Box-and-Whisker Plots
Standard
S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center... and spread... of two or more different data sets.
Goals for this section:
Students should be able to construct a box-and-whisker plot and interpret the information it conveys.
Quartiles
Think of quartiles as values that separate "sections".
- Quartile 2 (Q2), the median, divides the data set into upper and lower halves.
- Quartile 1 (Q1) is the median of the lower half.
- Quartile 3 (Q3) is the median of the upper half.
We also use minimum and maximum to describe the ends.
Example 1: Finding quartiles (even amount of data)
Given: 20, 20, 21, 22, 24, 25, 25, 25, 27, 28, 28, 30
Step 1
List your data in order! You must list the data set from least to greatest. This example is already in order.
Step 2
Find Q2. The middle.
This is the median of the data set.
20, 20, 21, 22, 24, 25, 25, 25, 27, 28, 28, 30
There's two values! So add and divide by 2 to get 25.
Q2 = 25
Step 3
Find Q1. The middle of the lower half.
This is the median of the lower half. Since this is an even set (n = 12), then you take the first 6 values as your lower half and find the middle.
20, 20, 21, 22, 24, 25
There's two values! So add and divide by 2.
(21 + 22) / 2 = 21.5
Q1 = 21.5
Step 4
Find Q3. The middle of the upper half.
This is the median of the upper half. This time, we take the upper 6 values.
25, 25, 27, 28, 28, 30
(27 + 28) / 2 = 27.5
Q3 = 27.5
Example 2: Finding quartiles (odd amount of data)
Given: 1, 6, 3, 5, 3, 3, 7, 2, 2, 8, 9
When it's odd, we do not include the median value (Q2) when finding Q1 and Q3 as demonstrated below.
Step 1
List your data in order!
1, 2, 2, 3, 3, 3, 5, 6, 7, 8, 9
Step 2
Find Q2. The middle.
1, 2, 2, 3, 3, 3, 5, 6, 7, 8, 9
Q2 = 3
Step 3
Find Q1. The middle of the lower half.
Notice that we're using the first 5 values and we do not include the median (Q2).
1, 2, 2, 3, 3
Q1 = 2
Step 4
Find Q3. The middle of the upper half.
Notice that we're using the upper 5 values and we do not include the median (Q2).
5, 6, 7, 8, 9
Q3 = 7
Box-and-Whisker Plot
This plot is a visual representation of your data that divides the data into 4 quartiles. Let's create a box-and-whisker plot for the previous problems.
Example 1
Create your range
The first example has a minimum of 20 and a maximum of 30.
Draw a long line for Q2
Draw a long line for Q1
Draw a long line for Q3
Connect the lines to form a box!
This is the "box" part of your box-and-whisker plot.
Create little lines for your minimum and maximum
So that's your smallest and biggest number
Connect those little lines to the box!
This is the "whisker" part of your box-and-whisker plot.
Example 2
Recall that for example 2, Q2 = 3, Q1 = 2, and Q3 = 7, with a minimum of 1 and a maximum of 9.
Notice how the median is closer to the left and suggests how the data is skewed.
Note: If there's outliers, it will usually be on the outside of the whiskers and represented with a dot.
Why use a box-and-whisker plot?
They are easy to interpret and are great for comparing data. For example, you can have box-and-whisker plots to compare test scores between different classes.
What can you say about 1st period? Their test scores aren't as great as 2nd period. Their median score is 80 as compared to 2nd period's median score at 85. Median is not necessarily the average, but you can see how the majority of the scores may be centered around Q2.