Warm Up 1) Find the mean & median for this data set: 7, 6, 5, 8, 10, 20, 4, 3 2) Which measure of center, the mean or median should be used to describe the data set? Why?

HW Check 2.4

Measures of Spread: Standard Deviation

Describing Data Two ways to describe data: Graphically Numerically Dot plot Histogram Boxplot Numerically Measures of Center: Median and Mean Measures of Spread 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 How much do values typically vary from the center? Range Interquartile Range (IQR) Standard Deviation Today’s focus will be on standard deviation.

Thinking about the Situation Consider the following test scores: Who is the best student? How do you know? Student Test 1 Test 2 Test 3 Test 4 Johnny 65 82 93 100 Will 86 89 83 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.

So who is the best?! Sometimes you need too look at more information than the measure of center… So we look at the SPREAD…

Measures of Spread How much do values typically vary from the center? One-Variable: Range Interquartile Range (IQR) Standard Deviation Remember, measures of spread address how far most of the data values typically fall from the center. We have already discussed the range. In this lesson we will talk about the mean absolute deviation and how it is related to the standard deviation. What does the word “deviation” mean? (1. The action of departing from an established course or accepted standard. 2. The amount that a single measurement differs from a fixed value such as the mean.)

Range To find the range of a set of data you take the largest value in the data set and subtract the smallest number in the data set BIG # - small # = range We will look at interquartile range in a different lesson.

Lets focus on STANDARD DEVIATION Standard deviation measures the dispersion of data. The greater the value of the standard deviation, the further the data tends to be dispersed from the mean. Luckily, we can find the standard deviation on the calculator.

Back to our students . . . Lets calculating each of their Standard Deviations Student Test 1 Test 2 Test 3 Test 4 Johnny 65 82 93 100 Will 86 89 83 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.

Steps Step 1: Enter the height data in L1. Commands: STAT  EDIT (Use 2nd QUIT to exit) Step 2: Find your “best friend” in this unit Commands: STAT  CALC  1: 1-Var Stats  Enter

What do they all mean… Number of data entries Mean The sum of your data  Ignore this Standard Deviation  Ignore this Number of data entries

Who is the best student? How do you know? Now calculate the standard deviation of each and discuss… Who is the best student? How do you know? Student Test 1 Test 2 Test 3 Test 4 Test Average Standard Deviation Johnny 65 82 93 100 85 13.2 Will 86 89 83 2.7 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.