## Five Number Summaries and Box Plots

Video from https://www.youtube.com/watch?v=MXIU5pQi3UM

**Overview:**

A five number summary is a descriptive statistic that provides information for a set of data. It includes the minimum, quartile 1, median, quartile 3, and maximum. Box and whisker plots are usually used when comparing multiple data sets that are related to one another, for example is test scores from multiple classes.

**How Constructing a Box Plot Can Help You Make a Five Number Summary**

When first constructing a box and whiskers plot you must arrange your values from least to greatest.

5, 13, 14, 15, 15, 15, 16, 19, 18, 20, 23, 30, 47

Then you must find the median, or the number in the middle. For this specific set that is the number 16. To find the lower quartile you’ll have to look at what the middle value is for the values to the left of the median. For this data set it’s 14.5. Finding the upper quartile is just like finding the lower quartile value but rather than looking at the data to the left of the median you’ll be looking at the values to the right of the median. For this specific data set the upper quartile is 21.5. The smallest and greatest values of that set are the minimum and maximum. For this specific data set it’s 5 and 47.

- Minimum: 5
- Lower Quartile: 14.5
- Median: 16
- Upper Quartile: 21.5
- Maximum: 47

Without actually drawing your box and whiskers plot you were able to find your five number summaries.

**Interquartile Range**

The interquartile range is simply the range of the upper and lower quartiles.

- IQR = Q3 – Q1

**Finding Outliers**

Now that you have the interquartile range you can see if any of your values are considered outliers. You will use the following two equations to find outliers.

- Upper Fence = Q3 + 1.5(IQR)
- Lower Fence = Q1 – 1.5 (IQR)

Any value that exceeds the upper and lower fence will be considered an outlier.

**Comparing Groups with Boxplots**

Just like any other chart or graph, boxplots can be used to compare data amongst various groups. Rather than constructing this yourself you can use your calculator. Enter the two data sets into L1 and L2. Now go to STATPLOT’s Plot 1 to make a boxplot for the first data set. Turn the plot ON. Select the first boxplot icon. For “Xlist:” put L1 and for “Freq:” put 1. Then select the mark you want to display outliers. Do the same thing for the second set of data under Plot2. Once you have finished inputting all the values press ZoomStat.

**Example 1:**Is it possible to find an outlier with the data given below? If yes, proceed and find the outliers.

- Minimum: 3
- Q1: 15
- Median: 23
- Q3: 30
- Maximum: 62

*Solution:*

*Although we weren’t given a list of data it is still possible to find outliers using the five number summaries.*

*IQR = Q3 - Q1 → 30 - 15 = 15*

*Upper Fence = Q3 + 1.5(IQR) → 30 + 1.5(15) = 52.5*

*Lower Fence =Q1 - 1.5 (IQR) → 15 - 1.5 (15) = -7*

*Since our maximum exceeds the upper fence we have enough evidence to state that there is at least one outlier in this data set.*

**Example 2:**Find the five number summaries and the interquartile range for the data below.

34, 28, 45, 37, 44, 19, 35, 42, 48, 6, 18

*Solution:*

*First you have to arrange the data in numerical order.*

*6, 18, 19, 28, 34, 35, 37, 42, 44, 45, 48*

*The median, middle number is 35. The median for the first half of the data also known as the quartile 1 is 19 and the median for the second half of the data, or quartile 3, is 44. The minimum is 6 and the maximum is 48. The interquartile range is the difference between 44 and 19 which is 25.*