# Lesson 2: What Does Mean Mean?

## Lesson 2: What Does Mean Mean?

### Objective:

Students will learn that values that gather around the center of a distribution show the typical value. This value is also referred to as the mean, or average.

### Materials:

1. Pennies on a Ruler handout (LMR_2.2_Pennies on a Ruler)

2. Markers (1 for each table)

3. Rulers (1 for each table)

4. Pennies (6 for each table group)

5. Tape

6. Exported, printed, and reproduced class’s Personality Color survey data

Advanced preparation required: The teacher must share students’ data on the IDS Home page (https://tools.idsucla.org) before it can be exported and printed. Students will keep for use in subsequent lessons.

7. Mr. Jones Mile Run Times handout (LMR_2.3_Mr. Jones Run Times)

### Vocabulary:

measures of central tendency (or center), typical, measures of variability (or spread), mean, average, balancing point

### Essential Concepts:

Essential Concepts:

The center of a distribution is the 'typical' value. One way of measuring the center is with the mean, which finds the balancing point of the distribution. The mean gives us the typical value, but does not tell the whole story. We need a way to measure the variability to understand how observations might differ from the typical value.

### Lesson:

1. In student pairs, ask students to discuss what they think the following terms mean:

1. Measures of central tendency. A value that shows the tendency of quantitative data to gather around a central, or typical, value. Also known as measures of center. Students will learn about two such measures: the mean and the median.

2. Measures of variability. Values that show how much the quantitative data varies. Also known as measures of spread. Note: This is not taught during this lesson, but will be addressed as part of Lesson 4.

2. Ask a pair to share what they think these two terms mean. Pairs who are listening must decide whether they agree or disagree with the pair that shared. Lead a discussion based on their statements of agreement or disagreement.

3. Communicate to the class that they will be learning more about these measures and what they tell us about data as we progress through this unit.

4. By a show of hands, ask students how many are familiar with finding the mean, or average.

5. Select a student to share his/her process for finding the mean. Possible answer: Add up all of the numbers. Then divide by how many numbers there are.

6. Another way to find the mean is to find the balancing point of a distribution. They will learn about the balancing point via the activity in Steps 7 & 8.

7. Distribute the Pennies on a Ruler handout (LMR_2.2) along with a marker, ruler, tape, and 6 pennies to each table group. If you prefer to not print the document, you can project it on the board instead.

8. Guide the students through the handout and have them share their findings throughout the activity. Be sure to emphasize the idea that the mean of a distribution can be identified by finding its balancing point.

9. Next, distribute the class’s Personality Color survey data to the students.

10. Have student pairs find the variable Blue (whether or not that was their predominant color) in the class’s printed data.

11. As a class, make a dot plot on the board to show the distribution of Blue values. Each student should come to the board and draw a dot to indicate where their value is in the distribution. Ask the students:

1. What do you think the typical Blue score is? Answers will vary by class. They should be driven to an answer in the center of the distribution.

2. Are the data roughly symmetric? Where is the balancing point of this distribution? Answers will vary by class. Once a value is chosen, indicate the location on the dot plots.

12. As a class, compute the mean Blue score for the entire class on the board and compare this value to the class’s prediction of the balancing point. Students may not remember exactly how to compute the mean, so you can remind them of the general algorithm or refer them back to their responses from Step 5 above.

13. Show the students the formula for calculating the mean:

$\bar{x}=\frac{&space;\sum_{x=1}^{n}&space;x_i}{n}$

14. Now that they have calculated the mean for the Blue score, ask them to identify each symbol in the formula with a step in their algorithm for finding the mean, and discuss the meaning of the symbols in the formula as a class. xi represents each individual data point and n represents the total number of observations.

15. Indicate the location of the calculated mean on the dot plots by drawing a vertical line at the value on the x-axis. Ask student pairs to engage in a conversation about how close the mean value is to their predicted balancing point and why their prediction was made that way. Select a pair to share their discussion with the whole class.

16. Using the Personality Color survey data from Step 9, ask student pairs to compute the mean score for each of the other three personality colors.

17. Inform the students that, during the next lesson, they will learn about another method that can be used for measuring the center of a distribution.

18. Now, you can inform the class about an even easier method of calculating the mean – using RStudio! Explain that the command RStudio uses to calculate the mean incorporates the algorithm of summing up all the data and dividing by the total number of observations. Students will be able to use this command for quick calculations now.

Note: If you have already “Exported, Downloaded, Imported” the class’s Personality Color campaign data, you can simply use the exact command below to calculate the mean Blue score:

> mean(~blue, data = colors)

In general, the function can be denoted as follows:

> mean(~variable, data = datafile)

So, for our specific example, blue is the variable we want to find the mean value of, and colors is the datafile.

19. Have the students Think-Pair-Share to discuss how the mean value of a group of data could be used to easily describe complicated things. For example, instead of giving someone the entire class’s Blue scores, we could just tell him/her the mean score and he/she would have a general idea about the class.

### Class Scribes:

One team of students will give a brief talk to discuss what they think the 3 most important topics of the day were.

### Homework

Students should complete the Mr. Jones Mile Rule Times handout (LMR_2.3) for homework. They can practice finding the mean of distributions by determining a balancing point for the data. Answers to the handout are below. Note: The mean values in part (3) do NOT need to be exact.

1. What kind of plots did Mr. Jones create for his classes? Histograms.

2. Where does each distribution balance? Find and label the balancing point of each distribution. The balancing point for all of these distributions is at the mean.

3. Based on the balancing points you found, what would you say the mean mile run time is for each class?

i. Period 1:     9.91

ii. Period 2:     8.48

iii. Period 3:     8.45

iv. Period 4:     8.17