EXPLORING descriptions of SPREAD

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 Measures of Center: Mean and Median review these descriptions in your notes Measures of Spread- Today we will focus on the numerical descriptions 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.

Measures of Spread describe how much values typically vary from the center These measures are: Range – a description of how far apart the distance from the highest to lowest data pieces; found using highest minus lowest data Interquartile Range (IQR) – a description of the middle 50% of the data ; found using Q3 – Q1 Mean Absolute Deviation briefly view description in your notes and then continue with the PPT.

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

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.

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.

NOW WE WILL TRY THIS METHOD. Mean Absolute Deviation (MAD) Add to notes 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) NOW WE WILL TRY THIS METHOD.

Mean Absolute Deviation Write the list of numbers shown on the number line and then find the Mean Absolute Deviation 0 1 2 3 4 5 6 7 8 9 10 11 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. STOP AND COMPLETE CHART on NOTES

So what exactly is deviation? -4 -3 +5 -1 +3 0 1 2 3 4 5 6 7 8 9 10 11 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

Mean Absolute Deviation -3.2 +3.2 0 1 2 3 4 5 6 7 8 9 10 11 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.

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.

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.

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