Introduction

Winny the Pooh meme, "mean, mode, and median" vs. "measures of central tendency"

We will learn to calculate some values in this section that are types of "measures of center" of data sets, or measures of central tendency. The main values we will discuss are mean, median, and mode. I will also briefly introduce range of a data set here, because it is very relevant!

All of these give slightly different information to us about what a data set looks like. Before we get to interpreting these values, let's find out how to identify them!

The mean (or average) of a data set is calculated by adding together all of the values and then dividing by the number of values there are. We also call this value of a sample " $LaTeX: \overline{x}$ " and pronounce this "x-bar." If we are talking about the mean of a population, then we refer to this value as " $LaTeX: \mu$ " and we pronounce this "mew".

For example:

The mean of the data set {1, 3, 3, 5, 8, 9} is found by adding all of these together (so 1+3+3+5+8+9) and then dividing by 6 (since there are six values listed). Then, we would get

$LaTeX: \overline{x}=\frac{1+3+3+5+8+9}{6}=\frac{29}{6}\approx4.83$

Notice the " $LaTeX: \approx$ " symbol above - this means that 4.83 is a rounded value. You can think of this symbol as saying "approximately" or "about". So, the mean of this sample is about 4.83.

The median of a data set is the second quartile, or LaTeX: Q_2 , which we learned about in Sections 2.3 and 2.4! This is the very middle of the data set (when it is in order), so 50% of the data is larger than this value and 50% of the data is less than this value!

In the data set above, there are two numbers that are in the "middle" of the data set, 3 and 5. We can only have one median, so when this happens, we find the mean of these two middle data values and name that average as the median of the data set.

For example, the median of the data set {1, 3, 3, 5, 8, 9} would be $LaTeX: \frac{3+5}{2}=\frac{8}{2}=4$ . So, the median of this data set is 4.

The mode of a data set is the most frequent data value(s) seen in the data set.

For example, the mode of the data set {1, 3, 3, 5, 8, 9} would be 3, because it is listed the greatest number of times.

*Note that we can have more than one mode (if there is more than one number listed the most frequently).

For example, let's say our data set was {1, 3, 3, 5, 5, 8, 9}, then there would be two modes, which are 3 and 5, because they are both listed twice (which is the most frequent number of times for any value). If this happens, we say that the data is bimodal (so, "bi" is a prefix that means "two", and "modal" refers to "modes" of a data set). I think we could assume if a data set has three modes that it would be considered trimodal(?).

We may also see a situation when we say there is "no mode", which happens if all numbers are listed the same number of times, or if the whole data set is the same value.

Although it is not considered a "measure of center", the range of the data set is the difference between the highest and lowest values in a data set. So, the range is equal to the maximum value minus the minimum value.

For example, the range of the data set {1, 3, 3, 5, 8, 9} would be LaTeX: 9-1=8 . So, the range of this data set is 8.

visual example of mean, median, and mode of a small data set

Fair Warning: A symbol you will start to see throughout the course is this " $LaTeX: \Sigma$ ". This symbol essentially means "to add up" values.

For example, when we find the mean of a data set and we are adding up all of the values in the data set, we can write this as $LaTeX: \Sigma i$ , where LaTeX: i is referring to each individual value in the data set.

What You Will Do

Watch this video that is meant to introduce the concepts of mean, median, and mode:

Watch this video from Khan Academy about mean, median, and mode:

Watch this video from Khan Academy where he works through an example of finding a missing data value if we are given the mean of a data set:

Read about mean, median, and mode in our textbook:

2.5 Measures of the Center of the Data

What's Next

You will learn about how we can talk about measures of center if we only have a frequency (or relative frequency) table or a histogram. You will also learn about skewness of data and how this relates to these measures of center! Then, you will take a Progress Check about the material so far.