MEAN, MEDIAN, MODE, AND RANGE

Mean Median

Mode Range

Concept: The words mean, median, mode, and range are used in the study of statistics. Perhaps an example will help to explain the meanings of these words.

Paul kept track of his blood sugar readings for one week. These were the readings. 105 95 110 85 90 100 90
If we arrange these readings from smallest to largest, they look like this. 85 90 90 95 100 105 110
The mean of a group of numbers is the average. To find the average of a group of numbers, add the integers together and divide the sum of the integers by the total number of integers. 1 2 3 4 5 6 7 85 + 90 + 90 + 95 + 100 + 105 + 110 = 675 675 ÷ 7 = 96.4 The mean in this group of numbers is 96.4.
The median of a group of numbers is the middle integer. 85 90 90 95 100 105 110
The median in this group of numbers is 95. When there are an even number of integers in a group, the median will be the average of the middle two integers.
The mode is the number which occurs more than any other number. 85 90 90 95 100 105 110 MODE The mode in this group of numbers is 90. There can be more than one mode in a group of numbers.
The range is the difference between the smallest number and the largest number. RANGE 85 90 90 95 100 105 110 110 – 85 = 25 The range is 25.

SPECIAL NOTE Solving Intermediate Median Problems often involves the use of multiple operations as well as the use of some basic algebraic equations. The Steps for Solving Intermediate Median Problems must be applied fluidly.

Steps for Solving Intermediate Median Problems

Step 1 | Arrange the list of numbers in ascending order (smallest to largest).
Step 2 | Count the number of integers in the list.
Step 3 | If the number of integers in the list is odd, determine the middle integer. That is the median.
Step 4 | If the number of integers in the list is even, determine the middle two integers and average them. The average of those two numbers is the median.
Step 5 | Write the final answer.

Example Omar bought chickens for 6 hours. Each hour he recorded how many chickens he bought. The median number he bought was 186.5 Alzheimer’s allowed him to remember his buys for only 5 hours. They were: 166, 178, 194, 197, 200. How many chickens did he buy in the other hour?

Step 1 | Arrange the list of numbers in ascending order (smallest to largest). |
| 166, 178, 194, 197, 200, and x. We do not know the value of x, so we do not know where it belongs in the order yet. |
Step 2 | Count the number of integers in the list. |
| 1 2 3 4 5 6 166 178 194 197 200 x | There are 6 numbers
Step 3 | If the number of integers in the list is odd, determine the middle integer. That is the median. | Does not apply.
Step 4 | If the number of integers in the list is even, determine the middle two integers and average them. The average of those two numbers is the median. | 6 is an even number. We must find the average of the two middle integers.
| We know the average of the two middle numbers is 186.5. That means that one number will be bigger than 186.5 and one number will be smaller than 186.5. We don’t know what the two numbers are, but since 186.5 falls between 178 and 194, we can conclude that one of the numbers used to find the average is 178 or 194. Let us try both 178 and 194 to see which number works. We are solving for X, and X must fall between 178 and 194. |
| We will try 178 first. |
| Next we will try 194. |
| 195 does not fall between 178 and 194, but 179 does, so 179 is our answer. To check it, substitute 179 for X. |
Step 5 | Write the final answer. | Omar bought 179 chickens in the other hour.