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.