T-tests are statistical hypothesis tests that analyze one or two sample means. When you analyze your data with any t-test, the procedure reduces your entire sample to a single value, the t-value. In this post, I describe how each type of t-test calculates the t-value. I don’t explain this just so you can understand the calculation, but I describe it in a way that really helps you grasp how t-tests work.

As usual, I’ll focus on ideas rather than formulas. However, I need to present a few easy equations to facilitate the analogy between how t-tests work and a signal-to-noise ratio.

## How 1-Sample t-Tests Calculate t-Values

The equation for how the 1-sample t-test produces a t-value based on your sample is below:

This equation is a ratio, and a common analogy is the signal-to-noise ratio. The numerator is the signal in your sample data, and the denominator is the noise. Let’s see how t-tests work by comparing the signal to the noise!

**The Signal – The Size of the Sample Effect**

In the signal-to-noise analogy, the numerator of the ratio is the signal. The effect that is present in the sample is the signal. It’s a simple calculation. In a 1-sample t-test, the sample effect is the sample mean minus the value of the null hypothesis. That’s the top part of the equation.

For example, if the sample mean is 20 and the null value is 5, the sample effect size is 15. We’re calling this the signal because this sample estimate is our best estimate of the population effect.

The signal portion of t-values is calculated so that when the sample effect equals zero, the numerator equals zero, which in turn means the t-value itself equals zero. The estimated sample effect (signal) equals zero when there is no difference between the sample mean and the null hypothesis value. For example, if the sample mean is 5 and the null value is 5, the signal equals zero (5 – 5 = 0).

The size of the signal increases when the difference between the sample mean and null value increases. The difference can be either negative or positive depending on whether the sample mean is greater than or less than the value associated with the null hypothesis.

A relatively large signal in the numerator produces t-values that are further away from zero.

**The Noise – The Variability or Random Error in the Sample**

The denominator of the ratio is the standard error of the mean, which measures the variation in the sample. The standard error of the mean represents how much random error is in the sample and how well the sample estimates the population mean.

As the value of this statistic increases, the sample mean provides a less precise estimate of the population mean. In other words, high levels of random error increase the probability that your sample mean is further away from the population mean.

In our analogy, random error represents noise. Why? Because when there is a greater amount of random error, you are more likely to see considerable differences between the sample mean and the null hypothesis in cases where *the null is true*. Noise appears in the denominator to provide a benchmark for how large the signal must be in order to be distinguishable from the noise.

**Signal-to-Noise ratio**

Our signal-to-noise ratio analogy equates to:

Both of these statistics are in the same units as your data. Let’s calculate a couple of t-values to see how to interpret them.

- If the signal is 10 and the noise is 2, your t-value is 5. The signal is 5 times the noise.
- If the signal is 10 and the noise is 5, your t-value is 2. The signal is 2 times the noise.

The signal is the same in both examples, but it is more distinguishable from the lower amount of noise in the first example. In this manner, t-values indicate how clear the signal is from the noise. If the signal is of the same general magnitude as the noise, it’s probable that random error causes the difference between the sample mean and null value rather than an actual population effect.

## Paired t-Tests Are Really 1-Sample t-Tests

I’ve seen a lot of confusion over how a paired t-test works and when you should use it. Pssst! Here’s a secret! Paired t-tests and 1-sample t-tests are the same hypothesis test incognito!

You use a 1-sample t-test to assess the difference between a sample mean and the value of the null hypothesis.

A paired t-test takes paired observations (like before and after), subtracts one from the other, and conducts a 1-sample t-test on the differences. Typically, a paired t-test determines whether the paired differences are significantly different from zero.

Download the CSV data file to check this yourself: T-testData. All of the statistical results are the same when you perform a paired t-test using the Before and After columns versus performing a 1-sample t-test on the Differences column.

Once you realize that paired t-tests are the same as 1-sample t-tests on paired differences, you can focus on the deciding characteristic —does it make sense to analyze the differences between two columns?

Suppose the Before and After columns contain test scores and there was an intervention in between. If each row in the data contains the same subject in the Before and After column, it makes sense to find the difference between the columns because it represents how much each subject changed after the intervention. The paired t-test is a good choice.

On the other hand, if a row has different subjects in the Before and After columns, it doesn’t make sense to subtract the columns. You should use the 2-sample t-test described below.

The paired t-test is a convenience for you. It eliminates the need for you to calculate the difference between two columns yourself. Remember, double-check that this difference is meaningful! If using a paired t-test is valid, you should use it because it provides more statistical power than the 2-sample t-test.

## How Two-Sample T-tests Calculate T-Values

Use the 2-sample t-test when you want to analyze the difference between the means of two independent samples. Like the other t-tests, this procedure reduces all of your data to a single t-value in a process that is much like the 1-sample t-test. The signal-to-noise analogy still holds true.

Here’s the equation for the t-value in a 2-sample t-test.

The equation is still a ratio, and the numerator still represents the signal. For a 2-sample t-test, the signal, or effect, is the difference between the two sample means. This calculation is straightforward. If the first sample mean is 20 and the second mean is 15, the difference or effect is 5.

Typically, the null hypothesis states that there is no difference between the two samples. In the equation, if both groups have the same mean, the numerator, and the ratio as a whole, equals zero. Larger differences between the sample means produce stronger signals.

The denominator again represents the noise for a 2-sample t-test. However, there are two different values you can use depending on whether you assume that the variation in the two groups is equal or not. Most statistical software let you choose which value to use.

Regardless of the denominator value you use, the 2-sample t-test works by determining how distinguishable the signal is from the noise. To ascertain that the difference between means is statistically significant, you need a high positive or negative t-value.

## How Do T-tests Use T-values to Determine Statistical Significance?

Here’s what we’ve learned about the t-values for the 1-sample t-test, paired t-test, and 2-sample t-test:

- Each test reduces your sample data down to a single t-value based on the ratio of the effect size to the variability in your sample.
- A t-value of zero indicates that your sample results match the null hypothesis precisely.
- Larger absolute t-values represent stronger signals, or effects, that stand out more from the noise.

For example, a t-value of 2 indicates that the signal is twice the magnitude of the noise.

Great … but how do you get from that to determining whether the effect size is statistically significant? After all, the purpose of t-tests is to assess hypotheses. To find out, read the companion post to this one: How t-Tests Work: t-Values, t-Distributions and Probabilities.

If you’d like to learn about the F-test using the same general approach, read: How F-tests Work in ANOVA.

harrison says

April 8, 2019 at 11:22 pmHi jim. First of all, I really appreciate your posts!

When I use t-test via R or scikit learn, there is an option for homogeneity of variance. I think that option only applied to two sample t-test, but what should I do for that option?

Should I always perform f-test for check the homogeneity of variance? or Which one is a more strict assumption?

Mel says

November 9, 2018 at 12:03 amThis blog is great. I’m at Stanford and can say this is a great supplement to class lectures. I love the fact that there aren’t formulas so as to get an intuitive feel. Thank you so much!

Jim Frost says

November 9, 2018 at 9:12 amThanks Mel! I’m glad it has been helpful! Your kind words mean a lot to me because I really strive to make these topics as easy to understand as possible!

Jessica Escorcia says

December 29, 2017 at 4:14 pmThank you so much Jim! I have such a hard time understanding statistics without people like you who explain it using words to help me conceptualize rather than utilizing symbols only!

Jim Frost says

December 29, 2017 at 4:56 pmThank you, Jessica! Your kind words made my day. That’s what I want my blog to be all about. Providing simple but 100% accurate explanations for statistical concepts!

Happy New Year!

Omkar says

October 22, 2017 at 2:38 pmHi Jim, sure, I’ll go through it…Thank you..!

Omkar says

October 22, 2017 at 4:50 amIn summary, the t test tells, how the sample mean is different from null hypothesis, i.e. how the sample mean is different from null, but how does it comment about the significance? Is it like “more far from null is the more significant”? If it is so, could you give some more explanation about it?

Jim Frost says

October 22, 2017 at 2:30 pmHi Omkar, you’re in luck, I’ve written an entire blog post that talks about how t-tests actually use the t-values to determine statistical significance. In general, the further away from zero, the more significant it is. For all the information, read this post: How t-Tests Work: t-Values, t-Distributions, and Probabilities. I think this post will answer your questions.

Jim

santhosh says

September 12, 2017 at 2:46 amExcellent explanation, appreciate you..!!

Jim Frost says

September 12, 2017 at 8:48 amThank you, Santhosh! I’m glad you found it helpful!