Measures of Central Tendency: Mean, Median, and Mode
- In This Quick Guide:
- Mean, or Average
- Median
- Mode
One broad category of descriptive statistics that is commonly used, measures of central tendency, describe the center point of our data set with a single value. It’s a valuable tool to help summarize many pieces of data with one number. In this guide, we will explore the many ways to measure the central tendency of your data.
Mean, or Average
The most common measure of central tendency is the mean or average, which we calculate by adding all the values in our data set and then dividing this result by the number of observations. The mathematical formula for the mean differs slightly depending on whether you’re referring to the sample mean or the population mean. The formula for the sample mean is as follows:
where:
= the sample mean
= the values in the sample (x1 = the first data value, x2 = the second data value, and so on)
= the sum of all the data values in the sample
n = the number of data values in the sample
The formula for the population mean is as follows:
where:
= the population mean (pronounced mu, as in “I hope you find this amusing”)
= the sum of all the data values in the population
N = the number of data values in the population
To demonstrate calculating measures of central tendency, let’s use the following example. As in many teenage households, video games are a common form of entertainment in our family room. Brian and John love to challenge me with a game and then clean my clock before I can ask “Which team is mine?” I suspected John of sticking me with the “bad” controller because it felt like a 10-second delay between pushing a button and the game responding. (Turns out the delay was really between my brain and my fingers.) Anyway, the following data set represents the number of hours each week that video games are played in our household.
3 7 4 9 5 4 6 17 4 7
Because this data represents a sample, we will calculate the sample mean:
It looks like I need some serious practice time to catch up to these guys.
Median
Another way to measure central tendency is by finding the median. The median is the value in the data set for which half the observations are higher and half the observations are lower. We find the median by arranging the data values in ascending order and identifying the halfway point.
Using our example with the video games, we rearrange our data set in ascending order:
3 4 4 4 5 6 7 7 9 17
Because we have an even number of data points (10), the median is the average of the two center points. In this case, that will be the values 5 and 6, resulting in a median of 5.5 hours of video games per week. Notice that there are four data values to the left (3, 4, 4, and 4) of these center points and four data values to the right (7, 7, 9, and 17).
To illustrate the median for a data set with an odd number of values, let’s remove 17 from the video games data and repeat our analysis.
3 4 4 4 5 6 7 7 9
In this instance, we only have one center point, which is the value 5. Therefore, the median for this data set is 5 hours of video games per week. Again, there are four data values to the left and right of this center point.
Mode
The last measure of central tendency on my mind is the mode, which is simply the observation in the data set that occurs the most frequently.
To illustrate the mode for a data set, let’s again use the original video game data.
3 4 4 4 5 6 7 7 9 17
The mode is 4 hours per week because this value occurs three times in the data set.
That wraps up all the different ways to measure central tendency of you data set. Armed with this information, you can now summarize your statistical data with ease. Good luck!
From The Complete Idiot’s Guide to Statistics, Second Edition, by Robert A. Donnelly, Jr., Ph.D.
