Introduction to Measures of Central Tendency
Measure what is measurable, and make measurable what is not so.
—GALILEO GALILEI (1564–1642)
This lesson will explore the basic measures of central tendency—mean, median, mode, and range.
When you are dealing with sets of numbers, there are measures used to describe the set as a whole. These are called measures of central tendency, and they include mean, median, mode, and range.
Mean
Mean is another way of saying average. To find the average, you total up all the values and then divide by the number of values.
Sound easy enough? Let's try a problem:
Find the mean of the following set: {17, 22, 18, 31, 27, 17}
Add up the six numbers in the set: 17 + 22 + 18 + 31 + 27 + 17 = 132.
Now, divide 132 by 6, the number of entries in the set: 132 ÷ 6 = 22.
The mean (or average) of the set is 22.
Let's try another.
The temperature, in degrees Fahrenheit, for the first week of July is as follows: 84, 88, 86, 87, 80, 84, and 86. What is the average temperature for the week?
Add up the seven temperatures: 84 + 88 + 86 + 87 + 80 + 84 + 86 = 595; 595 divided by 7, the number of days measured, is 595 ÷ 7 = 85. The average temperature is 85º Fahrenheit.
Tip:If you are asked to find the mean of a set of numbers, and the set is evenly spaced apart such as 2, 4, 6, 8, 10, 12, 14, the mean is the middle number in this set, because there is an odd number of data items.In this example, the mean is 8. If there is an even number of data items, there are two middle numbers: 4, 8, 12, 16, 20, and 24. In this case,the mean is the average of the two middle numbers: 12 + 16 = 28, and 28 divided by 2 is 14. |
Median
When you are considering a list of values in order (from smallest to largest), the median is the middle value. If there are two "middle" values, then you just take their average.
Let's find the median of 2, 8, 3, 4, 7, 6, and 6. First you put these numbers in order:
Next, circle the middle number:
What's the median of the following?
The numbers are already listed in order, so you don't have to worry about arranging them. Notice that this list of numbers has two middle terms:
In this case, you need to take the average of these two numbers to find the median.
The median is 5.
Mode
In a list of values, the mode is the number that occurs the most. If two numbers occur "the most," then you have two modes. This is called bimodal.
Let's find the mode of the following numbers: 35 52 17 23 51 52 18 32
In this series of numbers, you see that 52 appears twice.
35 52 17 23 51 52 18 32
So, the mode is 52.
Range
The range indicates how close together the given values are to one another in a set of data. To find the range, determine the difference between the largest and the smallest values in the set of data. Subtract the smallest value from the largest value in the set.
Let's see how this works.
Find the ranges of ages in the community play, given these ages in years: 68, 54, 49, 40, 39, 39, 24, 22, 20, 10, and 10.
The range of ages is 68 – 10 = 58 years. Now, find the range of this set: {42, 40, 45, 43, 43, 40, 45}
Find the largest and smallest values in the set. In this example, these are 45 and 40, respectively. The difference between 45 and 40, the range, is 45 – 40 = 5.
Find practice problems and solutions for these concepts at Measures of Central Tendency Practice Questions.
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