Appropriate Measure of Central Tendency

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Objective

SWBAT decide and explain which measure of central tendency most accurately describes a given data set, recognizing inappropriate uses of statistical measures.

Big Idea

Measures of Central Tendency describe the middle point of a distribution of a data set; however, some measures of central tendency better represent the data than others.

Think About It

7 minutes

Students work on the Think About It problem independently.  I give students access to calculators for this problem, so that the majority of their time is spent reasoning about the best measure to use.

One note:  the word 'representative' was challenging for my struggling readers.  Before this class began, I had the kids read and underline any words what were difficult to understand.  While students are working on the means and medians, I circulate around the room and look for underlined works so that I can help support students who need it.  

After 4 minutes of work time, I have students turn and quickly compare their means and medians with a partner before we start our whole class conversation.

I ask students to comment about what type of student they think Shemar is.  I then ask them what they think I, as the teacher, would put in the grade book.  

Intro to New Material

10 minutes

To start the Intro to New Material section, we walk through an example about a restaurant.  I ask students about the data - did people generally have a good experience on Tuesday night?  I ask them what the 1 represents.  Drawing from the learning in the previous lesson, I ask students what the 1 would do to the mean of the data set. 

I ask students if the mean or the median would be a better overall representation of the scenario.  In this example, we want an overall idea to know how most people feel about customer service. A precise measure is not as important because the data is based on peoples’ opinions and it’s more important to get an overall picture. An exact average, which would be the mean, would be skewed lower because one person had a bad experience at the restaurant.

We do not calculate the mean and median for this problem, but I will provide for students at the end of the conversation, so that they can see that the median in this example (9.5) is a better representation than the mean (8.5).

The key ideas that come out in conversation (and are recorded in the graphic organizer):

  • The mean is used when an exact, precise representation of ALL the data values is needed
  • The mean is used when the data is closely gathered
  • The mean is used if the data is binary (meaning that there are only two possible data points- yes/no, 1 and 2, 0 and 1, etc.)
  • The median is used when an outlier might skew the data (or there might be some observational error)
  • It is incredibly important to consider the context of the data. At times, you want an EXACT average, and want to include the outliers. At other times, you do not want to include these data points. 

 

 

Partner Practice

15 minutes

Students work in pairs on the Partner Practice problem set.  As they are working, I circulate around the room and check in with each group.  I am looking for:

  • Are students making accurate predictions as to the impact values will have on a data set?
  • Are students looking at data and extracting information in order to determine the appropriate measures of central tendency?
  • Are students considering the context of the data to make decisions?

I am asking:

  • How did you know that the mean/median would best represent this data set?
  • Why would the mean be a better representation that the median? (or vice versa)

 

I also will ask groups to calculate the mean and median of a data set once they've written their response, as a way to justify their thinking. 

After 10 minutes of partner practice time, students complete the Check for Understanding problem independently. Two CFU examples of student responses are provided. 

 

 

Independent Practice

15 minutes

Students complete the Independent Practice problem set.  

A number of the problems in this lesson ask students to draw a dot plot to represent the data.  This is a good review of material mastered in the previous unit.   It also gives students a visual to use, to decide whether or not there are outliers and to determine if the data is closely clustered.  

As I circulate, I check the work of problem 4.  This problem requires students to create a dot plot and the data set contains an outlier.  I can quickly see whether or not the visual representation is correct.  There are many opportunities for questions here.  I can ask what would happen if the next two quizzes were 40 and 45 - how would that change your thinking?  What if the teacher dropped the lowest score?  What measure would make the most sense then?  What happens to Rasheed's average, because of the 40%?  There are opportunities for me to ask a rage of questions, all at varying degrees of complexity.    

I also ask students to share with me their thinking for Problem 6, where students are asked to create their own data set.

 

Closing and Exit Ticket

10 minutes

After independent work time, I have students share their data sets for problem number 6 with their partners.  Each student will have a unique data set, so the pairs have the chance to talk through how they decided on the numbers they've used.  

Students then complete the Exit Ticket to close the lesson.  An exit ticket sample provides an example of what student thinking might look like.