How to interpret t test

Independent Samples T Test: Definition, Using & Interpreting

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.

This test assesses two groups.

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.

Independent Samples T Tests Hypotheses

Independent samples t tests have the following hypotheses:

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.

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.

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, you can waive this assumption 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 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 or has outliers. Either condition can cause the test results to be invalid. In this case, you might need to use a nonparametric 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.

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.

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!

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.

The negative values reflect the fact that Method A has a lower mean than Method B (i.e., Method A – Method B

Reader Interactions

Comments

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.

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.

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.

Comments and Questions Cancel reply

Primary Sidebar

Meet Jim

I’ll help you intuitively understand statistics by focusing on concepts and using plain English so you can concentrate on understanding your results.

How t-Tests Work: t-Values, t-Distributions, and Probabilities

T-tests are statistical hypothesis tests that you use to analyze one or two sample means. Depending on the t-test that you use, you can compare a sample mean to a hypothesized value, the means of two independent samples, or the difference between paired samples. In this post, I show you how t-tests use t-values and t-distributions to calculate probabilities and test hypotheses.

As usual, I’ll provide clear explanations of t-values and t-distributions using concepts and graphs rather than formulas! If you need a primer on the basics, read my hypothesis testing overview.

What Are t-Values?

The term “t-test” refers to the fact that these hypothesis tests use t-values to evaluate your sample data. T-values are a type of test statistic. Hypothesis tests use the test statistic that is calculated from your sample to compare your sample to the null hypothesis. If the test statistic is extreme enough, this indicates that your data are so incompatible with the null hypothesis that you can reject the null. Learn more about Test Statistics.

Don’t worry. I find these technical definitions of statistical terms are easier to explain with graphs, and we’ll get to that!

When you analyze your data with any t-test, the procedure reduces your entire sample to a single value, the t-value. These calculations factor in your sample size and the variation in your data. Then, the t-test compares your sample means(s) to the null hypothesis condition in the following manner:

Read the companion post where I explain how t-tests calculate t-values.

The tricky thing about t-values is that they are a unitless statistic, which makes them difficult to interpret on their own. Imagine that we performed a t-test, and it produced a t-value of 2. What does this t-value mean exactly? We know that the sample mean doesn’t equal the null hypothesis value because this t-value doesn’t equal zero. However, we don’t know how exceptional our value is if the null hypothesis is correct.

To be able to interpret individual t-values, we have to place them in a larger context. T-distributions provide this broader context so we can determine the unusualness of an individual t-value.

What Are t-Distributions?

A single t-test produces a single t-value. Now, imagine the following process. First, let’s assume that the null hypothesis is true for the population. Now, suppose we repeat our study many times by drawing many random samples of the same size from this population. Next, we perform t-tests on all of the samples and plot the distribution of the t-values. This distribution is known as a sampling distribution, which is a type of probability distribution.

If we follow this procedure, we produce a graph that displays the distribution of t-values that we obtain from a population where the null hypothesis is true. We use sampling distributions to calculate probabilities for how unusual our sample statistic is if the null hypothesis is true.

Luckily, we don’t need to go through the hassle of collecting numerous random samples to create this graph! Statisticians understand the properties of t-distributions so we can estimate the sampling distribution using the t-distribution and our sample size.

The degrees of freedom (DF) for the statistical design define the t-distribution for a particular study. The DF are closely related to the sample size. For t-tests, there is a different t-distribution for each sample size.

Use the t-Distribution to Compare Your Sample Results to the Null Hypothesis

T-distributions assume that the null hypothesis is correct for the population from which you draw your random samples. To evaluate how compatible your sample data are with the null hypothesis, place your study’s t-value in the t-distribution and determine how unusual it is.

The sampling distribution below displays a t-distribution with 20 degrees of freedom, which equates to a sample size of 21 for a 1-sample t-test. The t-distribution centers on zero because it assumes that the null hypothesis is true. When the null is true, your study is most likely to obtain a t-value near zero and less liable to produce t-values further from zero in either direction.

We know that our t-value of 2 is rare when the null hypothesis is true. How rare is it exactly? Our final goal is to evaluate whether our sample t-value is so rare that it justifies rejecting the null hypothesis for the entire population based on our sample data. To proceed, we need to quantify the probability of observing our t-value.

t-Tests Use t-Values and t-Distributions to Calculate Probabilities

Hypothesis tests work by taking the observed test statistic from a sample and using the sampling distribution to calculate the probability of obtaining that test statistic if the null hypothesis is correct. In the context of how t-tests work, you assess the likelihood of a t-value using the t-distribution. If a t-value is sufficiently improbable when the null hypothesis is true, you can reject the null hypothesis.

I have two crucial points to explain before we calculate the probability linked to our t-value of 2.

Additionally, it is possible to calculate a probability only for a range of t-values. On a probability distribution plot, probabilities are represented by the shaded area under a distribution curve. Without a range of values, there is no area under the curve and, hence, no probability.

t-Test Results for Our Hypothetical Study

The probability distribution plot indicates that each of the two shaded regions has a probability of 0.02963—for a total of 0.05926. This graph shows that t-values fall within these areas almost 6% of the time when the null hypothesis is true.

There is a chance that you’ve heard of this type of probability before—it’s the P value! While the likelihood of t-values falling within these regions seems small, it’s not quite unlikely enough to justify rejecting the null under the standard significance level of 0.05.

t-Distributions and Sample Size

The sample size for a t-test determines the degrees of freedom (DF) for that test, which specifies the t-distribution. The overall effect is that as the sample size decreases, the tails of the t-distribution become thicker. Thicker tails indicate that t-values are more likely to be far from zero even when the null hypothesis is correct. The changing shapes are how t-distributions factor in the greater uncertainty when you have a smaller sample.

You can see this effect in the probability distribution plot below that displays t-distributions for 5 and 30 DF.

Sample means from smaller samples tend to be less precise. In other words, with a smaller sample, it’s less surprising to have an extreme t-value, which affects the probabilities and p-values. A t-value of 2 has a P value of 10.2% and 5.4% for 5 and 30 DF, respectively. Use larger samples!

If you like this approach and want to learn about other hypothesis tests, read my posts about:

To see an alternative to traditional hypothesis testing that does not use probability distributions and test statistics, learn about bootstrapping in statistics!

T-Test

Adam Hayes, Ph.D., CFA, is a financial writer with 15+ years Wall Street experience as a derivatives trader. Besides his extensive derivative trading expertise, Adam is an expert in economics and behavioral finance. Adam received his master’s in economics from The New School for Social Research and his Ph.D. from the University of Wisconsin-Madison in sociology. He is a CFA charterholder as well as holding FINRA Series 7, 55 & 63 licenses. He currently researches and teaches economic sociology and the social studies of finance at the Hebrew University in Jerusalem.

What Is a T-Test?

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

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

Key Takeaways

T-Test

Understanding the T-Test

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

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

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

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

Using a T-Test

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

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

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

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

T-Test Formula

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

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

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

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

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

T-Score

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

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

Paired Sample T-Test

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

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

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

Equal Variance or Pooled T-Test

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

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

Complete Guide: How to Interpret t-test Results in R

A two sample t-test is used to test whether or not the means of two populations are equal.

This tutorial provides a complete guide on how to interpret the results of a two sample t-test in R.

Step 1: Create the Data

Suppose we want to know if two different species of plants have the same mean height. To test this, we collect a simple random sample of 12 plants from each species.

Step 2: Perform & Interpret the Two Sample t-test

Next, we will use the t.test() command to perform a two sample t-test:

Here’s how to interpret the results of the test:

data: This tells us the data that was used in the two sample t-test. In this case, we used the vectors called group1 and group2.

t: This is the t test-statistic. In this case, it is -2.5505.

df: This is the degrees of freedom associated with the t test-statistic. In this case, it’s 20.488. Refer to the Satterthwaire approximation for an explanation of how this degrees of freedom value is calculated.

alternative hypothesis: This tells us the alternative hypothesis used for this particular t-test. In this case, the alternative hypothesis is that the true difference in means between the two groups is not equal to zero.

sample estimates: This tells us the sample mean of each group. In this case, the sample mean of group 1 was 11.667 and the sample mean of group 2 was 14.75.

The two hypotheses for this particular two sample t-test are as follows:

H₀: µ₁ = µ₂ (the two population means are equal)

H_A: µ₁ ≠µ₂ (the two population means are not equal)

Because the p-value of our test (.01884) is less than alpha = 0.05, we reject the null hypothesis of the test. This means we have sufficient evidence to say that the mean height of plants between the two populations is different.

Notes

The t.test() function in R uses the following syntax:

t.test(x, y, alternative = “two.sided”, mu = 0, paired = FALSE, var.equal = FALSE, conf.level = 0.95)

In our example above, we used the following assumptions:

Feel free to change any of these arguments when you conduct your own t-test, depending on the particular test you want to perform.

Complete Guide: How to Interpret t-test Results in Excel

A two sample t-test is used to test whether or not the means of two populations are equal.

This tutorial provides a complete guide on how to interpret the results of a two sample t-test in Excel.

Step 1: Create the Data

Suppose a biologist want to know whether or not two different species of plants have the same mean height.

To test this, she collects a simple random sample of 20 plants from each species:

Step 2: Perform the Two Sample t-test

To perform a two sample t-test in Excel, click the Data tab along the top ribbon and then click Data Analysis:

If you don’t see this option to click on, you need to first download the Analysis ToolPak.

In the window that appears, click the option titled t-Test: Two-Sample Assuming Equal Variances and then click OK. Then e nter the following information:

Once you click OK, the results of the t-test will be displayed:

Step 3: Interpret the Results

Here is how to interpret each line in the results:

Mean: The mean of each sample.

Variance: The variance of each sample.

Observations: The number of observations in each sample.

Pooled Variance: The average variance of the samples, calculated by “pooling” the variances of each sample together using the following formula:

Hypothesized mean difference: The number that we “hypothesize” is the difference between the two population means. In this case, we chose 0 because we want to test whether or not the difference between the two populations means is 0.

df: The degrees of freedom for the t-test, calculated as:

t Stat: The test statistic t, calculated as:

In this case, p = 0.530047. This is larger than 0.05, so we fail to reject the null hypothesis. This means we do not have sufficient evidence to say that the two population means are different.

t Critical two-tail: This is the critical value of the test. This value can be found by using a t Critical value Calculator with 38 degrees of freedom and a 95% confidence level.

In this case, the critical value turns out to be 2.024394. Since our test statistic t is less than this value, we fail to reject the null hypothesis. Once again, this means we do not have sufficient evidence to say that the two population means are different.

Note #1: You will arrive at the same conclusion whether you use the p-value method or the critical value method.

Note #2: If you are performing a one-tailed hypothesis test, you will instead use the values for P(T

Additional Resources

The following tutorials provide step-by-step examples of how to perform various t-tests in Excel: