Frequentist Statistics

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What is frequentist statistics?

Frequentist statistics uses formal frameworks and rules to conduct hypothesis tests and find p-values and confidence Intervals. Probability is defined as long-term frequency in a repeatable, random experiment [1].

For every statistics problem, there’s data. And for every data set there’s a test with rigid formulas and rules.

For example, we might say: “If H0 is true, then we would expect to get a result as extreme as the one obtained from our sample 2.9% of the time. Since that p-value is smaller than our alpha level of 5%, we reject the null hypothesis in favor of the alternate hypothesis.”

Deviation from the rules is never allowed if you want your frequentist research to be considered statistically sound.

Frequentist statistics is the type of statistics you’ll come across in statistics 101. We usually drop the “frequentist” and just say statistics. However, there is another popular approach to statistics, called Bayesian statistics. With this approach, rules are more flexible and developed from the best known information available.

Bayesian vs frequentist

Bayesian and frequentist approaches are quite different and stem from different camps [2]:

  • Frequentist: Based on work by Jerzy Neymann (1894-1981), Karl Pearson (1857 – 1936) and Abraham Wald (1902 – 1950). This relatively orthodox view of statistics tells us sampling is infinite with — usually — sharp decision rules.
  • Bayesian: Based on work by Thomas Bayes (1702 – 1761), Pierre-Simon Laplace (1749 – 1827), and Bruno de Finetti (1906 – 1985). Unknown quantities are treated probabilistically and the state of the world is seen as always updatable.
bayesian vs frequentist statistics
Image: XKCD

Frequentist statistics is based on the idea that probability is the frequency of an event over many trials. In comparison, Bayesian statistics is based on the idea that probability is a measure of belief; It takes into account prior knowledge about a situation when calculating probabilities. In other words, while frequentist probability is in principle independent of the observer, Bayesian probability is a property of the observer and the system being observed [3].

Here is a simple analogy that may help to understand the difference between the two camps:

Imagine you are flipping a coin. A frequentist statistician would say that the probability of flipping heads is 50%, because if you flip the coin a large number of times, you will get heads about half the time. A Bayesian statistician, on the other hand, might start with a different probability if they know something about the coin. For example, if they know that the coin is weighted so that it is more likely to land on heads, they might start with a prior probability of 55%.

When you flip the coin, the frequentist statistician would update their probability estimate to reflect the outcome of the flip. The Bayesian statistician, on the other hand, would use Bayes’ theorem to update their prior probability in light of the new information.

Bayesian vs frequentist statistics demonstrated with coin flips.

The four pillars of frequentist statistics

In addition to the normal distribution, three other standardized probability distributions are known as the “big four” of frequentist statistics. These are the chi-squared-distribution, t-distribution, and f-distribution. Each of these three distributions is actually a family of distributions, with each family consisting of many instances of the same type of distribution [4]:

  • Normal distribution: used for sample statistics related to binomials and ranks.
  • T-distribution: for sample statistics related to means—such as average height of a sample of people. The t-distribution incorporates additional uncertainty into the equation. When there is no additional uncertainty, it is equivalent to the normal-distribution
  • Chi-squared distribution: For sample frequentist statistics related to variances—such as the variety of heights within a sample of people.
  • F-distribution: For comparing two sample variances—such as comparing the variety of heights between two samples of people.


[1] Orloff, J. & Bloom, J. The Frequentist School of Statistics. Retrieved October 25, 2023 from:

[2] Casella, G. Bayesians and Frequentists: Models, Assumptions, and Inference. Retrieved October 25, 2023 from:

[3] James, F. A Unified Approach to Understanding Statistics. CERN.

[4] J.E. Kotteman. Statistical Analysis Illustrated – Foundations . Published via Copyleft. You are free to copy and distribute the content of this article.

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