Lesson 4: How Far is it from Typical?

Objective

You 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 it 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.

Vocabulary

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

Essential Concepts

Lesson 4 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

You learned about two different measures of center during the previous two lessons: the mean and the median. Copy the sentence frames below into your IDS Journal and fill in the blanks with the appropriate word/words:
1. Mean – is used with _______ distributions.
2. Median – is used with _______ distributions, or when there are _______.
In today's lesson, you 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, you will use two different measures of spread: (1) the mean of absolute deviations (MAD), and (2) the interquartile range (IQR).
In your IDS Journal, write what you think the word deviation means, how it could relate to variability, and give at least three common synonyms like: fluctuation, modification, etc.
For the following activity, you will need the help of at least three people.

Find an open floor space in your house. Place a 4-5 foot long piece of masking tape (or painter’s tape) on the floor. Then read to the following scenario to your family member(s):

Your family 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 entire family consider the piece of tape (aka the rope) to be the “typical” path you all must take to finish the circus act. Since nobody wants to fall from the suspended tightrope while performing at the actual circus, you will all need to practice walking directly on the middle of the line at all times. If anyone deviates from the line, he/she will no longer be walking the “typical” path, and will likely fall.
You should select who will be the starting performer.
There should be at least four of you working together in this activity. One is the performer, two are measuring the distance of the deviation (one on each side of the tape), and one is the recorder.
As one family member walks and attempts to balance across the “rope”, 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.
The performer will walk the tightrope by looking straight up to the sky – first looking to place a foot on the line, then walking naturally while looking up to the sky, repeating this one step at a time for four steps, measuring after each step. Any time the performer missteps, this is considered a variation from the typical value. You will take turns so everyone gets a chance to balance, walk, measure and to record your observations in your IDS Journal.
Now that you have an idea about what it means to deviate from something you consider “typical,” you can start looking at distributions to see how data points vary from your typical value.
In this activity, you were observing deviations from typical while calculating actual differences between the rope and the performers' steps. When data are quantified with numbers, you 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). Consider the components of the MAD in math terms, and in your IDS Journal, write what the MAD value might represent.

mean – average

absolute – ______

deviation – ______
Using the three concepts in Step 12, the MAD measures the absolute distance of each data point from the mean, and then finds the average of all those distances.
Below is the formula for the MAD distribution:
$MAD=\frac{ \sum_{x=1}^{n} |x_i-\bar{x}|}{n}$
Copy the formula above and the table below into your IDS Journal. Then write what each symbol in the formula means and how we use it to perform the calculation (by completing the table):

x_i - represents each individual data point

x̄ - represents the mean value

n -________

Σ -________
To practice using this formula with actual data, you will calculate and compare the MAD values for two distributions.
The How Far Apart? handout contains two of the dotplots - plots (a) and (c) - from the Where is the Middle? handout used in Lesson 3. As before, the dotplots 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.

Here's how you might have calculated the MAD values for the two distributions:

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)

$\begin{align*} MAD &= \frac{3|0-2.53|+0|1-2.53|+4|2-2.53|+5|3-2.53|+5|4-2.53|}{17} \\ &= \frac{3(2.53)+0(1.53)+4(0.53)+5(0.47)+5(1.47)}{17}\\ &= \frac{7.59+0+2.12+2.35+7.35}{17}\\ &= \frac{19.41}{17}\\ &\approx 1.1418 \end{align*}$

Click on the document name to download a fillable copy of the How Far Apart? handout (LMR_2.6).
After you've completed the How Far Apart? handout, answer the following questions in your IDS Journal:
1. Which MAD value did you think would be larger based only on the look/shape of the distributions? Why?
2. Which MAD value was actually larger when you calculated it?
3. Did your prediction match the actual calculated values, or were you surprised by the results?
To continue exploring with the class’s Personality Color survey data, let's revisit the same dot plot visuals that we used in Lesson 2.

Compute the MAD value for the Blue scores.

Does the MAD value seem reasonable based on the dot plot?

Reflection

What are the essential learnings you are taking away from this lesson?

Homework & Next Day

Calculate the MAD values for each of the other three personality color scores and compare the values of the four color scores.

LAB 2A: All About Distributions

Complete Lab 2A prior to Lesson 5.