Lesson 4: How Far is it from Typical?

Objective:

Students will understand that the mean of the absolute deviations (MAD) is a way to assess the degree of variation in the data from the mean and adjusts for differences in the number of points in the data set (n). The MAD measures the total distance between all the data values from the mean and divides it by the number of observations in the data set.

Materials:

Masking tape (or painter’s tape) – approximately 4-5 feet long – one for each student team
How Far Apart? handout (LMR_2.6_How Far Apart) – will be used again in Lesson 17
Exported, printed, and reproduced class’s Personality Color survey data

Vocabulary:

measures of variability (or spread) deviation mean of absolute deviations (MAD)

Essential Concepts:

MAD measures the variability in a sample of data - the larger the value, the greater the variability. More precisely, the MAD is the typical distance of observations from the mean. There are other measures of spread as well, notably the standard deviation and the interquartile range (IQR).

Lesson:

Remind students that they learned about 2 different measures of center during the previous 2 lessons: the mean and the median. Have the students recall when it is appropriate to use each value based on the shape of the distribution.
1. Mean – use with symmetric distributions.
2. Median – use with skewed distributions or when there are outliers.
Inform the students that, during today’s lesson, they will learn about measures of variability – also known as measures of spread. These values show us how much the quantitative data varies from the center of a distribution. Similar to measures of center, we will use two different measures of spread: (1) the mean of absolute deviations (MAD), and (2) the interquartile range (IQR).

Note: IQR will be discussed in detail during Lesson 5.
Introduce the term deviation. Using Think, Pair, Share, ask students what they think this word means and how it could relate to variability. A deviation is the act of departing from an established course or accepted standard. Common synonyms include departure, detour, difference, digression, divergence, fluctuation, inconsistency, modification, shift, etc.
On the classroom floor next to each student team, place a 4-5 foot long piece of masking tape (or painter’s tape). Then, propose the following scenario:

Your team has been invited to guest star at the circus! You have been asked to perform as part of the tightrope act – a routine that requires tremendous focus and balance to walk across a tightly pulled rope that is suspended high in the air. In order to practice your balancing skills, the circus has provided your team with a line of tape that will represent the tightrope.
Have the students consider the piece of tape (aka the rope) to be the “typical” path they must take to finish the circus act. Since they do not want to fall from the suspended tightrope while performing at the actual circus, they will need to practice walking directly on the middle of the line at all times. If they deviate from the line, they will no longer be walking the “typical” path, and will likely fall.
Each team should select one student to be their starting performer.
In teams of 4, one student is the performer, two are measuring the distance of the deviation (one on each side of the tape), and one is the recorder.
Place a ruler perpendicular to the “rope” and measure the distance, in centimeters, from the path to the center of the back of their heel as the student walks and attempts to balance across the “rope.”
The performer will walk the tightrope by looking straight up to the sky – first they look to place a foot on the line, then walk naturally while looking up to the sky, and repeating one step at a time for 4 steps, measuring after each step. Any time the performer missteps, this is considered a variation from the typical value. You can have students take turns so everyone gets a chance to balance, walk, and to measure, depending on time in your class.
Now that the students have an idea about what it means to deviate from something they consider “typical,” they can start looking at distributions to see how data points vary from their typical value.
Inform students that they were observing deviations from typical while calculating actual differences between the rope and the performer’s steps. When data are quantified with numbers, we can then calculate how far away each value is from the center.
One such calculation that is popular among data scientists is the mean of absolute deviations (MAD). Ask students to consider the components of the MAD in math terms, and brainstorm what the MAD value might represent.

mean – an average

absolute – in mathematics, we talk about absolute value, the positive difference between 2 numerical values

deviation – as discussed earlier in the lesson, deviation represents how much things vary
Using the 3 components in Step 12, explain that the MAD measures the absolute distance of each data point from the mean, and then finds the average of all those distances.
Display the formula for the MAD distribution for the whole class to see.
$MAD=\frac{ &space;\sum_{i=1}^{n}&space;|x_i-\bar{x}|}{n}$
Discuss what each symbol in the formula means and how we use it to perform the calculation. x_i represents each individual data point, x̄ represents the mean value, and n represents the total number of observations. The symbol Σ represents the summation – this tells us to add up all the absolute distances from each point to the mean.
To practice using this formula with actual data, students will calculate and compare the MAD values for 2 distributions.
Distribute the How Far Apart? handout (LMR_2.6), which contains 2 of the dot plots - plots (a) and (c) from the Where is the Middle? handout (LMR_2.5) used in Lesson 3. As before, the dot plots depict the number of candies eaten by a group of 17 high school students on different days of the week. The means are also given.

LMR_2.6

The calculations for each plot are shown below for the teacher’s reference.

MAD for plot (a)

$\begin{align*} MAD &= \frac{1|0-2|+5|1-2|+6|2-2|+3|3-2|+2|4-2|}{17} \\ &= \frac{1(2)+5(1)+6(0)+3(1)+2(2)}{17}\\ &= \frac{2+5+0+3+4}{17}\\ &= \frac{14}{17}\\ &\approx 0.8235 \end{align*}$

MAD for plot (c) $\begin{align*} MAD &= \frac{3|0-2.53|+0|1-2.53|+4|2-2.53|+5|3-2.53|+5|4-2.53|}{17} \\ &= \frac{3(2.53)+0(1.53)+4(0.53)+5(0.47)+5(1.47)}{17}\\ &= \frac{7.59+0+2.12+2.35+7.35}{17}\\ &= \frac{19.41}{17}\\ &\approx 1.1418 \end{align*}$
Students may work in pairs to complete the handout. After all student pairs have come to an agreement on their answers, pose the following questions to the class as a whole:
1. Which MAD value did you think would be larger based only on the look/shape of the distributions? Why? Since plot (c) is skewed to the left, it probably has a larger MAD because more points will be further away from the mean than in plot (a).
2. Which MAD value was actually larger when you calculated it? The MAD value for plot (c) was larger (1.1418 > 0.8253).
3. Did your prediction match the actual calculated values, or were you surprised by the results? Yes. The distribution with the wider spread (more variability) had the larger MAD value.
To continue exploring with the class’s Personality Color survey data, student teams should calculate the MAD value for their Blue scores. Does the MAD value seem reasonable based on the dot plot they created during Lesson 2?

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 & Next Day

Students should calculate the MAD values for each of the other 3 personality color scores and compare the values of the 4 color scores.

Declutter your Environment Pane

Unit 2 utilizes new datasets so it's a good idea to declutter your Environment Pane.

Refer students to this video or demonstrate it on your own Environment.

LAB 2A: All About Distributions

Complete Lab 2A prior to Lesson 5.