2. Box-and-Whisker Plots

3. Today’s Learning Goal We will learn another way to show data in a visual way. We will continue to compare data sets by their centers and spreads.

4. Explaining Data Consider the data at the right that shows the amount of miles of coastline land for each state on the east coast. What is the minimum? State on East Coast Length of Coast (mi) Delaware Florida Georgia Maine Maryland Massachusetts New Hampshire New Jersey New York North Carolina Rhode Island South Carolina Virginia 28 580 100 228 31 192 13 130 127 301 40 187 112 What is the maximum? Yes…13 miles (NH). Yes…580 miles (FL). We can use the minimum and maximum data points to roughly explain the data by saying that east coast states’ coastlines range from 13 to 580 miles.

5. Explaining Data Knowing the minimum and maximum allows us to see how spread out the data is. State on East Coast Length of Coast (mi) Delaware Florida Georgia Maine Maryland Massachusetts New Hampshire New Jersey New York North Carolina Rhode Island South Carolina Virginia 28 580 100 228 31 192 13 130 127 301 40 187 112 Another way to explain the data is by using the center. 0100200 300 400 500 600 13580 Two measures of the center are the mean and median.

6. Review of Medians What do we need to do first to find the median of this data? State on East Coast Length of Coast (mi) Delaware Florida Georgia Maine Maryland Massachusetts New Hampshire New Jersey New York North Carolina Rhode Island South Carolina Virginia 28 580 100 228 31 192 13 130 127 301 40 187 112 0100200 300 400 500 600 13580 Awesome…put the data in order. 13,28,31,40,100,112,127,130,187,192,228,301, 580 There are 13 data points. How many data points will be below and above the median? Nice…6.

7. Review of Medians So, what is the median of this data set? State on East Coast Length of Coast (mi) Delaware Florida Georgia Maine Maryland Massachusetts New Hampshire New Jersey New York North Carolina Rhode Island South Carolina Virginia 28 580 100 228 31 192 13 130 127 301 40 187 112 0100200 300 400 500 600 13580 Great…127. 13,28,31,40,100,112,127,130,187,192,228,301, 580 127 Don’t forget that the median is the exact center of the data. With an odd number of data points, it is the exact center data point!

8. Explaining Data State on East Coast Length of Coast (mi) Delaware Florida Georgia Maine Maryland Massachusetts New Hampshire New Jersey New York North Carolina Rhode Island South Carolina Virginia 28 580 100 228 31 192 13 130 127 301 40 187 112 0100200 300 400 500 600 13580 13,28,31,40,100,112,127,130,187,192,228,301, 580 127 Notice from the picture above that half of the data is clumped between 13-127 and half of the data is spread out between 127-580. 50% What percent of the data fall below the median? Excellent…50% is below.

9. Quartiles 0100200 300 400 500 600 13 580 13,28,31,40,100,112,127,130,187,192,228,301, 580 127 The minimum, maximum, and median are three important points that can help explain a data set well. However, there are two other points that help explain the data even more precisely. They are the quartiles. The median splits the data into two equal parts. Quartiles split the data into four equal parts. (min) (med) (max)

10. Quartiles 13,28,31,40,100,112,127,130,187,192,228,301, 580 The median is the second quartile (denoted Q 2 ) similar to. To find the first quartile of the data, find the median of the bottom half of the data (not including median). Q2Q2 Will the median of the bottom half be 40? No…because that would put 3 below 40 and 2 above. 0100200 300 400 500 600 13 580127 Q2Q2

11. Quartiles 13,28,31,40,100,112,127,130,187,192,228,301, 580 What do we need to do to find the median of the bottom half? Notice that the first quartile is denoted Q 1 similar to ¼. Q2Q2 Correct…find the mean of 31 and 40. 31+40 2 35.5 Q1Q1 71 = 35.5 Also notice that the first two quartiles are equal in size because they have the same number of data points between them. 0100200 300 400 500 600 13 580127 Q2Q2 Q1Q1

12. Quartiles 13,28,31,40,100,112,127,130,187,192,228,301, 580 What do we need to do to find the median of the upper half? Notice that the third quartile is denoted Q 3 similar to ¾. Q2Q2 Great…find the mean of 192 and 228. 192+228 2 210 Q3Q3 420 = 210 Also notice that the last two quartiles are equal in size to the first two quartiles (they all have three data points). 0100200 300 400 500 600 13 580127 Q1Q1 Q3Q3 Q2Q2 35.5 Q1Q1

13. Box-and-Whisker 13,28,31,40,100,112,127,130,187,192,228,301, 580 Now we have a five-number summary for the data: With the five-number summary, we can make what is called a box-and-whisker plot. Q2Q2 210 Q3Q3 Simply make a box around Q 1 and Q 3, put a line down the box for the median, and connect the min and max with lines (whiskers). 0100200 300 400 500 600 13 580 35.5 Q1Q1 (i) the min, (ii) Q 1, (iii) median (Q 2 ), (iv) Q 3, and (v) the max. min max

14. Box-and-Whisker 13,28,31,40,100,112,127,130,187,192,228,301, 580 What percent of the data is in the box? As you can see from this picture, 75% of the data is between 13 and 210! The long whisker to the right shows that the last 25% of the data is spread out! Q2Q2 210 Q3Q3 0100200 300 400 500 600 13 580 35.5 Q1Q1 Wow…50% of the data is within the box! min max 25%

15. Box-and-Whisker Notice how the box plot gives us a picture of the data. It lets us visually see the following: The box give us an idea of the center and where half of the data falls. The whiskers let us see how spread out the data is. 0100200 300 400 500 600 Does a box plot let us see every data point like a stem-and-leaf plot does? No…it gives us an overall general picture of the data!

16. Explaining Data Now, consider the data at the right that shows the amount of miles of coastline land for each state on the west coast (minus Alaska). State on West Coast Length of Coast (mi) California Hawaii Oregon Washington 840 750 363 157 Let’s get the five-number- summary needed to make a box plot for this data. What is the minimum? What is the maximum? 157840 MinQ 1 Q 2 (Med) Q 3 Max

17. Medians Now we need the quartiles. Take a look at the data in order below. What do we need to do to get the median? State on West Coast Length of Coast (mi) California Hawaii Oregon Washington 840 750 363 157 840 157, 363, 750, 840 Yes…average 363 and 750. 363+750 2 1113 = 556.5 556.5 Q2Q2 MinQ 1 Q 2 (Med) Q 3 Max

18. Quartiles What do we need to do to get the first quartile? State on West Coast Length of Coast (mi) California Hawaii Oregon Washington 840 750 363 157 840 157, 363, 750, 840 Perfect…average 157 and 363. 157+363 2 520 = 260 556.5 Q2Q2 MinQ 1 Q 2 (Med) Q 3 Max 260 Q1Q1

19. Quartiles What do we need to do to get the third quartile? State on West Coast Length of Coast (mi) California Hawaii Oregon Washington 840 750 363 157 840 157, 363, 750, 840 Good…average 750 and 840. 750+840 2 1590 = 795 556.5 Q2Q2 MinQ 1 Q 2 (Med) Q 3 Max 260 Q1Q1 795 Q3Q3 Notice again how the quartiles split the data up into four equal parts!

20. Box-and-Whisker The box-and-whisker plot for the east coast states is shown below. 0100200 300 400 500 600700800900 We can put a box-and-whisker plot for the west coast states on the same number line to compare. 13 580 157840556.5 MinQ 1 Q 2 (Med) Q 3 Max 260795 157840

21. Box-and-Whisker Looking at the box-and-whisker plots below, what can we say about east states’ coastlines vs. west states’? 0100200 300 400 500 600700800900 13 580 157840 Fantastic…it is obvious that west coast states have a longer coastline than east coast states.

22. Box-and-Whisker Looking at the box-and-whisker plots below, which datsa appears to be more symmetrical? 0100200 300 400 500 600700800900 13 580 157840 Super…it appears that the west coast states’ data are more symmetrical. The east coast data have a maximum data point that is much different than the rest of the data.

23. Partner Work You have 30 minutes to work on the following questions with your partner.

24. For those that finish early In this lesson, we made box plots showing the lengths of coastline land for west coast states and one for east coast states. But, we did not include Alaska in the box plot for west coast states. 1) Go online and determine the length of Alaska’s coastline. 2) Explain why we probably did not include Alaska based on the length of its coastline.

25. Big Ideas from Today’s Lesson A box-and-whisker plot is another way to compare data sets. The box-and-whisker plot is nice because it shows five important numbers: Minimum Q 1 (1 st Quartile) Median Q 3 (2 nd Quartile) Max

26. Homework Complete Homework Worksheet Pgs. 619 – 621 (4 – 9, 16 – 19, 22, 23) If you want a challenge, please try #24 and #25 on page 621.