The five number summary is a descriptive statistic which provide information about a set of observation. The quartiles aids us to understand the spread of the data set given. The five number summary is useful to compare various set of observations which is then represented through the boxplot. Five number summary, consists of

  • Minimum
  • Maximum
  • Median
  • First quartile, and
  • Third quartile.

Generally a computer is used to calculate the five number summary. Given below are the few steps which will help you understand how it is calculated:
Step 1: The given data should be arranged from the smallest value to the largest value which is on the left and right respectively.

Step 2: Find the minimum and the maximum value from the given data.

Step 3: Median gives the middle number, if it is an even number then the median is the mean of the two middle numbers.

Step 4: Upper Quartile (Q3): It is the median of the upper half of the given data set $\frac{3}{4}$$(n+1)$ nth value where n is the number of data values in the data set.

Step 5: Lower Quartile (Q1): It is the median of the lower half of the given data set $\frac{1}{4}$$(n+1)$ nth value where n is the number of data values in the data set.


Solved Examples

Question 1: Compute the set { 6, 8, 9, 7, 2, 5, 4 }
Solution:
 
  • Arrange the given data 2, 4, 5, 6, 7, 8, 9
  • The minimum and maximum values are 2 and 9 respectively for the above problem.
  • From the given data we see that median is 6.
  • The lower half is {2, 4, 5} and the middle term of that half is 4. Therefore, the lower quartile is 4.
  • The upper half is {7, 8, 9}, and the middle term of that half is 8. Therefore, the upper quartile is 8.

 

Question 2: Suppose we wish to compute the five number summary for the set {11, 25, 29, 43, 56, 68, 72, 87, 92, 104}
Solution:
 
  1. Here the given data is already ordered { 11, 25, 29, 43, 56, 68, 72, 87, 92, 104 } so the minimum and maximum values are 11 & 104 respectively.
  2. The median is $\frac{56+68}{2}$ = 62, as 56 and 68 represents the middle values.
  3. Total number of observations for the given problem is 10, as (10 -1 = 9) is not evenly divisible by four, so the upper quartile is the median of the observations to the right of 62, therefore the upper half is  {68, 72, 87, 92, 104} and the median for this set is 87. Now  the upper quartile for the above problem (Q3) is 87.
  4. So now the lower quartile is the median of the observations to the left of 62, therefore the lower half is { 11, 25, 29, 43, 56 } and the median for this set is 29. Now the lower quartile for the above problem (Q1) is 29.