Type II errors.

So, you select the confidence interval depending on your selected confidence level. So, α = 1 – confidence level. The confidence interval is usually 0.95 since the most prevalent confidence interval is usually 0.05.

It is worth noting that the α section is typically highlighted in the t table as it is the most prevalently applied as a significance level.

For our sample above, you can choose an α of 0.05 to test your hypothesis, as it is the most common significance level used by researchers.

Step 4: Finding the critical value

Now that you have all the data you require to apply the t table, you can find the critical value:

• Apply the first t table when using a two-tailed test or finding a confidence interval.
• Apply the second table when using a one-tailed test.
• The dfs are listed on the left side of the table.
• Find the row with the df you found in the second step.
• Round the number down to the nearest smallest if it is not listed.
• Find the level of significance at the top of the t table.
• Find the significance level you chose in step three.
• Determine the critical value of t for your statistical test where the row and the column intersect.

Using the t table, you will discover that in a two-tailed test featuring a

Compare the critical value of t you find from the table to the t value you previously calculated for your sample to determine whether to reject your null hypothesis.

What is the t table used for in statistics?

The t table helps you determine the critical value of your sample and compare it with your t value to determine whether to reject your null hypothesis.

What are the two types of t-tests?

The two types of t-tests are one-tailed and two-tailed tests .

How can you determine whether to use one-tailed or two-tailed tests?

Use a one-tailed test when you have a directional alternative hypothesis. In contrast, use a two-tailed test when the alternate premise is non-directional.

What do you need to find the critical t value from a t table?

Firstly, you must choose between one-tailed or two-tailed tests, find the degrees of freedom, and choose a significance level. You can use this information to determine the critical value of t in the t table .

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S.3.3 hypothesis testing examples.

• Example: Right-Tailed Test
• Example: Left-Tailed Test
• Example: Two-Tailed Test

Brinell Hardness Scores

An engineer measured the Brinell hardness of 25 pieces of ductile iron that were subcritically annealed. The resulting data were:

The engineer hypothesized that the mean Brinell hardness of all such ductile iron pieces is greater than 170. Therefore, he was interested in testing the hypotheses:

H 0 : μ = 170 H A : μ > 170

The engineer entered his data into Minitab and requested that the "one-sample t -test" be conducted for the above hypotheses. He obtained the following output:

Descriptive Statistics

$\mu$: mean of Brinelli

Null hypothesis    H₀: $\mu$ = 170 Alternative hypothesis    H₁: $\mu$ > 170

The output tells us that the average Brinell hardness of the n = 25 pieces of ductile iron was 172.52 with a standard deviation of 10.31. (The standard error of the mean "SE Mean", calculated by dividing the standard deviation 10.31 by the square root of n = 25, is 2.06). The test statistic t * is 1.22, and the P -value is 0.117.

If the engineer set his significance level α at 0.05 and used the critical value approach to conduct his hypothesis test, he would reject the null hypothesis if his test statistic t * were greater than 1.7109 (determined using statistical software or a t -table):

Since the engineer's test statistic, t * = 1.22, is not greater than 1.7109, the engineer fails to reject the null hypothesis. That is, the test statistic does not fall in the "critical region." There is insufficient evidence, at the \(\alpha\) = 0.05 level, to conclude that the mean Brinell hardness of all such ductile iron pieces is greater than 170.

If the engineer used the P -value approach to conduct his hypothesis test, he would determine the area under a t n - 1 = t 24 curve and to the right of the test statistic t * = 1.22:

In the output above, Minitab reports that the P -value is 0.117. Since the P -value, 0.117, is greater than \(\alpha\) = 0.05, the engineer fails to reject the null hypothesis. There is insufficient evidence, at the \(\alpha\) = 0.05 level, to conclude that the mean Brinell hardness of all such ductile iron pieces is greater than 170.

Note that the engineer obtains the same scientific conclusion regardless of the approach used. This will always be the case.

Height of Sunflowers

A biologist was interested in determining whether sunflower seedlings treated with an extract from Vinca minor roots resulted in a lower average height of sunflower seedlings than the standard height of 15.7 cm. The biologist treated a random sample of n = 33 seedlings with the extract and subsequently obtained the following heights:

The biologist's hypotheses are:

H 0 : μ = 15.7 H A : μ < 15.7

The biologist entered her data into Minitab and requested that the "one-sample t -test" be conducted for the above hypotheses. She obtained the following output:

$\mu$: mean of Height

Null hypothesis    H₀: $\mu$ = 15.7 Alternative hypothesis    H₁: $\mu$ < 15.7

The output tells us that the average height of the n = 33 sunflower seedlings was 13.664 with a standard deviation of 2.544. (The standard error of the mean "SE Mean", calculated by dividing the standard deviation 13.664 by the square root of n = 33, is 0.443). The test statistic t * is -4.60, and the P -value, 0.000, is to three decimal places.

Minitab Note. Minitab will always report P -values to only 3 decimal places. If Minitab reports the P -value as 0.000, it really means that the P -value is 0.000....something. Throughout this course (and your future research!), when you see that Minitab reports the P -value as 0.000, you should report the P -value as being "< 0.001."

If the biologist set her significance level \(\alpha\) at 0.05 and used the critical value approach to conduct her hypothesis test, she would reject the null hypothesis if her test statistic t * were less than -1.6939 (determined using statistical software or a t -table):s-3-3

Since the biologist's test statistic, t * = -4.60, is less than -1.6939, the biologist rejects the null hypothesis. That is, the test statistic falls in the "critical region." There is sufficient evidence, at the α = 0.05 level, to conclude that the mean height of all such sunflower seedlings is less than 15.7 cm.

If the biologist used the P -value approach to conduct her hypothesis test, she would determine the area under a t n - 1 = t 32 curve and to the left of the test statistic t * = -4.60:

In the output above, Minitab reports that the P -value is 0.000, which we take to mean < 0.001. Since the P -value is less than 0.001, it is clearly less than \(\alpha\) = 0.05, and the biologist rejects the null hypothesis. There is sufficient evidence, at the \(\alpha\) = 0.05 level, to conclude that the mean height of all such sunflower seedlings is less than 15.7 cm.

Note again that the biologist obtains the same scientific conclusion regardless of the approach used. This will always be the case.

Gum Thickness

A manufacturer claims that the thickness of the spearmint gum it produces is 7.5 one-hundredths of an inch. A quality control specialist regularly checks this claim. On one production run, he took a random sample of n = 10 pieces of gum and measured their thickness. He obtained:

The quality control specialist's hypotheses are:

H 0 : μ = 7.5 H A : μ ≠ 7.5

The quality control specialist entered his data into Minitab and requested that the "one-sample t -test" be conducted for the above hypotheses. He obtained the following output:

$\mu$: mean of Thickness

Null hypothesis    H₀: $\mu$ = 7.5 Alternative hypothesis    H₁: $\mu \ne$ 7.5

The output tells us that the average thickness of the n = 10 pieces of gums was 7.55 one-hundredths of an inch with a standard deviation of 0.1027. (The standard error of the mean "SE Mean", calculated by dividing the standard deviation 0.1027 by the square root of n = 10, is 0.0325). The test statistic t * is 1.54, and the P -value is 0.158.

If the quality control specialist sets his significance level \(\alpha\) at 0.05 and used the critical value approach to conduct his hypothesis test, he would reject the null hypothesis if his test statistic t * were less than -2.2616 or greater than 2.2616 (determined using statistical software or a t -table):

Since the quality control specialist's test statistic, t * = 1.54, is not less than -2.2616 nor greater than 2.2616, the quality control specialist fails to reject the null hypothesis. That is, the test statistic does not fall in the "critical region." There is insufficient evidence, at the \(\alpha\) = 0.05 level, to conclude that the mean thickness of all of the manufacturer's spearmint gum differs from 7.5 one-hundredths of an inch.

If the quality control specialist used the P -value approach to conduct his hypothesis test, he would determine the area under a t n - 1 = t 9 curve, to the right of 1.54 and to the left of -1.54:

In the output above, Minitab reports that the P -value is 0.158. Since the P -value, 0.158, is greater than \(\alpha\) = 0.05, the quality control specialist fails to reject the null hypothesis. There is insufficient evidence, at the \(\alpha\) = 0.05 level, to conclude that the mean thickness of all pieces of spearmint gum differs from 7.5 one-hundredths of an inch.

Note that the quality control specialist obtains the same scientific conclusion regardless of the approach used. This will always be the case.

In our review of hypothesis tests, we have focused on just one particular hypothesis test, namely that concerning the population mean \(\mu\). The important thing to recognize is that the topics discussed here — the general idea of hypothesis tests, errors in hypothesis testing, the critical value approach, and the P -value approach — generally extend to all of the hypothesis tests you will encounter.

T-Table Calculator: A Versatile Tool for Statistical Analysis

Student t-value calculator, introduction, understanding the t-table, the role of t-table calculators, streamlining calculations, flexibility for various scenarios:, how t-table calculators work, ​practical examples, example 1: one-sample t-test, example 2: paired t-test, ​advantages of t-table calculators, ​accuracy and efficiency:, user-friendly interface:, accessibility:, versatility:.