Minds On
Painting survey
A group of 15 painters were surveyed about how much time they spend painting each week.
The following set of data represents the number of hours each painter spent painting in a week
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20, 15, 22, 10, 6, 16, 27, 18, 19, 21, 23, 25, 9, 6, 11 |
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Consider this question:
- How could you use this data to describe the amount of time one painter spends painting compared to the others?
Throughout this learning activity, you can record your thoughts digitally, orally, or in print.
Action
Mean, median, and mode
Mean, median, and mode are three kinds of number averages, or measures of central tendency, that we use in math and in everyday life.
We are now going to explore how to determine the mean and the median and how to identify the mode(s), if any.
Mean
The mean is the average of all numbers in a data set.
You determine the mean by adding up all the numbers in a data set and then dividing that total by the amount of numbers in the data set.
For example, the mean of 10, 20, and 60 is the sum of the data (10 + 20 + 60 = 90) divided by amount of numbers (90 ÷ 3 = 30). Therefore, the mean is 30.
Let’s explore another example:
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How to determine the mean of the following data set: 22, 28, 26, 35, 44. |
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Step 1 |
Add the numbers together |
22 + 28 + 26 + 35 + 44 = 155 |
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Step 2 |
Divide the total by the amount of numbers in the data set |
155 ÷ 5 = 31 |
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Step 3 |
Therefore, the mean is… |
31 |
Median
The median is the number that appears in the middle of a data set when its data is ordered from smallest to largest.
You identify the median by ordering the numbers in a data set from smallest to largest and then determining the number that appears in the middle.
You identify the median by ordering the numbers in a data set from smallest to largest and then determining the number that appears in the middle.
Let’s explore another example:
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How to identify the median of the following data set: 12, 24, 13, 15, 22. |
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Step 1 |
Organize the data set in order from smallest to largest |
12, 13, 15, 22, 24 |
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Step 2 |
Count the amount of numbers in the data set. If there is an odd number of data values, you can find the exact middle. |
There are five numbers, so you can find the exact middle. |
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Step 3 |
Therefore, the median is… |
15 |
Sometimes data sets have an even number of data values, which means that two numbers will appear in the middle. When this happens, in order to identify the median you must calculate the mean of the two middle numbers.
Let’s explore an example with a data set that has an even number of data values:
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How to identify the median of the following data set: 12, 24, 13, 15, 22, 17 |
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Step 1 |
Organize the data set in order from smallest to largest |
12, 13, 15, 17, 22, 24 |
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Step 2 |
Count the amount of numbers in the data set. If there is an odd number of data values, you can find the exact middle. If there is an even number of data values, you must calculate the mean of the two middle numbers |
There are six numbers, so you must calculate the mean of the two middle numbers. |
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Step 3 |
Identify the two middle numbers |
15 and 17 |
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Step 4 |
Determine the mean of the two middle numbers by calculating their sum and dividing by two (the total amount of numbers in the set of middle numbers) |
15 + 17 = 32 32 ÷ 2 = 16 |
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Step 5 |
Therefore, the median is… |
16 |
Mode
The mode is the number that appears the most amount of times in a set of data.
You identify the mode by counting the frequency of the different numbers in a data set.
For example, the mode of 3, 5, 6, 5, 6, 5, 4, and 5 is the number that appears the most. The numbers three and four each appear one time each, the number five appears four times, and the number six appears two times. Therefore, the mode is 5.
Let’s explore another example:
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How to determine the mode of the following data set: 7, 4, 8, 2, 8, 5, 4, 9, 1, 8 |
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Step 1 |
Organize the data set in order from smallest to largest |
1, 2, 4, 4, 5, 7, 8, 8, 8, 9 |
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Step 2 |
Count the frequency of the numbers in the data set. |
The numbers 1, 2, 5, 7, and 9 each appear one time. The number 4 appears two times. The number 8 appears three times. |
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Step 3 |
Therefore, the mode is… |
8 |
Sometimes data sets have more than one number that appears the most amount of times, which means that the mode will be more than one number.
Let’s explore an example with a data set that has a mode that is more than one number:
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How to determine the mode of the following data set: 10 ,7, 7, 2, 5, 1, 5, 7, 2, 10, 2, 3 |
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Step 1 |
Organize the data set in order from smallest to largest |
1, 2, 2, 2, 3, 5, 5, 7, 7, 7, 10, 10 |
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Step 2 |
Count the frequency of the numbers in the data set. If more than one number appears the most amount of times, then the mode is more than one number. |
The numbers 1 and 3 each appear one time. The numbers 5 and 10 each appear two times. The numbers 2 and 7 each appear three times. |
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Step 3 |
Therefore, the mode is… |
2 and 7 |
Determining mean and median and identifying mode
We are now going to review different data sets to determine the mean and the median and identify the mode.
Determining the mean
Explore the image below and think about the following question:
- If you were to move cubes in order to make all the bars have the same number of cubes, how many cubes would you have in each bar?
- What is the mean?
- How would you explain the concept of mean to a student who is learning about it for the first time?
- How could you use this activity to explain the concept of mean to them?
I see that the fourth bar has the most cubes.
I would take two cubes and add them to the second bar so it now has 6 cubes. I would take two more cubes and add them to the fifth bar so it also has 6 cubes. The fourth bar now has only 5 cubes so I would take one from the first bar so it has 6.
Now each bar has 6 cubes. This is the mean.
If I add up the cubes in all five bars, there are 30 cubes total. 30 divided evenly across the five bars is 6 cubes for each bar.
I could tell someone that when you are calculating the mean of a set of numbers, you are trying to find what the average is—you are trying to even out the numbers. I did this by taking some away from the bigger sets of cubes to add to the bars that had fewer cubes. You could then use how you evened out the numbers (or the bars, in this case) to start comparing.
I know that I had to add to the third and fifth bars, which means they were below the mean. I took quite a few from the fourth bar, which means it was above the mean. I didn’t do much to the first and second bars, which means they were already pretty close to the mean.
Determining the median
Explore the following video, “What is a Stem and Leaf Plot?” to review stem-and-leaf plots.
Now review this set of data and organize it into a stem-and-leaf plot:
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23, 36, 43, 75, 67, 36, 36, 22, 24, 25, 55, 62, 45, 47, 56, 60, 61 |
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The stem is recorded in the left column and lists the first digit (or digits) of a number.
The leaf is recorded in the right column and lists the last digit of the number. Each leaf can have only a single digit.
If there is more than one leaf on a stem the leaves are listed in ascending (increasing) order.
If there are no leaves, nothing is recorded beside the stem.
Stems are listed in order from least to greatest. Even if there aren't any leaves, the stem is still listed.
Record your responses to the following questions using a method of your choice:
- What is the median?
- How did the stem-and-leaf plot help you determine the median?
Identifying the mode
Review the following survey results and answer the questions below, using a method of your choice:
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Scenario 1: Ten ice cream shop employees participated in a survey to determine the most popular ice cream flavour. The results were: chocolate, vanilla, vanilla, mint chocolate chip, vanilla, chocolate, vanilla, strawberry, cookies and cream, chocolate. Does this data set have a mean, median, and mode? Explain your thinking. |
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Scenario 2: A transportation company surveyed its riders to find out how many minutes they spent on a bus each day. The results were: 0, 20, 25, 22, 25, 0, 22, 13, 15, 25. What is the mode of this data? |
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Scenario 3: A group of athletes recorded their long jump distances. The results were: 100cm, 105cm, 98cm, 102cm, 98cm, 96cm, 98cm, 101cm, 103cm, 98cm. What is the mode of this data? What information does the mode tell you about this group of athletes and their ability to perform the long jump? Is there a better measure of tendency that could provide you with more information? Explain your thinking. |
Press ‘Possible answers’ to reveal the suggested answers.
Scenario 1: There is no mean or median as there are no numbers to arrange from least to greatest or to add and divide. The mode is vanilla because it is the flavour that is repeated the most.
Scenario 2: The mode of the data is 25 because it appears the most often (three times).
Scenario 3: The mode of the data is 98cm because that distance appears 4 times.
5 jumps were farther than the mode (which was the most frequent) so I think this shows that the athletes can jump farther. I don't know how many athletes there were (it only says a group). If there are 10 people in the group, then that's different than if there are fewer. If there are a lot fewer athletes, then each person jumped more than once and that would mean that maybe 98 was their first jump and then after they were warmed up they could jump farther.
If you calculated the median, after the numbers are ordered, 99cm would be the median (the number between the 5th (98cm) and 6th (100 cm) distance). This measure tells me that half the people jumped below 99cm and half jumped above. But the ones below were only lower by 1cm and one person by 3cm while the ones above were higher by as much as 6cm!
The mean (the result when I add the numbers all together and divide by 10) is 99.9 or 100 if I round it so it's similar to the set of numbers since they were all whole numbers. I think it's more representative.
Consolidation
The mean, median, and mode of random numbers
You are now going to show what you know about mean, median, and mode.
Create your own data set by randomly selecting 7 to 11 numbers. Make sure all the numbers you pick are either all 2-digit numbers or 3-digit numbers.
Using the tool of your choice, record the 11 numbers in your data set and calculate answers to the following questions:
- What is the mean of your data set?
- What is the median of your data set?
- What is the mode of your data set?
Mean and median Worksheet
Complete the Mean and Median Worksheet in your notebook or using the following fillable and printable document.
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Mean and Median Worksheet |
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1) Find the median of the following data sets. Don’t forget to put the numbers in order if you need to.
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2) Two people are discussing the median of the following data set: 12, 12, 13, 26, 25, 27, 29. One person says that the median of this data set is 25, and the other person says it is 26. How could you explain to them who is correct? Explain: |
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3) Determine the mean and median of the following data sets:
a) Describe how the mean changes as the last data value changes. b) Describe how the median changes as the last data value changes. |
Press the ‘Activity’ button to access Mean and Median Worksheet.
Reflection
How do you feel about what you have learned in this activity? Which of the next four sentences best matches how you are feeling about your learning? Press the button that is beside this sentence.
I feel…
Now, record your ideas using a voice recorder, speech-to-text, or writing tool.