A t-test is a statistical tool used to assess the difference between two data groups. In particular, it helps you determine whether the differences in means (averages) are statistically significant or not. If that probability is minimal, you can be confident that the difference is meaningful (or statistically significant).
The t-statistic, the t-distribution values, and degrees of freedom determine whether or not there is a statistically significant difference between the two samples.
The null hypothesis assumes that the two populations are the same and there is no meaningful difference between them. A t-test will prove or disprove your assumptions about the null hypothesis.
While t-tests are very insensitive to assumptions being broken, they do imply the following:
- Continuous data is being collected.
- The sample data were drawn at random from a population.
- There is variance homogeneity.
- The distribution is in the neighborhood of ordinary.
Independent samples are required for two-sample t-tests. A paired t-test may be acceptable if the samples are not separate.
How can you use t-tests?
T-tests are used to compare the means of two groups of data, and the test determines if the means are statistically different from each other. You can use this information to make business decisions about how to move forward with a product or service.
Several different types of t-tests can be used, depending on the situation. When studying one group, a paired or one-sample t-test can be used to compare the mean over time or after an intervention. When studying two groups, a nonparametric test can be used to compare the difference in group means without assuming any particular distribution of observations in either group.
A two-sample t-test compares two groups of people, and a one-sample t-test compares a single population to a standard value. A paired t-test compares a single population before and after some experimental intervention or at two different points in time.
T-tests should not be used to measure differences among more than two groups because the error structure will underestimate the actual error when many groups are compared.
How to run a t-test
You can do it in Excel, SAS, or SPSS software programs. There are two types of t-tests: independent samples and paired samples. The independent samples t-test is used when the two groups’ data are randomly sampled from populations with different variances. The paired samples t-test is used when the data for the two groups being compared are from the same population and are matched (e.g., before and after measurements).
There are a few things you need to know to run a t-test:
- The difference in mean values between your data sets (known as the mean difference)
- The standard variation for each (that is, the degree of variation)
- The total amount of data values included inside each group.
- A value for 𝝰 (alpha). This component indicates how much risk you are willing to accept for getting it wrong. An alpha value of 0.05 indicates a 5% risk.
- Manual calculations need a critical value table to aid in the interpretation of the findings. These are readily available online, including on university websites.
You can either manually perform the t-test using equations, or you may use a statistical software tool such as SPSS or Minitab to calculate your findings.
This is composed of two parts: the difference in the means of your two groups and their variance. These two variables are defined in terms of a ratio. If it is small, there is little distinction between the groups, and if it is larger, the difference is more significant.
Degrees of freedom
This pertains to the sample size and the degree to which the sample’s values can change while still retaining the same average. In terms of mathematics, it is the sample size minus one. Additionally, you might think about it as the number of values that you would need to discover to decide all of the values.
With these two figures in hand, you may use your critical value table to determine:
This is the critical value — it indicates the possibility of your t-value occurring by chance. The lesser the p-value, the more confident you may be that your results are statistically significant.
Interpreting test results
When you run a statistical test, you get back a t-value and degrees of freedom. The t-value is the distance from 0 (in other words, how far away the sample mean is from the hypothesized mean), and the degrees of freedom tells us how many “free” data points we have to make comparisons. We only care about the absolute value of the difference or how far away the sample mean is from the hypothesized mean.
2e-16 (or 2.2 with 15 zeros) means that the probability you would see a t-value that extreme by chance is less than 1 in 100,000. This number is called our p-value, and it tells us how likely it is that our results were just due to chance. The smaller the p-value, the more convinced we are that our results are not due to chance alone.
The confidence interval for this test is the range of numbers within which the actual difference in mean lengths will be 95% of the time. In other words, there’s a 95% chance that the actual difference in means falls within this range.
How to present the results of a t-test
When you have completed your data analysis, you will want to present the results of your t-test. This includes the t -value, p-value, and degrees of freedom (DF) for the test. Additionally, including the mean and standard deviation provides a snapshot of what has happened in the data analysis process thus far.
If you want to include mean and standard deviation in your presentation, use descriptive statistics to summarize your data first. This will help you better understand what has happened in your data analysis so far.
When should you use a t-test?
If there are two or fewer groups, the t-test is utilized. If your sample size exceeds two, another technique, such as ANOVA (Analysis of Variance), may be more appropriate.
There are a few additional requirements for utilizing a two-sample t-test, including the following:
- Your data is presented on an ordinal or interval scale (such as ranking or numerical scores)
- The two groups you’re comparing are unrelated (one does not affect the other). This is not applicable if you are performing a paired t-test.
- Your sample is chosen at random.
- You have a normal distribution.
- Each group has a comparable degree of variation.
Additionally, you must have a large enough sample size to ensure the validity of the results. However, one advantage of the t-test is how it enables you to work with minimal amounts of data, as it is based on the sample mean and variance rather than the population mean and variance.