*Written by Tim Bushnell, Ph.D*

The ultimate goal of any experiment is to analyze data and determine whether it supports or disproves a given hypothesis. To do that, scientists turn to statistics.

*Statistics** is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e.g., a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied.*

One of the first important concepts to take from this definition is the idea of a population. An example population might be all the people in the world who have a specific disease.

It is time and cost prohibitive to try to study all of these people, so the scientist must sample a subset of the population, such that this sample represents (as best as possible) the whole population. How big the population is and what fraction is sampled in the experiment contributes to the power of the experiment, a topic for another day.

**Figure 1**: Relationship of population, sample size, and statistics.

This sample size, and how it is obtained, should be described before one begins any experiments, as getting the population sampling correct is a critical component of improving reproducibility. Consequences of poor sample design can be found throughout history, such as the issues surrounding the use of Thalidomide in pregnant women.

The second critical component is to identify the question(s) that the experiments are designed to to test. This will lead the researcher to state the Null hypothesis (H_{O}), which is what statistics are designed to test.

An additional factor that should be addressed at the beginning of the experimental process is the significance level (α value) — the probability of rejecting the null hypothesis when it is actually true (a Type I statistical error).

At the conclusion of the experiments, we collect the data to generate a P value, which we compare to the α value.

If the P value is less than the α value, the null hypothesis is rejected, and the findings are considered statistically significant. On the other hand, if the P value is greater than or equal to the α value, the null hypothesis cannot be rejected.

Once the experiments are done and the primary analysis is completed, it is time for the secondary analysis.

There are a host of different tests available, depending on what comparisons are being made and the distribution of the data (i.e. normally distributed, or not.) There is an excellent resource at the Graphpad Software website, makers of Graphpad Prism.

If we wish to compare either a single group to a theoretical hypothesis, or two different groups, and these groups are normally distributed, the test of choice is the Student’s t-Test, a method developed by William Gosset while working at Guinness Brewery.

Using the t-Test, the t-statistic is calculated on the distributions, which is an intermediate step on the way to calculating the P value. The P value is then compared to the threshold to determine if the data is statistically significant.

## Assumptions About the Data

The t-Test assumes that the data comes from a normal (Gaussian) distribution. That is to say, the data observes a bell-shaped curve.

**Figure 2**: A normal distribution.

**Although the t-Test was originally developed for small samples, it is also resistant to deviations from the normal distribution with larger sample sizes.**

If the data doesn’t follow a normal distribution, a non-parametric test, such at the Wilcoxon or Mann-Whitney test, is best. Non-parametric tests rank the data and perform a t-Test on the ranked data, with the assumption that the ranked data is randomly distributed.

## Performing a t-Test

The minimum information needed to perform a t-Test is the means, standard deviations, and number of observations for the two populations. As shown below :

**Figure 3**: Calculating a t-Test in Graphpad Prism (ver. 7) with input values calculated elsewhere.

The data is collected elsewhere, and the mean, standard deviation, and N are entered into the software. For visualization, a bar graph showing the average and standard deviation is plotted.

Using the analysis feature in the software, the appropriate statistical parameters are chosen (un-paired t-Test, threshold to 0.05 discussed below). The Welch correction is applied because the N’s are different between the two samples.

Prism generates a summary table and shows details in the red box. In this case, the experimental sample is statistically significantly different from the control, and we may reject the null hypothesis.

Another way to perform this test is to enter the data into your preferred program and let the software do the work, as shown below for Prism.

**Figure 4**: Calculating a t-Test in Graphpad Prism (ver. 7) by entering the data.

This second plotting method has the advantage of letting the reader see all the data points in the analysis.

## Final Tips for Performing a t-Test

There are a few variations of the t-Test, based on sample size and variance in the data. One can perform a one- or two-tailed t-Test. The decision to use one versus the other is related to the hypothesis.

If the expected difference is in one direction, the one-tailed t-Test is performed. If it is not known, or the expected difference could be an increase or a decrease, the two-tailed t-Test is performed.

**Figure 5**: The null hypothesis for either a one-tailed (left) or two-tailed (right) t-Test.

In conclusion, to perform the t-Test, it is critical to start from the beginning of the experiment to establish several parameters, including the type of test, the null hypothesis, the assumptions about the data, the number of samples to be analyzed (Power of the experiment), and the threshold.

**The experiments are performed, and only then, after the primary analysis is completed, is statistical testing performed.**

Each software package has its specific methods of performing these tests, and we have shown you one (Graphpad Prism). It is recommended that you consult your local statistical community and see what they are using for their analysis.

By establishing the statistical plan at the beginning of the experiment, the planning for the rest of the experiment become easy. Likewise, one does not begin to chase a hypothesis with the data, rather the data stands alone to support or reject the hypothesis.

**To learn more about How To Perform A Flow Cytometry t-Test, and to get access to all of our advanced materials including 20 training videos, presentations, workbooks, and private group membership, get on the Flow Cytometry Mastery Class wait list.**

### Tim Bushnell

My other passions include grilling, wine tasting, and real food. To be honest, my biggest passion is flow cytometry, which is something that Carol and I share. My personal mission is to make flow cytometry education accessible, relevant, and fun. I’ve had a long history in the field starting all the way back in graduate school.

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