Before you use numerical hypothesis tests, you need to be sure that the following things are true:

1. The samples should each be normally distributed…ish

Data analysts in the real world often still perform hypothesis tests on datasets that aren’t exactly normally distributed. What is more important is to recognize if there is some reason to believe that a normal distribution is especially unlikely. If your dataset is definitively not normal, the numerical hypothesis tests won’t work as intended.

For example, imagine you have three datasets, each representing a day of traffic data in three different cities. Each dataset is independent, as traffic in one city should not impact traffic in another city. However, it is unlikely that each dataset is normally distributed. In fact, each dataset probably has two distinct peaks, one at the morning rush hour and one during the evening rush hour. The histogram of a day of traffic data might look something like this:


In this scenario, using a numerical hypothesis test would be inappropriate.

2. The population standard deviations of the groups should be equal

For ANOVA and Two Sample T-Tests, using datasets with standard deviations that are significantly different from each other will often obscure the differences in group means.

To check for similarity between the standard deviations, it is normally sufficient to divide the two standard deviations and see if the ratio is “close enough” to 1. “Close enough” may differ in different contexts, but generally staying within 10% should suffice.

3. The samples must be independent

When comparing two or more datasets, the values in one distribution should not affect the values in another distribution. In other words, knowing more about one distribution should not give you any information about any other distribution.

Here are some examples where it would seem the samples are not independent:

  • the number of goals scored per soccer player before, during, and after undergoing a rigorous training regimen
  • a group of patients’ blood pressure levels before, during, and after the administration of a drug

It is important to understand your datasets before you begin conducting hypothesis tests on them so that you know you are choosing the right test.



Use the base R hist() function to display the histograms for dist_one, dist_two, dist_three, and dist_four.


Do the distributions look normal?

One of these distributions would probably not be a good choice to use in an ANOVA comparison. Create a variable called not_normal and set it equal to the distribution number (1, 2, 3, or 4) that would be least suited for use in an ANOVA test.


Calculate the ratio of standard deviations between dist_two and dist_three, and store the value in a variable called ratio. View ratio. Is this “close enough” to perform a numerical hypothesis test between the two datasets?

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