Minds On
Mean, median, and mode
Dog heights
Dogs come in all shapes and sizes. The weight of several small dogs was recorded at Parkview Veterinary Clinic. Here are the 12 results, recorded in kilograms:
9.5, 11.4, 10.2, 8, 7.5, 9.2, 7.5, 11.9, 10.5, 8.4, 8.7, 7.
Calculating mean, median, and mode
Calculate the mean, median, and mode of the dog-height data.
Press each type below for the steps you need to follow.
To determine the mean of a set of numbers, add all the numbers together and divide by how many numbers there are. To determine the mean of this data, we have to complete two steps:
- Find the sum by adding all the numbers together.
- Divide this sum by the total number of dogs.
The median is the middle value of an ordered list. To find the median of this data, we have to complete two steps:
- Arrange the data from the smallest value to the largest value.
- Determine the number in the middle of this list of data. This is the median.
Hint: When the data has an even number of items, you will have to determine the mean of the two values in the middle of the list.
The mode is the value that is appears the most often.
- It is possible for there to be no mode in a set of data.
- A set of data can have two modes.
Throughout this learning activity, you can record your ideas digitally, orally, or in print.
Action
Task 1: Outliers
An outlier is a data value or a data point that lies outside of the overall pattern of the data. Removing outliers can change the way data is shaped.
Grade 7 results: Math test
Here are the results of a math test in percentages:
50, 72, 78, 72, 80, 85, 81, 75, 78, 100
- What do you notice about the pattern of data in this set?
- Which values stand out most to you?
Calculating mean, median, and mode
You will now calculate the mean, median, and mode of the math-test data. Then you’ll see how these are affected if you remove the smallest or the largest value from the data set.
Complete Test Results: Mean, Median, and Mode in your notebook or use the following fillable and printable document. If you prefer, use another method to record your ideas.
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Original data (10 values) |
Data with lowest value removed (9 values) |
Data with highest value removed (9 values) |
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|---|---|---|---|
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Data set |
50, 72, 78, 72, 80, 85, 81, 75, 78, 100 |
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Mean |
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Median |
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Mode |
Press the ‘Activity’ button to access Test Results: Mean, Median, and Mode.
Press ‘Answers’ to reveal the solution.
| Test Results: Mean, Median, and Mode | |||
|---|---|---|---|
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Original data (10 values) |
Data with lowest value removed (9 values) |
Data with highest value removed (9 values) |
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Data set |
50, 72, 78, 72, 80, 85, 81, 75, 78, 100 |
72, 78, 72, 80, 85, 81, 75, 78, 100 | 50,72, 78, 72, 80, 85, 81, 75, 78 |
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Mean |
77.1 | 80.1 | 74.6 |
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Median |
78 | 78 | 78 |
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Mode |
72, 78 | 72, 78 | 72, 78 |
Student Success
Manipulating measures of central tendency
Examine your organized results. Consider the following questions:
- What is the impact on the mean, median, and mode when you remove the highest value?
- What is the impact on the mean, median, and mode when you remove the lowest value?
Note to teachers: See your teacher guide for collaboration tools, ideas and suggestions.
Graphing data
A student is interested in which movie genres are the most popular. They find the following table:
| Top 50 movies | ||
|---|---|---|
| Movie genre | Frequency in top 50 | Relative frequency (as a percentage) |
| Mystery | 12 | 24% |
| Adventure | 18 | 36% |
| Documentary | 5 | 10% |
| Animated | 3 | 6% |
| Drama | 7 | 14% |
| Comedy | 5 | 10% |
| Total | 50 | 100% |
Source: Adapted from “All Things Movies.”
The student wishes to make a circle graph of the data. To calculate the size of each “slice,” they convert the percentages into degrees:
| Top 50 movies | ||
|---|---|---|
| Movie genre | Relative frequency (as a percentage) | Degrees |
| Mystery | 24% | 0.24 × 360 = 86° |
| Adventure | 36% | 0.36 × 360 = 130° |
| Documentary | 10% | 0.10 × 360 = 36° |
| Animated | 6% | 0.06 × 360 = 22° |
| Drama | 14% | 0.14 × 360 = 50° |
| Comedy | 10% | 0.10 × 360 = 36° |
| Total | 100% | 360° |
The student uses the number of degrees to create the following circle graph:
Adding data
The student finds a table with additional data. This one breaks down the top 100 movies instead of the top 50. Examine the following table and the corresponding circle graph.
| Top 100 movies | ||
|---|---|---|
| Movie genre | Frequency in top 50 | Relative frequency (as a percentage) |
| Mystery | 14 | 14% |
| Adventure | 22 | 22% |
| Documentary | 12 | 12% |
| Animated | 10 | 10% |
| Drama | 17 | 17% |
| Comedy | 14 | 14% |
| Musical | 1 | 1% |
| Fantasy | 7 | 7% |
| Romance | 2 | 2% |
| Western | 1 | 1% |
Source: Adapted from “All Things Movies.”
Examine the following circle graph made from the table of data you previously accessed:
Brainstorm
Compare the data sets
Compare the original data set with the expanded set. What is the impact of adding the new data? Look for similarities and differences between the two data sets (original and expanded).
To guide your thinking, complete the Comparing Data Sets in your notebook or use the following fillable and printable document. If you prefer, use another method to record your ideas.
| Compare the original data set (Top 50 movies) with the expanded data set (Top 100 movies. |
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1. What impact does the addition of the new data have on the mode? |
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2. What movies are the most popular with the addition of the new data? How does this compare with the original data? |
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3. What new genres appear with the addition of the new data? |
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4. How has the circle graph changed with the addition of the new data? |
Press the ‘Activity’ button to access Comparing Data Sets.
Consolidation
Height of basketball players
A local Grade 7 basketball team wanted to figure out the typical height of their players. They measured their heights in centimetres and recorded the results.
Test Your Skills!
Mean, median, and mode
Find the mean, median and mode for this data set. Record your responses digitally, orally, or in print.
|
Player |
Height (cm) |
|---|---|
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1 |
169 |
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2 |
175 |
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3 |
182 |
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4 |
181 |
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5 |
181 |
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6 |
181 |
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7 |
175 |
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8 |
172 |
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9 |
175 |
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10 |
170 |
Adding two NBA players
One day, two special guests show up for practice. They are former NBA players, each measuring 231 cm. Their heights are added to the data set.
- Recalculate the mean, median and mode with this additional data. Note how each of these measures has been affected by the addition of the new data.
This exercise has given you some experience with outliers. Why is it important to consider outlier data?
Press ‘Hint’ to reveal the suggested answer.
Reflection
As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.
I feel…
Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.
When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.
Press ‘Discover More’ to extend your skills.
Discover MoreIs the data always right?
Whenever data is collected through a survey, test, or experiment, there is always an opportunity for something to impact the results.
For example, someone was recording the height of the school flag and recorded one flag as 182 cm when they meant to write down 82 cm. In this case, removing the outlier will make the set of data more accurate, while leaving it in could have a big impact on the set of data.
Think about what might happen if data is manipulated with the intention of misleading people.
Create a situation where data has been recorded incorrectly, either through manipulation or by accident. Complete Dealing with Incorrect Data in your notebook or use the following fillable and printable document. If you prefer, use another method to record your ideas.
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Describe a situation where data has been recorded incorrectly, either through manipulation or by accident: |
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Record the data set, including the error: |
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What is the result of this error? |
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How would the data appear without the error? |
Press the ‘Activity’ button to access Dealing with Incorrect Data.
Connect with a TVO Mathify tutor
Think of TVO Mathify as your own personalized math coach, here to support your learning at home. Press ‘TVO Mathify’ to connect with an Ontario Certified Teacher math tutor of your choice. You will need a TVO Mathify login to access this resource.
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