August 4, 2010

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In this section of the lesson, students will be introduced to measures of central tendency through a series of example spreadsheets.

- Mathematics > General
- Mathematics > Applied Mathematics
- Mathematics > Data Analysis & Probability
- Mathematics > Number Sense & Operations
- Mathematics > Problem Solving
- Mathematics > Statistics

- Grade 6
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- Grade 8

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**Materials needed for this lesson:**

1. *Example Class Data Set Spreadsheet* (see Lesson 8 Resources folder).

2. Calculators.

**Procedure:**

Begin this lesson by reminding students of the end of the last unit, where they had to report the results of their individual families' financial lives to other people. Lead a brief discussion to have the students remind you of the various statistical tools they used to do this reporting (bar, line, and circle graphs, etc.). Now, tell the students that the goal of this lesson is to be able to report on *the class as a whole* (rather than on each family individually). That is, if we wanted to be able to describe how life is financially for the whole class (is the class rich? poor? average?) as well as giving certain mathematical measures, or *indicators*, for the class, how would we do it?

Pull up the *Example Class Data Set* on the screen. Use this as an example class to introduce the notions of mean, median, and mode. Pull up the *Example #1* sheet in the *Example Class Data *spreadsheet. This sheet has ten different net monthly incomes arranged in random order. Ask the class to attempt to describe this class. Is it rich? Is it poor? How could you tell? What's the "overall" picture of the incomes in this class? How could we tell? Attempt to elicit that one way to tell would be to see what the most frequent or common income is, or what the "center" of the data set would look like. Also, ask if there would be something we could do that would make it easier to get a "picture" of the data. Elicit that ordering the data would help gain a better picture of relative frequencies of salaries.

Once this answer is elicited, use the copy and paste function of Excel to copy the column of incomes under the "Incomes Arranged..." heading. Then, use the *Sort* function of Excel (it's a button with an A and Z on it) to arrange the incomes from least to greatest. Ask students if this helps. Lead a discussion where they decide what a good number would be to represent this example class "overall". Their answer should be around $2,800 or $2,900. (As the mean of this set is $2765 and the median is $2767, there is no mode.)

Next, lead a discussion to help students understand why it would be important to have a set "kind" of number that you could find for any set of data to describe its general properties, rather than just relying on intuition. Don't you think mathematicians would want something precise, rather than something "fuzzy"?! At this point introduce the three **measures of central tendency**: **the mean, the median, and the mode**. Instruct the students as to the means of finding each, highlighting the absolute importance of arranging the data from least to greatest before beginning the analysis*.* Have the students work together in pairs to find the mean, median, and mode of the example data set.

Once student pairs have all finished, briefly review their findings and correct any lingering misunderstandings. Point out that there is no mode of this data set, because no two numbers occur twice in the data. Next, pull up the *Example #2* sheet on the spreadsheet, and have student pairs work through it to find the mean, median, and mode. Stop them after they have arranged the data from least to greatest to check their work (do this by right-clicking on columns "B" and "D" and using the "unhide" function to reveal the hidden column "C"). Have the students continue. When everyone is done (or close), use the same "unhide" function to reveal columns "E" and "F" simultaneously). These columns contain the values for the measures of central tendency. Lead a discussion as to which of the three is the most accurate measure. (In actuality, they are all three accurate and appropriate given the almost normal distribution of the percentages.)

Next, lead a discussion to highlight the different functions of the three measures. Ask the class to predict what would happen to the mean if one of the percentages was significantly higher than the others. Then, in the "Arranged" column, change the *1%* value to *100%*. Note what happens to the mean. Also, highlight the fact that neither the median nor the mode changed. Why not?

Lead a brief discussion to help the students decide which of the measures would be most appropriate now (answer: *the median, because of the outlier*). Change a few more of the values to demonstrate how changing values on the margins of the data set does not change the median at all.

To highlight this, pull up the *Example #3* sheet on the spreadsheet. This is a list of annual gross incomes for a fictitious community, as well as the measures of central tendency that describe them. Highlight how the mean hides the true nature of the very stratified community, whereas the median gives a much more accurate representation. Highlight how this might be a "factory town", where the owner of the factory lives on a hill in a mansion, making over $1 million a year, whereas all the factory workers live in a slum at the bottom, making less than the poverty line. Given this, does the mean income of over $140,000 a year reflect either the poor majority or the very wealthy minority? Of course not! However, the median is a more accurate reflection of the overall state of the community.

Ask the class: *If you were a real estate agent trying to convince a middle-class family to move into this town (without ever having visited), which measure of central tendency for incomes would you quote them? Why? Would you be telling the truth or lying?*

Point out that this is the very reason that in Lesson 1 of this unit, when they were deciding whether or not their family qualified for a government program, the program used a cutoff that was the *Area Median*

Finally, to highlight how the mode is hardly ever used (or useful), display the *Example #4* sheet. Highlight that this is about the only situation in which the mode would be a useful measure of central tendency.

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