Box and Whisker Plots

A box plot summarizes data using the median, upper and lower quartiles, and the extreme (least and greatest) values. It allows you to see important characteristics of the data at a glance.

Example Problem: The gas mileages in miles per gallon (mpg) of 4-cylinder manual transmission cars are in a table on the next slide

MPG of 4-cylinder cars

28 32 42 37

30 25 44 38

24 32 33 44

38 34 30 44

31 28 31 29

39 29 32 29

24 25 28 28 29 29 29 30 30 31 31 32 32 32 33 34 37 38 38 39 42 44 44 44

To make a box plot, organize the data by either making a stem and leaf plot or just arranging the data in order least to greatest.

24 25 28 28 29 29 29 30 30 31 31 32 32 32 33 34 37 38 38 39 42 44 44 44

Find the median of the data. It is 32This divides the data in half. The lower half : 24 25 28 28 29 29 29 30 30 31 31 32 and the upper half: 32 32 33 34 37 38 38 39 42 44 44 44

Find the median of the top half of the data.

32 32 33 34 37 38 38 39 42 44 44 44This is called the high median, upper quartile or quartile 3. It is 38.

Take the lower half of the data and find the median of it.

24 25 28 28 29 29 29 30 30 31 31 32

This data, 29, is called the low median, lower quartile or quartile 1.

Next, find the lower and upper extremes. This simply means the lowest data, 24, and the highest data, 44.Let’s organize all 5 pieces of data together so we can see them.

Lower extreme = 24Lower quartile(Q1) =29Median (Q2) = 32Upper quartile(Q3) =38Upper extreme(Q4)=44

The data is now divided into quartiles(4ths) so each quartile represents one-fourth of the data.

Next, make a number line that will best display the 5 pieces of data (24 29 32 38 44)

20 24 28 32 36 40 44

Place a dot above the number line to show the lower extreme and one for the upper extreme.

20 24 28 32 36 40 44

Put a vertical slash above the number line for the median and one for the lower and upper quartiles.

20 24 28 32 36 40 44

Enclose the vertical slashes into a box. Draw a line from the right center of the box to the upper extreme and one from the lower end of the box to the lower extreme, forming the whiskers.

20 24 28 32 36 40 44

You must label the number line to tell what the data represents.

Miles per gallon (mpg)

20 24 28 32 36 40 44

All graphs must have a title that clearly represents what your graph is showing.

Miles per Gallon of 4-cylinder Cars

20 24 28 32 36 40 44

Miles per gallon (mpg)

Interpreting the Box Plot:

Study your Box and Whisker Plot to determine what it is telling you. Make a statement about what it is saying, then support the statement with facts from your graph.

You should include the following in your interpretation:• Range or spread of the data and what it

means to your graph• Quartiles—compare them. What are

they telling you about the data?• Median- this is an important part of the

graph, and should be an important part of the interpretation.

• Percentages should be used to interpret the data, where relevant.

We will now interpret the data we have on mpg of 4-cylinder cars. We will do this step-by-step, then put all the interpretation together as our final summary.

Miles per Gallon of 4-cylinder Cars

20 24 28 32 36 40 44

Miles per gallon (mpg)

Make a statement about what it is saying, then support the statement with facts from your graph:

The Box and Whisker Plot clearly shows that there is a lot of different gas mileages on various 4-cylinder vehicles

Miles per Gallon of 4-cylinder Cars

20 24 28 32 36 40 44

Miles per gallon (mpg)

Range or spread of the data and what it means to your graph

The mileage ranged from 24 miles per gallon(mpg) to a high of 44 mpg. This is a 20 miles per gallon spread, which in car mileage is quite a bit of difference.

Quartiles—compare them. What are they telling you about the data?

The first quartile reads as 32 mpg which means that 75% of the vehicles in this study got 32 mpg or more.

The 3rd quartile tells us that 25% of these cars got 38 mpg or higher which is really good mileage.

Median- this is an important part of the graph, and should be an important part of the interpretation.

The median cuts the data in half. The median is 32 mpg. Therefore half the cars in the study received 32 mpg or higher.

Put all the data together in a summary that is clearly stated, uses facts based on the graph, and is easy to follow.

The Box and Whisker Plot clearly shows that there is a lot of different gas mileages on various 4-cylinder vehicles.

The mileage ranged from 24 miles per

gallon(mpg) to a high of 44 mpg. This is a 20 miles per gallon spread, which in car mileage is quite a bit of difference.

The first quartile reads as 32 mpg which means that 75% of the vehicles in this study got 32 mpg or more.

The 3rd quartile tells us that 25% of these cars got 38 mpg or higher which is really good mileage.

The median cuts the data in half. The median is 32 mpg. Therefore half the cars in the study received 32 mpg or higher.

From this study, we can conclude that there is a wide range of gas mileage that should be considered when traveling or purchasing a vehicle.