The 5 Number Summary
The five number summary is another name for the visual representation of the box and whisker plot.
Five number summary consists of:
- The median ( 2nd quartile).
- The 1st quartile
- The 3rd quartile.
- The maximum value in a data set.
- The minimum value in a data set.
A five number summary provides the five descriptive measures of the given data set. It consists of the smallest value (XS), first quartile (Q1), median (Md), third quartile (Q3) and the largest value (XL) of the data set. Five number summary is very helpful to determine the shape of the distribution.
Example: The following is the five number summary of a data set with 30 observations.
Xsmallest = 19, Q1 =25, Md = 29.5, Q3 = 34, XLargest = 42
What is your observation with this five number summary? Comment about the shape of the distribution.?
Solution:
The distance from smallest value to median = 29.5 – 19 = 10.5
The distance from the median to the largest value = 42- 29.5 = 12.5
The distance from smallest value to Q1 = 25 – 19 = 6
The distance from Q3 to the largest value = 42 – 34 = 8
Here the distance from the smallest value to the median is slightly less than the distance from median to the largest value (i.e., 10.5 < 12.5).
Similarly, the distance from the smallest value to Q1 is also slightly less than the distance from Q3 to the largest value (i.e., 6 < 8).
This shows that the data is slightly right skewed.
Box and Whisker plot
A box and whisker plot is a graphical presentation of the data that displays a five number summary of a data set
Steps for the construction of Box and whisker plot
Step 1 – Find the median.
Remember, the median is the middle value in a data set.
18, 27, 34, 52, 54, 59, 61, 68, 78, 82, 85, 87, 91, 93, 100. 68 is the median of this data set.
Step 2 – Find the lower quartile.
The lower quartile is the median of the data set to the left of 68.
(18, 27, 34, 52, 54, 59, 61,) 68, 78, 82, 85, 87, 91, 93, 100
Here, 52 is the lower quartile.
Step 3 – Find the upper quartile.
The upper quartile is the median of the data set to the right of 68.
18, 27, 34, 52, 54, 59, 61, 68, (78, 82, 85, 87, 91, 93, 100)
87 is the upper quartile.
Step 4 – Find the maximum and minimum values in the set.
The maximum is the greatest value in the data set.
The minimum is the least value in the data set.
18, 27, 34, 52, 54, 59, 61, 68, 78, 82, 85, 87, 91, 93, 100
18 is the minimum and 100 is the maximum.
Step 5 – Find the inter-quartile range (IQR).
The inter-quartile (IQR) range is the difference between the upper and lower quartiles.
Upper Quartile = 87
Lower Quartile = 52
87 – 52 = 35
35 = IQR
The 5 number summary of above example is;
Organize the 5 number summary
- Median – 68
- Lower Quartile – 52
- Upper Quartile – 87
- Max – 100
- Min – 18
