When you have a set of data, one of the first things you might want to know is, "What's a typical value?" If you have a list of test scores for a class, for example, you'd want a single number that represents how the class did overall. This single, representative number is what statisticians call a measure of central tendency.

There are three main ways to measure central tendency: the mean, the median, and the mode. Each gives you a different way of looking at the data's center, and the best one to use depends on the situation.

The most common measure of central tendency is the mean, which is just the arithmetic average. You've probably calculated it many times. You simply add up all the values in your data set and then divide by the number of values.

If we represent our data set as $x_1, x_2, ..., x_n$, the formula for the mean (represented by the Greek letter $\mu$, pronounced 'myoo') is:

Let's say a student's scores on five quizzes are 88, 92, 85, 95, and 90. To find the mean, you add them up ($88 + 92 + 85 + 95 + 90 = 450$) and divide by the number of quizzes (5). The mean score is $450 / 5 = 90$.

The mean is great because it uses every value in the dataset, giving a well-rounded picture. However, it has one major weakness: it's very sensitive to extreme values, also known as outliers.

The median is the middle value in a dataset that has been arranged in order from smallest to largest. It's the number that splits the data in half: 50% of the values are above it, and 50% are below it.

To find the median, you just line up your numbers and find the one in the middle. If you have an even number of data points, the median is the average of the two middle numbers.

Because the median focuses on position, it isn't affected by outliers. Let's look at those quiz scores again, including the low score: 20, 85, 88, 90, 92, 95. The two middle numbers are 88 and 90. The median is their average, which is 89. This is much closer to the student's typical performance than the mean of 78.3.

This resistance to outliers makes the median a very useful measure, especially for skewed data like income or housing prices, where a few very high values can make the average misleadingly high.

The mode is the simplest measure of central tendency. It's the value that appears most frequently in a dataset.

A dataset can have one mode, more than one mode (bimodal or multimodal), or no mode at all if every value occurs only once. For example, in the dataset {Red, Blue, Blue, Green, Red, Blue}, the mode is Blue because it appears three times, more than any other color.

The mode is the only measure of central tendency that can be used for categorical data (data that isn't numerical). You can't calculate the average favorite color, but you can find the most popular one.

So, which measure should you use? It depends entirely on your data and what you want to communicate.

Imagine we're looking at the salaries of 10 employees at a small company:

$45k, $50k, $50k, $55k, $60k, $65k, $70k, $75k, $80k, $250k

• Mean: $80,000. The high salary of the CEO ($250k) pulls the average up significantly.
• Median: $62,500. This is the average of the two middle salaries ($60k and $65k) and gives a better sense of a 'typical' employee's salary.
• Mode: $50,000. This tells us the most common salary in the company.

In this case, the median is likely the best measure to describe the central tendency of salaries. The mean is skewed by the outlier, and the mode only represents the lowest-paid group.

Ready to test your understanding of these core statistical concepts?

Understanding mean, median, and mode is the first step in summarizing and making sense of data. They are the fundamental building blocks of descriptive statistics.