Probabilistic World

The Variance: Measuring Dispersion

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Small pawn-shaped magnets of different color, representing varianceA few posts ago I introduced you to the “three M’s” of statistics — the concepts of mean, mode, and median. Today I want to talk to you about a related concept called variance.

While the three M’s measure the central tendency of a collection of numbers, the variance measures their dispersion. That is, it measures how different the numbers are from each other.

Measuring dispersion is another fundamental topic in statistics and probability theory. On the one hand, it tells you how much you can trust the central tendency measures as good representatives of the collection. High variance usually means a lot of the numbers in the collection will be far away from those measures.

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The Mean, the Mode, And the Median

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Mean, median, and mode of a distribution shown on a Spaghetti Western background

The concepts of mean, median, and mode are fundamental to statistics, probability theory, and anything related to data analysis as a whole. Being this important, they deserve their own introduction.

In statistics, these 3 concepts are examples of measures of central tendency. This is a fancy way of saying that they are single values that summarize collections of values.

Let’s unpack the last sentence. What exactly do I mean by ‘values’ and ‘collections of values’?

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Frequentist and Bayesian Approaches in Statistics

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Dennis Lindley vs. Ronald Fisher

What is statistics about?

Well, imagine you obtained some data from a particular collection of things. It could be the heights of individuals within a group of people, the weights of cats in a clowder, the number of petals in a bouquet of flowers, and so on.

Such collections are called samples and you can use the obtained data in two ways. The most straightforward thing you can do is give a detailed description of the sample. For example, you can calculate some of its useful properties:

  • The average of the sample
  • The spread of the sample (how much individual data points differ from each other), also known as its variance
  • The number or percentage of individuals who score above or below some constant (for example, the number of people whose height is above 180 cm)
  • Etc.

You only use these quantities to summarize the sample. And the discipline that deals with such calculations is descriptive statistics.

But what if you wanted to learn something more general than just the properties of the sample? What if you wanted to find a pattern that doesn’t just hold for this particular sample, but also for the population from which you took the sample? The branch of statistics that deals with such generalizations is inferential statistics and is the main focus of this post.

The two general “philosophies” in inferential statistics are frequentist inference and Bayesian inference. I’m going to highlight  the main differences between them — in the types of questions they formulate, as well as in the way they go about answering them.

But first, let’s start with a brief introduction to inferential statistics.

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Coin Bias Calculation Using Bayes’ Theorem

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An ancient Greek coin with an unusual shape and lopsidedness

Why do people flip coins to resolve disputes? It usually happens when neither of two sides wants to compromise with the other about a particular decision. They choose the coin to be the unbiased agent that decides whose way things are going to go. The coin is an unbiased agent because the two possible outcomes of the flip (heads and tails) are equally likely to occur.

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