Lesson Plan: Mean, Median and Mode: An "Idol" Analysis
Grade Level:
9-12
Time Span:
4 class sessions
Massachusetts Frameworks:
- Select, create, and interpret an appropriate graphical representation (e.g., scatterplot, table, stem-and-leaf plots, box-and-whisker plots, circle graph, line graph, and line plot) for a set of data and use appropriate statistics (e.g., mean, median, range, and mode) to communicate information about the data. Use these notions to compare different sets of data.
- Describe and explain how the relative sizes of a sample and the population affect the validity of predictions from a set of data.
- Design surveys and apply random sampling techniques to avoid bias in the data collection.
- Select an appropriate graphical representation for a set of data and use appropriate statistics (e.g., quartile or percentile distribution) to communicate information about the data.
Sequence of Events
Day 1: Introduction of Mean, Media and Mode
This unit begins with a 30-40 minute Q&A session led by the teacher. During this session the teacher will introduce the three topics: mean, median and mode.
Define: Mean -- "a value that is computed by dividing the sum of a set of terms by the number of terms" (from the Merriam-Webster Dictionary).
Mean example: A teacher gives a test to six students and calculates the grades.
- Student 1: 90
- Student 2: 92
- Student 3: 88
- Student 4: 95
- Student 5: 60
- Student 6: 72
To find the mean, you need to add all the test scores (90 + 92 + 88 + 95 + 72 + 60) and then divide the sum by the total number of tests (6). Your work would look like this:
90
92
88
95
72
+60
497 (this is the sum)
Then, you would take 425 and divide it by the total number of tests -- 6:
497/6 = 82.83333333333
The mean is 82.83333333333 (round this to 82.8)
Define: Median -- "a value in an ordered set of values below and above which there is an equal number of values or which is the arithmetic mean of the two middle values if there is no one middle number" (from the Merriam-Webster Dictionary).
The definition is confusing, but "median" is actually qute simple. To find it, you sort your data points from lowest to highest and then look for the data point that falls directly in the middle.
Median example: If we take the data points from the "mean" example, we can find the median by first sorting the data points from lowest to highest:
60, 72, 88, 90, 92, 95
This example is a little tricky because, technically, the middle point in this data falls between the third figure (88) and the fourth figure (90). When this happens, we need to add the two data points (88 + 90 = 178) and then divide the sum by 2. This would give us:
178/2 = 89
So, our median, in this case, is 89.
Median example 2: Let's say there are only five test scores to choose from (60, 72, 88, 90, 92). If the median is defined as the middle of your data points, which number in this set would be the median? (Answer: 88).
Median example 3: The median gets a little tricky when your data set includes the same numbers. For example, let's say the test scores were:
60, 72, 72, 72, 88, 92
Because we have an even number of test scores (6), we know that the median of this data set falls between the third test score (72) and the fourth test score (72). As such, we can determine that the median of this data set is 72. We don't do any addition or division, because the numbers located around the median are the same.
Define: Mode -- "the most frequent value of a set of data" (from the Merriam-Webster Dictionary).
The mode is the data point that repeats most often. Using our test score example, let's say the teacher gave 10 tests and the scores were:
60, 66, 77, 82, 82, 90, 90, 90, 91, 95
Which number repeats most often in this data set? (Answer: 90 -- it shows up 3 times).
90 is our mode.
Mode example 2: You can have multiple modes in a data set. For example, if the test scores came out to ...
60, 66, 82, 82, 82, 90, 90, 90, 91, 95
... we would have two modes in this set: 82 (it repeats 3 times) and 90 (it also repeats 3 times).
Mode example 3: Sometimes, a data set does not have a mode. This occurs when a value doesn't repeat:
60, 66, 82, 90, 91, 95
Final 10-15 minutes of class: Students will use the supplied worksheet to practice finding the mean, median and mode of data sets (see mean-median-mode-worksheet.doc).
Day 2: Data gathering
In this example, students must take on the role of "American Idol Investigators." This is an important job that will influence millions of votes (and, presumably, lift an unknown singer into the heights of superstardom).
The "Idol Investigators" must do the following:
- Develop a survey that allows them to scientifically analyze the popularity of five "Idol" contestants.
- Examine survey data by analyzing the mean, median and modes.
- Develop three charts to represent their data.
- Use their survey data to predict who will win the competition.
In the second day of this lesson, students will work in 3-person teams to develop their surveys. The teacher can guide the survey creation process by asking questions and nudging students along. Questions could include:
- How do you quantify popularity on a survey?
- Should you ask people who they vote for?
- Should you ask people to list their favorite contestants?
- Should you ask people to rank their favorite contestants?
Students should also use Web research to develop their surveys. They can obtain qualitative data by looking through past "Idol" results. Possible research could include:
- Which current contestants have been in the bottom three?
- What predictions are "Idol" experts making? ("Entertainment Weekly" does a nice job with this).
Students must also customize the attached survey spreadsheet (survey-spreadsheet.xls) so they can properly gather and organize their survey results.
Student teams must show their survey to the teacher before the end of class. The teacher needs to sign off on the survey before it can be administered.
Day 3: Surveying
Students will use the third class session to survey fellow classmates. The teacher may want to keep things moving by setting a survey time limit (i.e. surveys must be conducted in three minutes or less).
Groups should pay careful attention to survey results. These results form the foundation of their analysis, so they need to methodically track the incoming information.
Day 4: Data Analysis
On the final day, students will take their data and develop three graphs:
- A "Mean" graph should quantify results for each "Idol" contestant included in the students' surveys. These results should then be plotted on a bar graph or pie chart. (The chart can be created in Excel or the students can draw it out in a paint program or by hand).
- A "Median" graph should quantify results for each "Idol" contestant included in the students' surveys.
- A "Mode" graph should quantify results for each "Idol" contestant included in the students' surveys.
Students will then use these graphs to answer the following questions:
- Based on your survey data, who are the most popular "Idol" contestants?
- Who do you predict will win the competition?
Teams must reveal their data to the class. Graphs, surveys and survey data sheets must be handed in to the teacher at the conclusion of the lesson.
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