Mean, Median, Mode, Range

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Mean, Median, Mode, Range

Introduction

When you are given a list of numbers, mean, median, mode, and range are all useful tools for describing the numbers, and helping you get a sense of what the numbers signify.

For example, if you were a teacher, and your class of 6 students got the following scores out of 10 in a Math test:

$10, 7, 3, 6, 7, 9$

You can use the mean, median, mode, and range to understand whether the test was too hard or too easy, whether your students were evenly spread, and how you can organize your lessons and tests in future.

The mean, median, and mode are all different ways of finding the 'middle' of a block of numbers, and the range tells you the difference between the highest number and the smallest. Each tells you something slightly different about a group of numbers and knowing how to find each is important.

Terms

Mean - The mean is defined as the sum of all values divided by the number of values. Also known as Average

Median - The median is the middle value in a list when values are sorted.

Mode - The mode is the number which appears most frequently.

Range - The range is the difference between the highest and lowest values.

Lesson

Let's take the list of numbers above from the results of the Math test. The students got the following scores:

$10, 7, 3, 6, 7, 9$

The first thing we need to do is to put the numbers in ascending order of value:

$3, 6, 7, 7, 9, 10$

We can then work out the different calculations to find out the mean, median, mode, and range.

Mean

The 'mean' of a group of numbers is what we normally think of as the average. To find the mean, we add up all the values and divide it by how many values we have.

$3 + 6 + 7 + 7 + 9 + 10 = 42$

We have six scores, so we divide 42 by 6:

$42/6 = 7$

Therefore, the mean score is 8. Even though no student got 8, those who scored, 9 and 10 are above the mean, those who scored 3 and 6 were below, and the two students who scored 7 and 7 were exactly on the mean. To put it another way, if a new student joined the class and took the test, we would expect him to score 7 out of 10 on it.

Median

The median is the middle value of a group of numbers. The easiest way to find the median is to write the numbers in ascending order of value and then scratch out the highest and lowest values. Then scratch out the remaining highest and lowest values until you have only the middle number left.

$Êncel{3}, 6, 7, 7, 9, Êncel{10}$

$Êncel{3}, Êncel{6}, 7, 7, Êncel{9}, Êncel{10}$

In this case, we are left with two middle numbers. When you have two middle numbers, you need to find the mean of them. Adding the two numbers up and dividing by two will give us the following:

$7 + 7 = 14$

$14/2 = 7$

Therefore, 7 is the median number.

Mode

Mode means the most common number in a list. Finding it is simple: you just look for which number occurs more than the others. In our list of numbers:

$3, 6, 7, 7, 9, 10$

7 occurs twice, so is the mode.

Range

The range of a group of numbers tells you how far apart the biggest and smallest values are. To find the range, subtract the smallest number from the largest numbers. So, in our list of test scores, we'd do the following:

$10 - 3 = 7$

The range of the numbers is therefore 7.

Try picking six of your own random numbers and finding the mean, median, mode, and range.

Examples

1, 2, 3, 4, 5

To find the mean, first add all the numbers together:

1 + 2 + 3 + 4 + 5 = 15

There are five numbers in the list

1, 2, 3, 4\text{ and }5

so we divide by five:

\dfrac{15}{5} = 3

The mean of the set is 3

Find The Mean

7, 8, 5, 2, 1

Right now the numbers are out of order, so it is difficult to tell which number will be in the middle of the list. So first put the numbers in order:

1, 2, 5, 7\text{ and }8

We can see that 5 is in the middle of the list. There are two numbers less than 5, and two numbers greater than 5.

The median of this set is 5

Find the Median

5, 4, 4, 1, 7, 6

Right now the numbers are out of order, so it is difficult to tell which number appears most often. So first put the numbers in order:

1, 4, 4, 5, 6\text{ and }7

The number that appears most often is 4, so 4 is the mode of the set

Find the Mode

5, 3, 1, 11, 19, 20, 14

Right now the numbers are out of order, so it is difficult to tell which number is the largest or the smallest. So first put the numbers in order:

1, 3, 5, 11, 14, 19\text{ and }20

Now it is easier to see that the smallest number in the list is 1 and the largest number is 20

To find the range, subtract 1 from 20:

20 - 1 = 19

The range of the set is 19