1. U4D3 Warmup:
Find the mean (rounded to the nearest tenth) and median for the following data: 73, 50, 72, 70, 70, 84, 85, 89, 89, 70, 73, 70, 72, 74 Mean: 74.4 Median: 72.5
…..Mean Median and Mode Toads
…..Rap
…. British explanation

2. Homework Check: Document Camera
How to calculate one variable statistics on the
Ti-83+
Ti-84+ or on next slide

4. Math IB U4D3: One-Variable Statistics Boxplots, Interquartile Range, and Outliers; Choosing Appropriate Measures

5. Objective Students will be able to…
Interpret data based on the shape of a data distribution
Choose the appropriate measures of center (mean or median) and spread (standard deviation or interquartile range) to describe the distribution.
Interpret summary statistics for center and spread in the context of the data.

6. Describing Data Graphically
Quantitative Data
Dotplot
Histogram
Boxplot
S-ID.1  Represent data with plots on the real number line (dot plots, histograms, and box plots).
Set the stage for today’s lesson – we have covered dotplots and histograms, today we are going to talk about boxplots.

7. Describing Data Numerically
Measures of Center – mean, median
Measures of Spread – range, interquartile range, standard deviation
We are also going to talk more about the median and learn about the interquartile range as a measure of spread.
S-ID.2  Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

8. Boxplots!
Measures of Spread - YouTube

9. Boxplots Min Q1 Median Q3 Max Lower Upper Quartile Quartile
Discuss how to draw a box plot by hand. First, you must know the “five-number summary” of the data. Use the max and min to determine an appropriate scale to use.
Minimum (min)
Lower Quartile (Q1)
Median (M)
Upper Quartile (Q3)
Maximum (max)
We use these 5 numbers to construct the boxplot. The quartiles form the edges of the “box,” the median is a line inside the box, and the max and min are attached to the sides of the box with “whiskers.” Thus this graph is sometimes called a box-and-whisker plot.
Discuss how to find shape, center and spread from the boxplot. For example: The shape of our buttons distribution is skewed right since the right whisker is a lot longer than the left whisker. The center is the line in the middle of the box that corresponds to the median. For the spread we can easily see how far out each whisker reaches (the range). We can also look at the length of the box. To find the length, we can subtract Q3 – Q1. This difference is known as the interquartile range (IQR for short), because it measures how spread out the quartiles are.

10. Practice!
Below is a stem and leaf plot of the amount of money spent by 25 shoppers at a grocery store.
Stem
Leaf 1 2 3 4 5 6 7 8 9 10 11
3 6
0 5
2 6
Guided practice: Ask if students know how to read a stem and leaf plot. Have a student explain to the rest of the class how to read the plot.
Key: 42 = $42

11. Practice! Calculate the mean and median.
Stem
Leaf 1 2 3 4 5 6 7 8 9 10 11
3 6
0 5
2 6
Calculate the mean and median.
Calculate the lower and upper quartiles and IQR.
Determine which, if any, values are outliers.
Write several sentences to describe this data set in context.
Name some factors that might account for the extreme values, and the much lower measure of center.
median $31, mean$37.64
lower quartile $18.50, upper quartile $47.50, IQR 29
LQ – 1.5 (IQR) = -25
No outliers
UQ +1.5(IQR) = 91
97 & 113 are outliers
Ex – note low center, big spread, two extreme values, on upper end of data
Ex – extreme values – larger family, shopping for entire week
Lower values – quick tips
Key: 42 = $42

12. 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.

13. 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.

14. 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 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.

15. Measures of Spread How much do values typically vary from the center?
One-Variable: Range Interquartile Range (IQR) Mean Absolute Deviation (MAD) 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 The amount that a single measurement differs from a fixed value such as the mean.)

16. 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.

17. Side-by-Side Boxplots
Using technology, create side-by-side boxplots.
Show students how to create side-by-side boxplots in the calculator (set up Plot1 to graph the data in List1 and Plot2 to graph the data in List2 and turn both “ON”). Have them transfer sketches of the box plots to graph paper, using a single number line scale. Be sure to label which box plot is male and which is female.

18. Creating Histograms
Using technology, create a histogram of each set of data. Make sure you use the same scale for each!
Show students how to create histograms using the same window (that works for both graphs) on the calculator. Have them transfer sketches of the histograms to graph paper. Be sure to label which box plot is male and which is female.

19. CW: U2D8 Comparing Types of Peanut Butter
Have students complete the Comparing Data Sets Practice activity in pairs or groups. Discuss answers. Have a couple of groups share their descriptions and discuss the quality of their narratives. The rest of class time can be spent on the Poster Project.

20. Comparing Types of Peanut Butter
Natural
Regular
Statistic
Natural Peanut Butter
Regular Peanut Butter
Min 34 11 Q1 57 31 M
61.5 40 Q3 69 54
Max 89 83
IQR 12 23
mean
61.2
42.7
standard deviation
13.6
18.8
Natural Peanut Butter Outliers:
34, 89
Which measure of center and spread would be most appropriate to use to describe these two sets of data? Explain.
It would be most appropriate to use the median and the IQR since the Natural Peanut Butter Quality Ratings data set has outliers. Note that since there are outliers on both ends of the data set, they balance out their effects on the mean and thus the value of the mean, 61.2, is close to the value of the median, However, the outliers do inflate the value of the standard deviation, 13.6, as compared to the IQR of 12.
Compare the two data sets in context. Be sure to address shape, center, spread, and outliers. Which type of peanut butter is better?
The distribution of quality ratings for natural peanut butter is slightly skewed to the left as evidenced by the longer whisker on the left side of the box. The distribution of the quality ratings for regular peanut butter is fairly symmetrical. Natural peanut butter has higher quality ratings than regular peanut butter with a median of 61.5 points for natural vs. 40 points for regular. The ratings for regular peanut butter are more spread out, ranging from 11 points to 83 points, and therefore less consistent than the ratings for natural peanut butter which ranges from 34 to 89 points. This can also be seen in the spread of the middle 50% of the data: the spread of the middle for regular peanut butter is 23 points but the spread of the middle for natural peanut butter is 12 points. Overall, natural peanut butter has higher quality ratings.

21. Snowboards