# Mean, Median, and Mode

Finding the mean, also known as averaging numbers, is a very useful thing to know how to do, especially when you need a precise estimate or a very accurate generalization. Means and medians are not exact numbers; however, they are based on a series of exact numbers, therefore they are precise. Finding the mean (average) is used most often when figuring out students’ grades, the price of an item, and other things such as the daily temperature. Median is used to find the mid-section of a group of numbers, and mode is used to find the most popular term of the series (the number that appears the most often).

In this lesson, we’ll take you through how to find the mean (average), median, and mode of a series of numbers, as well as give you several examples of instances when you would use each.

## Mean

There are two steps to finding the mean, or averaging numbers; they are, first, to add up the series of numbers you have and, second, to divide the sum by the number of numbers in the series. Let’s go through an example so that you can see each step. Here is our series:

7, 9, 2, 4, 13, 9, 17.

Now, we’re going to find the mean of these numbers. The first step in averaging is to add up the numbers we have, so we do the following: 7 + 9 + 2 + 4 + 13 + 9 + 17 = 61.

Next, we have to divide 61 by the number of numbers we added together. Counting them, we see that we have 7 numbers that we added together, therefore we’re going to divide 61 by 7. Notice that we did not finish our long division, we stopped after we reached the thousandths’ place value. Sometimes, you will be asked to round your answer to a certain place value; in this case, we are going to round our answer to the hundredths’ place value, making our answer 8.71. Normally, you would finish out the division unless you are told to stop and round to a certain place value.

Let’s try this one more time. This time we’ll give you the numbers, so that you can practice finding the mean on your own. After you’re done, type your answer into the box and check it with ours to see if you did it correctly. Here are the numbers you need to average:

17, 29, 26, 15, 23, 21, 19, 20.

When you add the series of numbers together, the sum is 170. Then, you would divide 170 by 8, because you added 8 numbers together. 170 divided by 8 = 21.25. Thus, the average of the series of numbers you were given is 21.25.
21.25

Many students like to be able to average because then they can figure out their overall grade. For example, most teachers give their students percentage grades, ranging from 0 to 100. In order to figure out students’ final grades, teachers average the percentages together and the averaged number is the grade. We’ll take you through an example of averaging grades so that you can see how that would work.

Amy’s math grades are: 99, 89, 94, 85, 79, 83, 95, 93, 100, 94, 80, 100.

First, you would add together her grades. When you add her grades together, you get 1,091. Next, you would divide 1,091 by 12 because 12 numbers were added together. You’re going to get a decimal answer, but we’re going to round to the nearest percent. When you divide, you would get 90.916667. In order to round to the nearest percent, you would look at the digit in the tenths’ place, which in this case is a 9. 9 rounds up to the next whole percent, so your final answer would be 91.

Now, you can try it. We’ll give you the numbers so that you can average them, and then you can type your answer into the box to check it.

Here are Zach’s Language Arts grades: 100, 76, 92, 79, 88, 85, 93, 82, 100, 68, 97, 89. When you average, round to the nearest whole number.

First, you would add all of Zach’s grades together. When you add them together, you get a total of 1,049. The next step would be to divide 1,049 by 12 because we added 12 numbers together to get 1,049. When you divide, you get 87.416667. More than likely, we’ll want to round to the nearest percent, so we look at the digit in the tenths’ column. In this case, the tenths’ digit is a 4, so we would round down. Thus, our final rounded answer is 87.
87

### Averaging Money

Averaging money is used to figure out a general amount an item costs, or, a general amount spent over a period of time. For example, you could average your weekly grocery bills to see how much you normally spend on groceries in a week. It wouldn’t be an exact number—but the average would help you plan to save the amount that you’ll probably need.

Here’s an example. Katy is trying to figure out how much 32" TVs cost. She knows that they don’t all cost the same price, but she wants a general idea of how much she needs to save in order to buy one, so she decides to look up the top 4 brands of TVs and average the amounts together. Brand A costs \$339.99. Brand B costs \$359.99. Brand C is on sale for \$319.95. And, Brand D costs \$349.99. What is the general amount she should probably save for a 32" TV?

First, add the four TV prices together, like this: \$339.99 + \$359.99 + \$319.95 + \$349.99 = \$1,369.92. Then, you would divide the total by 4, since you are comparing 4 different brands of TVs. When you divide \$1,369.92 by 4, you get \$342.48. In this case, you would want to report 2 decimal places (that’s all there were in this problem) because money has two decimal places. So, in the end, Katy knows that she should try to save at least \$342.48 in order to buy her TV.

Now, we’ll give you an example to try. When you’re done, type your answer into the box to see if it’s right! (Don’t forget your dollar sign, \$, and decimal point.)

Jacqui is trying to figure out how much she spends per week on groceries. She has her grocery store receipts from the past month, and she wants to use these amounts in order to figure out how much she spends. The first week, she spent \$98.57. The second week, she spent \$105.92. The third week, she spent \$89.48, and the fourth week she spent \$100.39. How much does she spend per week on groceries, on average?

First, you would add together the four amounts, and you would get \$394.36. Then, you would divide that number by 4, because you are averaging 4 numbers together. When you divide \$394.36 by 4, you get \$98.59 as the average amount she spends per week. Notice that the average is in the middle of the original numbers, it is not at one end or the other. This is true for all answers to averaging problems.
{\$98.59| 98.59}

## Median

Finding the median of a set of numbers is similar to finding an average, but it relies less on averaging and more on where the middle number is in the series. In order to find the median of a series of numbers, you first need to order the series from least to greatest. Then, you would count how many numbers there are in the series. If there is an odd number of numbers, find the middle number (for example, if there are 7 numbers, the 4th number would be the middle, because there would be 3 numbers on either side of it). If there were an even number of numbers, you would find the middle two numbers (for example, if you had 8 numbers, the middle two numbers would be the 4th and 5th numbers, because there would be 3 numbers on either side of them). Once you have the middle two numbers picked out, you would average them together—add them together and divide by 2. This middle number, in both cases, is the median of the series of numbers.

Let’s practice this. The first series of numbers is: 2, 6, 9, 5, 7, 5, 3, 9, 10, 4, 8. The first step is to order to the numbers from least to greatest. When we re-order them, we place any doubles right next to each other, like this: 2, 3, 4, 5, 5, 6, 7, 8, 9, 9, 10. Notice that the two 5s and two 9s are right next to each other in numerical order. Now, our next step is to see how many numbers we have, and if it is an odd or even number. After counting up the numbers, we see that we have 11 numbers. 11 is an odd number, so we can figure out which number is in the middle. We count and see that 6 is in the middle, with 5 numbers on each side. We look at the series that is now re-ordered (from least to greatest) and find the 6th number on the list, which is 6. Thus, 6 is our median of this series.

We’ll try it one last time. This time we’ll give you the numbers and you can find the median on your own, and then type it into the box to check your answer with ours!

Here is the series of numbers: 12, 43, 29, 38, 59, 28, 73, 13, 45, 47, 96, 84. Find the median of the series.

First you would re-order the series of numbers from least to greatest. The list looks like this: 12, 13, 28, 29, 38, 43, 45, 47, 59, 73, 84, 96. Then, you count how many numbers there are; in this case we have 12. 12 is an even number, so you have to find the two middle numbers and average them together. We find that 6 and 7 are our two middle numbers, and there are 5 numbers on each side of them. So, we count and find that our 6th number is 43 and our 7th number is 45. Now, we have to average those two numbers together. First, we add them, and get 88. Then we have to divide 88 by 2, and we would get 44. Thus, our median of this series of numbers is 44.
44

## Mode

The mode of a series of numbers is the number that appears the most often. Some series do not have a mode, because there are no repeating numbers. However, other series have a mode—the number that occurs most often.

Let’s look at a series of numbers: 3, 2, 6, 4, 9, 8, 4, 7, 10. Notice that the series has one of each number, but has two 4s. Thus, 4 is the mode because it occurs more often than any other number.

There are sometimes exceptions to this rule. For example, take the series from the first example of the last section. The series was 2, 6, 9, 5, 7, 5, 3, 9, 10, 4, 8. Note that there are two 5s and two 9s. In this example there are actually two modes—5 and 9.

Here are a couple examples you can try on your own. When you’re done, type the answer into the box so you can check your answer with ours.

Series: 43, 23, 88, 45, 56, 67, 56, 58, 48, 93, 84, 48, 28, 29, 28, 39, 28, 30. Find the mode.

You could first put them in order from least to greatest; this tends to be easier to help spot multiple instances of the same number. Here is the newly ordered list: 23, 28, 28, 28, 29, 30, 39, 43, 45, 48, 48, 56, 56, 58, 67, 84, 88, 93. Then, look at the numbers that occur more than once. We see that 56 occurs twice, 48 occurs twice, and 28 occurs three times. 28 is the most often occurring number, therefore 28 is our mode.
28

Series: 77, 78, 73, 75, 77, 83, 74, 82, 71, 70, 80. Find the mode.

First, you can order the numbers from least to greatest, like this: 70, 71, 73, 74, 75, 77, 77, 78, 80, 82, 83. Next, look for any repeating numbers. In this series, there is only one number that repeats: 77. Thus, our answer is 77.
77

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