4.11 Measures of Central Tendency

Learning Outcomes

  • Calculate the mean, median, and mode of a set of data

Practice makes perfect

You’ve seen mean, median, and mode already and practiced finding them. Now, you’ll see how they relate to different statistical situations.  These 3 are collectively called Measures of Central Tendency.

Mean, Median, and Mode (Measures of Central Tendency)

Maybe you’ve seen mean, median, and mode already. Now, you’ll see how they relate to different statistical situations.  These 3 are collectively called Measures of Central Tendency.

Silver sphere which has red smaller spheres clustered on its left side, with magnetic attraction

Let’s begin by trying to find the most “typical” value of a data set.

Note that we just used the word “typical” although in many cases you might think of using the word “average.” We need to be careful with the word “average” as it means different things to different people in different contexts.  One of the most common uses of the word “average” is what mathematicians and statisticians call the arithmetic mean, or just plain old mean for short.  “Arithmetic mean” sounds rather fancy, but you have likely calculated a mean many times without realizing it; the mean is what most people think of when they use the word “average.”

Mean

The mean of a set of data is the sum of the data values divided by the number of values.

The next measure of center is the median.

Median

The median of a set of data is the value in the middle when the data is in order.

  • To find the median, begin by listing the data in order from smallest to largest, or largest to smallest.
  • If the number of data values, N, is odd, then the median is the middle data value. This value can be found by rounding N/2 up to the next whole number.
  • If the number of data values is even, there is no one middle value, so we find the mean of the two middle values (values N/2 and N/2 + 1)

The average is one number in a set of numbers that is somehow typical of the whole set of numbers. The mean and median are both often called the average. Yes, it can be confusing when the word average refers to two different numbers, the mean and the median! In fact, there is a third number that is also an average. This average is the mode. The mode of a set of numbers is the number that occurs the most. The frequency, is the number of times a number occurs. So the mode of a set of numbers is the number with the highest frequency.

Mode

The mode is the element of the data set that occurs most frequently.

The mode is fairly useless with data like weights or heights where there are a large number of possible values. The mode is most commonly used for categorical data, for which median and mean cannot be computed.

It is possible for a data set to have more than one mode if several categories have the same frequency, or no modes if each every category occurs only once.

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