Question

What is a five number summary?

Answer 1

This type of question depends on the variety of descriptive statistics. The five number summaries is a convenient way to combine five descriptive statistics. The five number summaries help us to know about the centre of our data as well as how the data points are spread out.

Complete step by step solution:
Here, we have to note that the five number summaries consist of following:
The minimum – The smallest value in our data set.
The first quartile – This number is denoted by ${{Q}_{1}}$ and $25\%$ of our data falls below the first quartile.
The median – This is the middle point of the data and $50\%$ of our data falls below the median.
The third quartile – This number is denoted by ${{Q}_{3}}$ and $75\%$ of our data falls below the third quartile.
The minimum – The largest value in our data set.
For Example:
Given the following set of data, we will report the five number summaries –
1, 1, 2, 2, 3, 4, 6, 6, 7, 7, 8, 11, 12, 14, 15, 15, 17, 18, 19, 20
We can see that there are a total of 20 points in the data set. Hence, we can say that the median is the average of the tenth and eleventh data values.
$\Rightarrow \text{Median = }\dfrac{(7+8)}{2}=7.5$
We also know that the median of the bottom half of the original data is the first quartile.
The bottom half is: 1, 1, 2, 2, 3, 4, 6, 6, 7, 7
$\Rightarrow \text{First Quartile = }{{Q}_{1}}=\dfrac{(3+4)}{2}=3.5$
We can write that the median of the top half of the original data is the third quartile.
The top half is: 8, 11, 12, 14, 15, 15, 17, 18, 19, 20
$\Rightarrow \text{Third Quartile = }{{Q}_{3}}=\dfrac{(15+15)}{2}=15$
Hence, we can write the five number summaries for above data set is 1, 3.5, 7.5, 15, 20

Note: While solving this problem students make mistakes if the data is not well organised. So students have to arrange the given data in ascending order if it is not. Also if the data set is large then students have to do the calculations carefully.