1. Warm Up – 5/15 - Thursday
Consider the following test scores: Answer the following in complete sentences: A) Who is the best student? B) How do you know?
Student
Test 1
Test 2
Test 3
Test 4
Johnny 65 82 93
100
Will 86 89 84
Anna 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.

2. 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

3. Investigation 1: Deviation from the Mean
Score
Mean
Deviation from
the Mean
Johnny
Test 1
Test 2
Test 3
Test 4
Have students work through Investigation 1: Deviation from the Mean
2) The sum of the deviations for each student is zero. The mean is the balance point, so the distances between the mean and each point balance in either direction. Go through next slides to solidify this concept.

4. Houston, we have a problem!
Measure of Spread
Sum of deviations =
(-4)+(-3)+(-1)+(+5)+(+3)
Average of the deviations= = 0
An average deviation of zero means that there is no variability!
Remember that we want to have some kind of measure of each student’s consistency. So let’s try to find a way to measure how much each student’s test grades vary from their average test grade. In order to have a numerical measure of spread, it would make logical sense to find the average distance, or deviation of each value from the mean. So to find the average we would add up the deviations and divide by the number of data values that we have.
Houston, we have a problem!

5. How can we fix our problem?
Take the absolute value of each distance/deviation and then find the average
So the average distance or deviation from the mean is about 3 points (above or below).
This is called the Mean Absolute Deviation, or MAD
Ask students to give their ideas on fixing the problem of the deviations adding up to zero. Although they may not state using the “absolute value”, they may state the concept by suggesting that we ignore the signs and just add up the numbers.

6. Mean Absolute Deviation
-3.2
+3.2
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.

7. This is called the standard deviation.
Square each deviation and then find
the average of the squared deviations.
This is called the standard deviation.
What else can we do? We could square each distance to make it positive and then find the average of the squared deviations.
But wait, 12 is a lot bigger than the 3.2 we found by calculating the mean average deviation.
Why is this? What happened?
We need to “undo” our squaring by taking the square root – then we will have a number that makes sense.

8. Back to Johnny, Will and Anna . . .
Investigation 3: Calculating the Standard Deviation
Student
Test 1
Test 2
Test 3
Test 4
Johnny 65 82 93
100
Will 86 89 84
Anna 80 99 73 88
Have students complete Investigation 3. You may want to walk them through Johnny’s data and then have them do Will’s and Anna’s in their groups and answer the questions.
Discussion Questions:
Why is the sum of the third column always equal to zero? The sum of the deviations from the mean is always zero because the mean is the balance point of the data.
Translate into words: The sum of the squared deviations from the mean. Or from inside parentheses out, subtract the mean from each value, square these differences and add them up.
Interpret Anna’s standard deviation in context. Anna’s test grades typically vary by about ten points from her average score of 85. So her test scores typically vary from about 75 to 95.

9. Who is the best student? How do you know?
Test 1
Test 2
Test 3
Test 4
Test
Average
Standard Deviation
Johnny 65 82 93
100 85
13.2
Will 86 89 84
2.2
Anna 80 99 73 88
9.7
Let students discuss this in their groups and make a decision. Have them share out their final verdict.

10. Standard Deviation
Here is the formula for standard deviation. Hopefully now it doesn’t look so scary!
Show how to find all measures using 1-Var stats on the calculator (last page of handout).