Lesson Objectives

- Learn how to find the mean (average) for a group of numbers
- Learn how to find the median for a group of numbers
- Learn how to find the mode for a group of numbers

## How to find the Mean, Median, & Mode

In many cases, we want to find the middle value for a group of numbers. This can be useful for estimating various things.
There are many methods that can be used to find the middle value. The two most commonly employed methods are the mean (average) and the median.
We will see mean (average) and median data plastered everywhere.
We might be interested in a certain profession. We can look up the average pay to get an estimate of our earnings for the position.
We may also be interested in purchasing a particular car or buying a house in a certain neighborhood.
Many websites can be used to look up the average or median price paid. We may use this pricing information to evaluate offering prices and negotiate the best deal.

Example 1: Solve the following problem.

The following table represents the grades for three students: Jen, Pam, and Ben. Each took a different number of tests for the year. Find the mean (average) grade for each student.

Let's begin by finding the mean (average) grade for Jen. She has the following test scores:

81, 100, 93, 100, 91, 96, & 90

First, we sum all of the numbers in the group (all of the grades):

81 + 100 + 93 + 100 + 91 + 96 + 90 = 651

We divide this sum by the number (quantity) of numbers in the group. There are 7 test scores, so we will divide 651 by 7:

651 ÷ 7 = 93

Jen's mean (average) grade was a 93.

Next, let's find the mean (average) grade for Pam. She has the following test scores:

79, 73, 80, 65, 67

First, we sum all of the numbers in the group (all of the grades):

79 + 73 + 80 + 65 + 67 = 364

We divide this sum by the number (quantity) of numbers in the group. There are 5 test scores, so we will divide 364 by 5:

364 ÷ 5 = 72.8

Pam's mean (average) grade was a 72.8

Lastly, let's find the mean (average) grade for Ben. He has the following test scores:

99, 100, 89

First, we sum all of the numbers in the group (all of the grades):

99 + 100 + 89 = 288

We divide this sum by the number (quantity) of numbers in the group. There are 3 test scores, so we will divide 287 by 3:

288 ÷ 3 = 96.

Ben's mean (average) grade was a 96

Example 2: Solve the following problem:

Over the course of the previous two business weeks (ten days), closing stock prices were recorded for Maztoning (Symbol » MZT), Xaylon (Symbol » XAO), and Zerturo (Symbol » ZTU). In some instances, the stock had no activity for the day and no price was recorded. Find the median stock price for the ten-day period for each company:

Let's begin by finding the median closing stock price for Maztoning (MZT). The stock traded on 7 out of 10 days. The closing prices were:

694, 688, 642, 675, 622, 629, & 680

We begin by arranging our numbers from least to greatest:

622, 629, 642, 675, 680, 688, 694

We can use our formula (n + 1)/2 to find the median position:

n = 7 » (7 + 1)/2 = 8/2 = 4

This tells us our middle number is in the 4th position:

1 » 622

2 » 629

3 » 642

4 » 675

5 » 680

6 » 688

7 » 694

We can see that the number 675 is in the 4th position.

The median closing stock price for Maztoning over the ten-day period was 675.

Next, let's find the median closing stock price for Xaylon (XAO). The stock traded on 10 out of 10 days. The closing prices were:

45, 49, 51, 53, 50, 48, 41, 43, 39, & 35

We begin by arranging our numbers from least to greatest:

35, 39, 41, 43, 45, 48, 49, 50, 51, 53

We can use our formula (n + 1)/2 to find the median position:

n = 10 » (10 + 1)/2 = 11/2 = 5.5

This tells us our middle number is the mean (average) of the number in the 5th position and 6th position.

1 » 35

2 » 39

3 » 41

4» 43

5 » 45

6 » 48

7 » 49

8 » 50

9 » 51

10 » 53

We will find the average of 45 and 48 » 45 + 48 = 93 » 93/2 = 46.5

The median closing stock price for Xaylon (XAO) over the ten-day period was 46.5.

Lastly, let's find the median closing stock price for Zerturo (ZTU). The stock traded on 6 out of 10 days. The closing prices were:

103, 141, 205, 180, 123, 139

We begin by arranging our numbers from least to greatest:

103, 123, 139, 141, 180, 205

We can use our formula (n + 1)/2 to find the median position:

n = 6 » (6 + 1)/2 = 7/2 = 3.5

This tells us our median number is the mean (average) of the number in the 3rd position and 4th position.

1 » 103

2 » 123

3 » 139

4 » 141

5 » 180

6 » 49

7 » 205

We will find the average of 139 and 141 » 139 + 141 = 280 » 280/2 = 140

The median closing stock price for Zerturo (ZTU) over the ten-day period was 140.

Example 3: Solve each problem:

Bowling scores for twelve games for Dan, Ivy, & Kim are presented in a table. Find the mode for each person's bowling scores (Dan, Ivy, & Kim):

Let's begin by looking at Dan's scores:

300, 300, 295, 300, 290, 291, 280, 295, 277, 280, 300, 295

It is helpful to arrange the numbers from least to greatest. This will allow us to see duplicates more easily.

277, 280, 280, 290, 291, 295, 295, 295, 300, 300, 300, 300

Looking through our list, we see that 300 appears most frequently (four times).

The mode for Dan's scores is 300.

Next, let's take a look at Ivy's scores:

100, 117, 120, 115, 125, 100, 105, 100, 115, 115, 100, 115

Let's arrange our numbers from least to greatest.

100, 100, 100, 100, 105, 115, 115, 115, 115, 117, 120, 125

Looking through our list, we can see that we have two modes (115) and (100). These two numbers each occur four times in our list. This group of numbers is said to be bimodal, meaning we have two modes.

The two modes for Ivy's scores are 100 and 115.

Lastly, let's take a look at Kim's scores:

206, 118, 155, 206, 130, 290, 209, 130, 155, 227, 200, 155

Let's arrange our numbers from least to greatest.

118, 130, 130, 155, 155, 155, 200, 206, 206, 209, 227, 290

Looking through our list, we see that 155 appears most frequently (three times).

The mode for Kim's scores is 155.

### How to find the Mean (Average) for a Group of Numbers

The mean of a group of numbers is more popularly known as the average. We all see this in school as we receive a final grade for the quarter, semester, or year based on the average of all of our test scores and other activities. In order to find the mean or average for a group of numbers:- Find the sum of all the numbers in the group
- Divide the sum by the number (quantity) of numbers in the group

Example 1: Solve the following problem.

The following table represents the grades for three students: Jen, Pam, and Ben. Each took a different number of tests for the year. Find the mean (average) grade for each student.

Test Scores | |||
---|---|---|---|

Name | Jen | Pam | Ben |

#1 | 81 | 79 | 99 |

#2 | 100 | 73 | 100 |

#3 | 93 | 80 | 89 |

#4 | 100 | 65 | |

#5 | 91 | 67 | |

#6 | 96 | ||

#7 | 90 |

81, 100, 93, 100, 91, 96, & 90

First, we sum all of the numbers in the group (all of the grades):

81 + 100 + 93 + 100 + 91 + 96 + 90 = 651

We divide this sum by the number (quantity) of numbers in the group. There are 7 test scores, so we will divide 651 by 7:

651 ÷ 7 = 93

Jen's mean (average) grade was a 93.

Next, let's find the mean (average) grade for Pam. She has the following test scores:

79, 73, 80, 65, 67

First, we sum all of the numbers in the group (all of the grades):

79 + 73 + 80 + 65 + 67 = 364

We divide this sum by the number (quantity) of numbers in the group. There are 5 test scores, so we will divide 364 by 5:

364 ÷ 5 = 72.8

Pam's mean (average) grade was a 72.8

Lastly, let's find the mean (average) grade for Ben. He has the following test scores:

99, 100, 89

First, we sum all of the numbers in the group (all of the grades):

99 + 100 + 89 = 288

We divide this sum by the number (quantity) of numbers in the group. There are 3 test scores, so we will divide 287 by 3:

288 ÷ 3 = 96.

Ben's mean (average) grade was a 96

### How to find the Median for a Group of Numbers

We can also use the median to think about the middle value for a group of numbers. If a group of numbers is arranged from least to greatest, the median is the middle value. Median values are not impacted by outliers (very high or very low values). Outliers can impact an average significantly and in some cases misrepresent the middle value. Since the median is not impacted by outliers, it is normally considered more desirable for use with pricing data. We will often see economic reports with median: house prices, salary data, or monthly family budgets by region. In order to calculate the median for a group of numbers:- Arrange the numbers from least to greatest (equal values are placed next to each other)
- If the number (quantity) of numbers in the group is:
- odd - the median is the middle number
- even - the median is the average of the two middle numbers

- There is no formula to calculate the actual median value. We can use a formula to obtain the position of the median value:
- When the number (quantity) of numbers in the group is odd, the median position is given as:
- (n + 1)/2 » n represents the number (quantity) of numbers in the group

- When the number (quantity) of numbers in the group is even, we use the same formula with one additional step:
- (n + 1)/2 » n represents the number (quantity) of numbers in the group
- The formula here gives us a position number with .5 at the end, we want to find the average of the numbers that are .5 units above and .5 units below

- When the number (quantity) of numbers in the group is odd, the median position is given as:

Example 2: Solve the following problem:

Over the course of the previous two business weeks (ten days), closing stock prices were recorded for Maztoning (Symbol » MZT), Xaylon (Symbol » XAO), and Zerturo (Symbol » ZTU). In some instances, the stock had no activity for the day and no price was recorded. Find the median stock price for the ten-day period for each company:

Stock Prices | |||
---|---|---|---|

Symbol | MZT | XAO | ZTU |

#1 | 694 | 45 | 103 |

#2 | 688 | 49 | 141 |

#3 | 51 | ||

#4 | 642 | 53 | 205 |

#5 | 675 | 50 | |

#6 | 48 | 180 | |

#7 | 622 | 41 | |

#8 | 629 | 43 | 123 |

#9 | 680 | 39 | |

#10 | 35 | 139 |

694, 688, 642, 675, 622, 629, & 680

We begin by arranging our numbers from least to greatest:

622, 629, 642, 675, 680, 688, 694

We can use our formula (n + 1)/2 to find the median position:

n = 7 » (7 + 1)/2 = 8/2 = 4

This tells us our middle number is in the 4th position:

1 » 622

2 » 629

3 » 642

4 » 675

5 » 680

6 » 688

7 » 694

We can see that the number 675 is in the 4th position.

The median closing stock price for Maztoning over the ten-day period was 675.

Next, let's find the median closing stock price for Xaylon (XAO). The stock traded on 10 out of 10 days. The closing prices were:

45, 49, 51, 53, 50, 48, 41, 43, 39, & 35

We begin by arranging our numbers from least to greatest:

35, 39, 41, 43, 45, 48, 49, 50, 51, 53

We can use our formula (n + 1)/2 to find the median position:

n = 10 » (10 + 1)/2 = 11/2 = 5.5

This tells us our middle number is the mean (average) of the number in the 5th position and 6th position.

1 » 35

2 » 39

3 » 41

4» 43

5 » 45

6 » 48

7 » 49

8 » 50

9 » 51

10 » 53

We will find the average of 45 and 48 » 45 + 48 = 93 » 93/2 = 46.5

The median closing stock price for Xaylon (XAO) over the ten-day period was 46.5.

Lastly, let's find the median closing stock price for Zerturo (ZTU). The stock traded on 6 out of 10 days. The closing prices were:

103, 141, 205, 180, 123, 139

We begin by arranging our numbers from least to greatest:

103, 123, 139, 141, 180, 205

We can use our formula (n + 1)/2 to find the median position:

n = 6 » (6 + 1)/2 = 7/2 = 3.5

This tells us our median number is the mean (average) of the number in the 3rd position and 4th position.

1 » 103

2 » 123

3 » 139

4 » 141

5 » 180

6 » 49

7 » 205

We will find the average of 139 and 141 » 139 + 141 = 280 » 280/2 = 140

The median closing stock price for Zerturo (ZTU) over the ten-day period was 140.

### How to find the Mode for a Group of Numbers

In some cases, we want to know which number occurs most frequently throughout a group of numbers. The mode for a group of numbers is the number or numbers that occur most often. Let’s look at an example.Example 3: Solve each problem:

Bowling scores for twelve games for Dan, Ivy, & Kim are presented in a table. Find the mode for each person's bowling scores (Dan, Ivy, & Kim):

Bowling Scores | |||
---|---|---|---|

Name | Dan | Ivy | Kim |

#1 | 300 | 100 | 206 |

#2 | 300 | 117 | 118 |

#3 | 295 | 120 | 155 |

#4 | 300 | 115 | 206 |

#5 | 290 | 125 | 130 |

#6 | 291 | 100 | 290 |

#7 | 280 | 105 | 209 |

#8 | 295 | 100 | 130 |

#9 | 277 | 115 | 155 |

#10 | 280 | 115 | 227 |

#11 | 300 | 100 | 200 |

#12 | 295 | 115 | 155 |

300, 300, 295, 300, 290, 291, 280, 295, 277, 280, 300, 295

It is helpful to arrange the numbers from least to greatest. This will allow us to see duplicates more easily.

277, 280, 280, 290, 291, 295, 295, 295, 300, 300, 300, 300

Looking through our list, we see that 300 appears most frequently (four times).

The mode for Dan's scores is 300.

Next, let's take a look at Ivy's scores:

100, 117, 120, 115, 125, 100, 105, 100, 115, 115, 100, 115

Let's arrange our numbers from least to greatest.

100, 100, 100, 100, 105, 115, 115, 115, 115, 117, 120, 125

Looking through our list, we can see that we have two modes (115) and (100). These two numbers each occur four times in our list. This group of numbers is said to be bimodal, meaning we have two modes.

The two modes for Ivy's scores are 100 and 115.

Lastly, let's take a look at Kim's scores:

206, 118, 155, 206, 130, 290, 209, 130, 155, 227, 200, 155

Let's arrange our numbers from least to greatest.

118, 130, 130, 155, 155, 155, 200, 206, 206, 209, 227, 290

Looking through our list, we see that 155 appears most frequently (three times).

The mode for Kim's scores is 155.

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