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11.1: Mean, Median and Mode

Difficulty Level: At Grade Created by: CK-12

Introduction

The Iditarod

The students in Mr. Hawkins class are studying about the Iditarod. Some of the students had never heard of the Iditarod before, so Mr. Hawkins started off his class by showing them this video about the race.

The students sat through the video in awe. When it was over, the room was so quiet that you could have heard a pin drop. Marcus was the first one to raise his hand.

“How far is it?” he asked.

“That is a great question,” Mr. Hawkins said. “The race is 1,150 miles from Anchorage Alaska to Nome Alaska. Men and women have finished it and won it too. This year, there were 10 men who finished on top. One finished in 8 days and the rest in 9 days.”

“How fast did they go?” Karen asked from the back of the room. “I mean, you can’t go very fast on a dog sled, right?”

“Well, for you and me it might not seem fast, but for those dogs I am sure that it is. This leads us to a great math problem. Here are the speeds of the top 10 finishers. What is the average speed here?”

Mr. Hawkins wrote these speeds on the board.

4.81 mph, 4.79 mph, 4.76 mph, 4.67 mph, 4.66 mph, 4.64 mph, 4.62 mph, 4.6 mph, 4.58 mph, 4.55 mph.

“Take out a piece of paper and figure this out.”

Marcus took out a piece of paper, but he couldn’t remember how to figure out the average score.

If you remember how to do it, figure it out now. Then go through this lesson to begin learning all about data and statistics. When you are finished this lesson, you can check your work with Marcus’ work and see if you have the correct average speed. Look for this problem again at the end of the lesson.

What You Will Learn

In this lesson, you will learn how to complete the following skill tests:

• Find the mean of a set of data.
• Find the median of a set of data.
• Find the mode of a set of data.
• Identify the range of a set of data.
• Select among mean, median, mode and range to describe a set of data and justify the choice for a particular situation.

Teaching Time

I. Find the Mean of a Set of Data

Data is one of those words that we hear all the time, especially in math class. We hear about collecting data, organizing data, analyzing data, etc. But....

What is data?

Data is numerical information collected in a set. When we look at data in math and science, we look at information that has been gathered over time or that has been gathered to evaluate a topic. Learning to look at data is part of the work of people in math and science. Analyzing data can help scientists predict future events too.

We can analyze numerical data several different ways. We can look for the mean, the median, the mode and the range of a set of data.

The mean is sometimes referred to as the average of a set of data. The mean is the sum of the data values divided by the number of data values. You often hear about averages with grades or speeds. What is your average grade in math class? That number determines your final grade or the grade that you received on a test or quiz. We use averages all the time. Let’s look at the steps to figuring out the mean or average through the following example.

Example

Find the mean for the set of data: 47, 56, 51, 45, and 41.

Step 1: Add the data values to determine the sum.

\begin{align*}47 + 56 + 51 + 45 + 41 = 240\end{align*}

Step 2: Divide the sum by the number of data values in the set. In this case, there are five numbers in the data set, therefore divide the sum by five.

\begin{align*}240 \div 5 = 48\end{align*}

The mean is 48.

Take a few minutes and write these steps for finding the mean of a set of values down in your notebook.

You can see that when we follow the steps, we can find the mean or average of a set of numbers quite easily. Let’s look at another example.

Example

The chart below depicts the daily temperature in San Diego for the first seven days in August. Calculate the mean temperature for the first seven days in August.

Date: Temperature:
Sunday 8/1 \begin{align*}88^\circ F\end{align*}
Monday 8/2 \begin{align*}83^\circ F\end{align*}
Tuesday 8/3 \begin{align*}87^\circ F\end{align*}
Wednesday 8/4 \begin{align*}89^\circ F\end{align*}
Thursday 8/5 \begin{align*}82^\circ F\end{align*}
Friday 8/6 \begin{align*}79^\circ F\end{align*}
Saturday 8/7 \begin{align*}87^\circ F\end{align*}

Step 1: Add to determine the sum of the data values.

\begin{align*}88 + 83 + 87 + 89 + 82 + 79 + 87 = 595\end{align*}

Step 2: Divide the sum, 595, by 7 since there are seven numbers in the given data set.

\begin{align*}595 \div 7 = 85\end{align*}

The mean temperature for the first week in August was \begin{align*}85^\circ F\end{align*}.

This was a real life example of how averages help us figure out weather. If you think about the weather forecast, you will often hear "average" being mentioned. The meteorologist will talk about average snowfall or average temperature or average rainfall.

Sometimes, an average will not be a whole number. When this happens, you may need to round to the nearest whole number.

11A. Lesson Exercises

Now it’s your turn to practice. Find the mean for each data set below.

1. 11, 13, 14, 15, 16, 22, 24, 25, 30, 32
2. 34, 36, 38, 41, 43, 44, 50, 53, 50, 50, 62, 66
3. 8, 16, 24, 32, 40

Take a few minutes to check your work with a friend.

II. Find the Median of a Set of Data

Now that you have learned about the mean of a set of data, let’s move on to the median. If you think about the word “median” you can think about the median in a road or street. The median of a street is in the middle of the street. Just like the median of a road, the median of a set of data is the middle value of the set of numbers.

The median is the middle number when the values are arranged in order from the least to the greatest.

Notice that a key to finding the median is that the values must be arranged in order from least to greatest. If they are not arranged in this way, you will not be able to determine an accurate median score!!!

Example

Find the median for the set of data: 47, 56, 51, 45, and 41.

Step 1: Arrange the data values in order from least to greatest.

41, 45, 47, 51, 56

Step 2: Determine the number in the middle of the data set. 47 is the median because it is the data value in the middle of the data set.

The median of the data set is 47.

Example

The chart below depicts the daily temperature in San Diego for the first seven days in August. Determine the median temperature.

Date: Temperature:
Sunday 8/1 \begin{align*}88^\circ F\end{align*}
Monday 8/2 \begin{align*}83^\circ F\end{align*}
Tuesday 8/3 \begin{align*}87^\circ F\end{align*}
Wednesday 8/4 \begin{align*}89^\circ F\end{align*}
Thursday 8/5 \begin{align*}82^\circ F\end{align*}
Friday 8/6 \begin{align*}79^\circ F\end{align*}
Saturday 8/7 \begin{align*}87^\circ F\end{align*}

Step 1: Arrange the temperatures in order from least to greatest.

79, 82, 83, 87, 87, 88, 89

Step 2: Determine the data value in the middle of the data set. In this case, 87 is the median temperature.

The answer is \begin{align*}87^\circ F\end{align*}.

Example

Katie’s first four test scores are 75%, 81%, 80%, and 84%. Determine the median of Katie’s test scores.

Step 1: Arrange the test scores in order from least to greatest.

75, 80, 81, 84

Step 2: In this case, there are two data values in the middle of the data set. To find the median, find the average of the two data values. Recall that to find the mean, determine the sum of the numbers and then divide by two.

\begin{align*}80 + 81 &= 161\\ 161 \div 2 &= 80.5\end{align*}

The median of Katie’s test scores is 80.5%.

Sometimes, you will have median scores that are not whole numbers. When this happens, you will likely need to include the decimal in your answer. This means that the median score is between two whole numbers.

Take a few minutes and write the steps to figuring out the median score down in your notebook.

11B. Lesson Exercises

Find the median score for each data set.

1. 12, 14, 15, 16, 18, 20
2. 14. 18, 19, 34, 32, 30, 41, 50
3. 5, 10, 23, 20, 7, 9, 11, 18, 35, 16, 22

III. Find the Mode of a Set of Data

You have learned how to find the mean and the median, now we can find the mode of a set of data too. The mode is the value that occurs the most times in a set of data. Mode sounds like most and that can be an easy way to remember it. Let’s look at how to find the mode.

Example

The chart below depicts the daily temperature in San Diego for the first seven days in August. Determine the mode for this set of data.

Date: Temperature:
Sunday 8/1 \begin{align*}88^\circ F\end{align*}
Monday 8/2 \begin{align*}83^\circ F\end{align*}
Tuesday 8/3 \begin{align*}87^\circ F\end{align*}
Wednesday 8/4 \begin{align*}89^\circ F\end{align*}
Thursday 8/5 \begin{align*}82^\circ F\end{align*}
Friday 8/6 \begin{align*}79^\circ F\end{align*}
Saturday 8/7 \begin{align*}87^\circ F\end{align*}

To find the mode, look for the number that occurs most often. In this case, 87 is the mode because it appears twice in the set of data.

The mode is \begin{align*}87^\circ F\end{align*}.

Example

Katie’s first four test scores are 75%, 81%, 80%, and 84%. Determine the mode of Katie’s test scores.

In this case, there is not one number that occurs more than another. Therefore, there is no mode.

Mode = none

Example

Miguel measured the height of eight students in his class in inches. Determine the mode of his classmate’s heights.

\begin{align*}51 \quad 45 \quad 47 \quad 51 \quad 50 \quad 56 \quad 52 \quad 50\end{align*}

In this case, there are two data values that appear most often, 51 and 50.

The modes of Miguel’s data are 51 and 50.

11C. Lesson Exercises

Name the mode of each data set.

1. 2, 3, 3, 5, 6, 7, 8, 9, 3
2. 12, 15, 67, 45, 44, 88, 90
3. 13, 13, 45, 45, 67, 89, 13, 67, 67, 67, 13

IV. Identify the Range of a Set of Data

The range of a set of data is the difference between the greatest and least values.

To find the range of a set of data, subtract the smallest data value from the largest data value.

Example

Determine the range for the set of data: 47, 56, 51, 45, and 41.

Subtract the smallest data value, in this case 541 from the largest data value of 56.

\begin{align*}56 - 41 = 15\end{align*}

The difference between the largest and smallest number is 15, therefore the range for this set of data is 15.

Example

The chart below depicts the daily temperature in San Diego for the first seven days in August. Identify the range in temperatures.

Date: Temperature:
Sunday 8/1 \begin{align*}88^\circ F\end{align*}
Monday 8/2 \begin{align*}83^\circ F\end{align*}
Tuesday 8/3 \begin{align*}87^\circ F\end{align*}
Wednesday 8/4 \begin{align*}89^\circ F\end{align*}
Thursday 8/5 \begin{align*}82^\circ F\end{align*}
Friday 8/6 \begin{align*}79^\circ F\end{align*}
Saturday 8/7 \begin{align*}87^\circ F\end{align*}

Subtract the smallest value 79 from the largest value 89.

\begin{align*}89 - 79 = 10\end{align*}

The range in temperatures is \begin{align*}10^\circ F\end{align*}.

11D. Lesson Exercises

Find the range of each data set.

1. 12, 14, 15, 16, 18, 20
2. 14, 18, 19, 34, 32, 30, 41, 50
3. 5, 10, 23, 20, 7, 9, 11, 18, 35, 16, 22

Be sure that you have copied some notes about mean, median, mode and range in your notebook as well as how to find each one.

V. Select among Mean, Median, Mode and Range to Describe a Set of Data

Different situations require different analysis. Sometimes it makes more sense to describe data using one term versus of another term.

For example, if you are looking for the difference between speeds or times, using the range would make the most sense. The average wouldn’t help you to understand the difference.

When analyzing the daily temperatures for the month of August, the best way to examine the data would be to calculate the mean because there shouldn’t be much variance in the daily temperature.

When surveying a group of middle school students about their favorite soda, the mode is the most appropriate measure since no numerical data is involved.

If you are looking for the middle value in, say, sales for a store, then the median would be the best way to analyze the data.

Let’s look at some examples.

Example

Should mean, median, mode, or range be determined to best analyze the following set of data?

Number of minutes it took five students to complete a math test.
Student: Number of minutes:
1 29
2 32
3 40
4 33
5 38

Since there is not a number that occurs most often, the mode is not the best way to analyze the data.

Calculating the range of this data just allows one to see the difference between the students who finished the exam first and last.

In this case, determining the mean is the best way to analyze this data. The mean gives the average amount of time it took for the five students to complete the exam. In this case, the mean is 34.4 minutes. It can also be helpful to look at the median of this data, 33. In this case, the mean and median are close in value.

Remember to think about what the data describes and what your objective is in analyzing the data and this will help you to choose the best method for analyzing the data.

Real Life Example Completed

The Iditarod

Here is the original problem once again. Reread it and then compare your answer for the mean with the given solution.

The students in Mr. Hawkins class are studying about the Iditarod. Some of the students had never heard of the Iditarod before, so Mr. Hawkins started off his class by showing them this video about the race.

The students sat through the video in awe. When it was over, the room was so quiet that you could have heard a pin drop. Marcus was the first one to raise his hand.

“How far is it?” he asked.

“That is a great question,” Mr. Hawkins said. “The race is 1,150 miles from Anchorage Alaska to Nome Alaska. Men and women have finished it and won it too. This year, there were 10 men who finished on top. One finished in 8 days and the rest in 9 days.”

“How fast did they go?” Karen asked from the back of the room. “I mean, you can’t go very fast on a dog sled, right?”

“Well, for you and me it might not seem fast, but for those dogs I am sure that it is. This leads us to a great math problem. Here are the speeds of the top 10 finishers. What is the average speed here?”

Mr. Hawkins wrote these speeds on the board.

4.81 mph, 4.79 mph, 4.76 mph, 4.67 mph, 4.66 mph, 4.64 mph, 4.62 mph, 4.6 mph, 4.58 mph, 4.55 mph.

“Take out a piece of paper and figure this out.”

Marcus took out a piece of paper, but he couldn’t remember how to figure out the average score.

By now you understand that the average is the same thing as the mean. The students have been asked to find the mean speed of the dog sleds on the 2010 Iditarod. They have been given the speeds of the top ten finishers. This is the data that we will use to figure out the mean.

First, add up all of the speeds.

\begin{align*}4.81 + 4.79 + 4.76 + 4.67 + 4.66 + 4.64 + 4.62 + 4.6 + 4.58 + 4.55 = 46.68\end{align*}

Next, we divide this sum by 10 because there were ten dog sleds, so there were 10 different speeds.

\begin{align*}46.68 \div 10 = 4.668\end{align*} rounds to 4.67

The average speed is 4.67 mph.

Now that you understand mean, median, mode and range, you can figure out the median, mode and range of the data as well. Take a few minutes to do that.

The median is the middle speed. This is between 4.66 and 4.64. We can say that the median speed is 4.65 mph.

There isn’t a mode for this data.

The range is the difference between the fastest speed and the slowest.

\begin{align*}4.81 - 4.55 = .26 \ mph\end{align*}

You can see that there wasn’t a huge difference between the fastest time and the slowest time. But that .26 was enough to make the difference between first and tenth place!

Vocabulary

Here are the vocabulary words that are found in this lesson.

Data
pieces of numerical information collected in a set.
Mean
the average value of a set of data.
Median
the middle value or score of a set of data.
Mode
the value that appears the most in a set of data.
Range
the difference between the highest value and the lowest value of a set of data.

Technology Integration

Other Videos:

1. http://www.mathplayground.com/howto_mode.html – This is a video that shows you how to find the mean, median and mode of a set of data.

Time to Practice

Directions: Analyze each data set.

\begin{align*}32 \quad 29 \quad 40 \quad 35 \quad 42 \quad 25 \quad 40\end{align*}

1. Mean –

2. Median –

3. Mode –

4. Range –

The weekly paychecks for ten part-time employees are: $140,$132, $200,$150, $175,$200, $180,$95, $145, and$155.

5. Mean –

6. Median –

7. Mode –

8. Range –

The data set below depicts the number of points LeBron James scored in the last five NBA games.

\begin{align*}25 \quad 27 \quad 25 \quad 29 \quad 22\end{align*}

9. Mean –

10. Median –

11. Mode –

12. Range –

Here is the number of days that it took the mushers of the Iditarod to finish the race in 2010.

8 days, 9 days, 9 days, 9 days, 9 days, 9 days, 9 days, 9 days, 9 days, 9 days

13. Mean –

14. Median –

15. Mode –

16. Range –

A red lantern is the award given to the last musher to finish the Iditarod. The longest finish took place in 32 days. If the fastest time in 2010 was 8 days, what is the range between the last and the first?

17. Range –

The scores that Marc earned on his math quizzes were 78%, 85%, 88%, 88% and 90%

18. What is the mean of the data?

19. What is the mode of the data?

20. What is the median of the data?

21. What is the range of the data?

22. The table below depicts the amount of calories in various fast food choices. Which measure of the data will prove most reliable?

\begin{align*}& \text{Burger} && 375\\ & \text{Fries} && 210\\ & \text{Chicken Sandwich} && 300\\ & \text{Shake} && 215\\ & \text{Soda} && 159\end{align*}

23. Describe a situation where a mean would be the best way to analyze data.

24. Describe a situation where the range would be the best way to analyze data.

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