2. Mean Absolute Deviation

3. Describing Data There are two ways to describe a set of data:
Graphically
Dot plot, Bar Graph, Histogram , Boxplot
Numerically
Using a single number to describe the relationship of the data
Mean, Median, Mode, Range, IQR
Use this slide to remind students of where they are in the study of one-variable statistics. This lesson continues our study of describing data numerically. Now we turn to measures of spread.

4. We describe the measure of center using:
Measures of Center
We describe the measure of center using:
Mean: A numerical measure that is the arithmetic “average” of the data. Use the mean to describe a data set if there is NO outlier! Outliers greatly affect the mean!
Median: Describes the middle of a data set. Not usually affected by outliers or affected very little. 50% of data is above and 50% of data is below the median
Use this slide to remind students of where they are in the study of one-variable statistics. This lesson continues our study of describing data numerically. Now we turn to measures of spread.

5. Measures of Spread/Variability describe
how much values typically vary from the center
These measures are:
Range – describes the distance from the highest to lowest data pieces; found using maximum minus minimum
Interquartile Range (IQR) – describes the middle 50% of the data ; found using Q3 – Q1
Mean Absolute Deviation (MAD) – shows how much data values vary from the mean. Low MAD indicates data points are not very spread out. High MAD indicates data points are not close together, are very spread out!

6. A low mean absolute deviation indicates
Highlight the following in your notes:
A low mean absolute deviation indicates
that the data points tend to be very close
to the mean and not spread out very far
so the mean is an accurate description of
“typical”, and a high mean absolute deviation
indicates that the data points are spread out
over a large range of values.

7. Now let’s explore “deviations from the mean” as a way to determine how accurately the mean may describe “typical”.

8. Thinking about the Situation
Consider the following test scores: Who is the best student? How do you know? Take a few minutes to decide with your partner
Student
Test 1
Test 2
Test 3
Test 4 Li 65 82 93
100
Bessie 86 89 83
Jamal 80 99 73 88
Let students discuss this in their groups and make a decision. Have them share out their ideas.
Answers:
What is the mean score for each student? 85 points
Based on the mean, who is the best student? Let them share their opinions. This may end up running into the third question. If it doesn’t come up, say “They all have the same mean, so they are equal as students.” What do they think?
If asked to select one student, who would you pick as the best student? Explain. Accept their explanations without judgment. Some may look at Johnny’s increasing trend over time. Some may look at Will’s consistency. Some may note Anna’s one low grade. Be sure that arguments for each student being the “best” come out, even if you have to be the one who makes them. The point is, just using the mean to describe each student is not enough. I think that we can all agree that they are not “equal” in their test performance. We need more information than just the typical test score. One thing to look at is how consistent each student is, and measures of spread will give us that information.

9. So with all students having the same mean, let’s see if we can dig a little deeper in our comparison. The MEAN ABSOLUTE DEVIAITON (MAD) will help us. Let’s take a few minutes to explore MAD.

10. Mean Absolute Deviation (MAD)
STEP 1: Find the mean
STEP 2: Subtract the mean from
each piece of data
STEP 3: Find the absolute value of
each difference
STOP and do example on the back of this paper with students using the data set 1, 2, 4, 8, and 10
STEP 4: Find the mean of the new
differences (deviations)

11. Mean Absolute Deviation
Step 1: Mean =
Step 2:
Step 3:
Data Values
Difference
Data Minus Mean
Absolute Value 1 2 4 8 10
Let’s go back to our number line to visualize this. Most of the data falls within plus or minus 3.2 points of 5.
Total of Values:
Step 4:

12. So what exactly is deviation?-4 -3 +5 -1 +3
Remember that the mean is the balance point for a set of data:
For example, the mean of 1, 2, 4,8, and 10 is 5 because it balances the distances on a number line from the mean to each data value.
(-4) + (-3) + (-1) = -8
(+5 ) + (+3) = +8

13. Mean Absolute Deviation
-3.2
+3.2
Notice that our Mean Absolute Deviation or MAD
was 3.2 and most of our original data does fall
within plus or minus 3.2 points of the mean of 5.
Let’s go back to our number line to visualize this. Most of the data falls within plus or minus 3.2 points of 5.

14. Now take a few minutes to go back to our original question about the best student. Find the MAD score for each student and then make a decision based on all of your data about the best student. Be prepared to discuss.
Student
Test 1
Test 2
Test 3
Test 4 Li 65 82 93
100
Bessie 86 89 83
Jamal 80 99 73 88
After a few minutes to determine the MAD ask students to make their call about the best student and give a reason. There is a space on the notes for students to record their work.

15. Li Data Values Difference Data Minus Mean Absolute Value 65 82 93 100
Step 1: Mean =
Step 2:
Step 3:
Data Values
Difference
Data Minus Mean
Absolute Value 65 82 93
100
Let’s go back to our number line to visualize this. Most of the data falls within plus or minus 3.2 points of 5.
Total of Values:
Step 4:

16. Bessie Data Values Difference Data Minus Mean Absolute Value 82 86 89
Step 1: Mean =
Step 2:
Step 3:
Data Values
Difference
Data Minus Mean
Absolute Value 82 86 89 83
Let’s go back to our number line to visualize this. Most of the data falls within plus or minus 3.2 points of 5.
Total of Values:
Step 4:

17. Jamal Data Values Difference Data Minus Mean Absolute Value 80 99 73
Step 1: Mean =
Step 2:
Step 3:
Data Values
Difference
Data Minus Mean
Absolute Value 80 99 73 88
Let’s go back to our number line to visualize this. Most of the data falls within plus or minus 3.2 points of 5.
Total of Values:
Step 4:

18. Now you try the CLASSWORK found on your handout.
When you and your partner have completed the work individually, check to see if you agree.
We will review this section together in a few minutes

19. Classwork Data Values Difference Data Minus Mean Absolute Value 30 38
Step 1: Mean =
Step 2:
Step 3:
Data Values
Difference
Data Minus Mean
Absolute Value 30 38 40 42 48
Let’s go back to our number line to visualize this. Most of the data falls within plus or minus 3.2 points of 5.
Total of Values:
Step 4: