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Box-and-Whisker Plots
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.
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
28580
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.
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
28580
100
228
31
192
13
130
127
301
40
187
112
Another way to explain the data is by using the center.
0 100 200 300 400 500 600
13
580
Two measures of the center are the mean and median.
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
28580
100
228
31
192
13
130
127
301
40
187
112
0 100 200 300 400 500 600
13
580
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.
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
28580
100
228
31
192
13
130
127
301
40
187
112
0 100 200 300 400 500 600
13
580
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!
Explaining DataState 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
28580
100
228
31
192
13
130
127
301
40
187
112
0 100 200 300 400 500 600
13
580
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% 50%
What percent of the data fall below the median? Excellent…50% is below.
Quartiles
0 100 200 300 400 500 600
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)
Quartiles
13, 28, 31, 40, 100, 112, 127, 130, 187, 192, 228, 301, 580
The median is the second quartile (denoted Q2) similar to .
To find the first quartile of the data, find the median of the bottom half of the data (not including median).
42
21
Q2
Will the median of the bottom half be 40?
No…because that would put 3 below 40 and 2 above.
0 100 200 300 400 500 600
13
580127Q2
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 Q1 similar to ¼.
Q2
Correct…find the mean of 31 and 40. 31+4
0 2
35.5
Q1
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.
0 100 200 300 400 500 600
13
580127Q2Q1
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 Q3 similar to ¾.
Q2
Great…find the mean of 192 and 228. 192+228
2
210
Q3
420= 210
Also notice that the last two quartiles are equal in size to the first two quartiles (they all have three data points).
0 100 200 300 400 500 600
13
580127Q1 Q3Q2
35.5
Q1
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.
Q2
210
Q3
Simply make a box around Q1 and Q3, put a line down the box for the median, and connect the min and max with lines (whiskers).
0 100 200 300 400 500 600
13
580
35.5
Q1
(i) the min, (ii) Q1, (iii) median (Q2), (iv) Q3, and (v) the max.
min max
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!
Q2
210
Q3
0 100 200 300 400 500 600
13
580
35.5
Q1
Wow…50% of the data is within the box!
min max
25% 25% 25% 25%
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.
0 100 200 300 400 500 600
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!
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?
157
840
Min Q1 Q2 (Med) Q3 Max
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
157
840
157, 363, 750, 840
Yes…average 363 and 750. 363+750
21113
= 556.5
556.5
556.5
Q2
Min Q1 Q2 (Med) Q3 Max
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
157
840
157, 363, 750, 840
Perfect…average 157 and 363.
157+3632
520= 260
556.5
556.5
Q2
Min Q1 Q2 (Med) Q3 Max
260
Q1
260
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
157
840
157, 363, 750, 840
Good…average 750 and 840.
750+8402
1590= 795
556.5
556.5
Q2
Min Q1 Q2 (Med) Q3 Max
260
Q1
260
795
Q3
795
Notice again how the quartiles split the data up into four equal parts!
Box-and-Whisker The box-and-whisker plot for the east coast
states is shown below.
0 100 200 300 400 500 600 700 800 900
We can put a box-and-whisker plot for the west coast states on the same number line to compare.
13
580
157
840
556.5
Min Q1 Q2 (Med) Q3 Max 260 795
157 840
Box-and-Whisker Looking at the box-and-whisker plots below, what can
we say about east states’ coastlines vs. west states’?
0 100 200 300 400 500 600 700 800 900
13
580
157 840
Fantastic…it is obvious that west coast states have a longer coastline than east coast states.
Box-and-Whisker Looking at the box-and-whisker plots below, which
datsa appears to be more symmetrical?
0 100 200 300 400 500 600 700 800 900
13
580
157 840
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.
Partner Work You have 30 minutes to work on the following
questions with your partner.
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.
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 Q1 (1st Quartile) Median Q3 (2nd Quartile) Max
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.