Five number summary is used in statistics to make a summary of the given set of numbers. This summary is very helpful for performing many tasks. In statistics, to calculate the maximum number or minimum number, or to calculate the quartiles, a five-number summary is used in this regard. 

In this post, we will learn the basics of the five numbers summary, and all the five involved terms in it, with a lot of examples.

Five Number Summary

Five number summaries are used to find the minimum or maximum values, first and third quartile, and the median. The observations having the range from sample minima and maxima of the terms, also the direction of these terms, all these terms are carried by five number summaries. 

A step-by-step way to calculate the five number summaries is mentioned below.

1. Minimum value present in the given data.
2. Q₁ = first quartile
3. Median or simply middle value or Q₂.
4. Q₃ = third quartile
5. Maximum value present in the given data.

How to apply a Five Number Summary?

Five number summary is applied to the set of data having integers. When the data is given then apply the five terms of a five number summary step-by-step. Now let’s discuss these terms one by one.

Minimum value present in the given data

When the set of numbers is given select the least one from the set and give it a name as the minimum value of the set. It is the smallest value present in the given set of data. 

Example

Find the smallest value present in the given set of the data, 4, 33, 21, 48, 12, 34, 15?

Solution 

Step 1: Arrange the data in ascending order.

4, 12, 15, 21, 33, 34, 48

Step 2: The very first value of the ascending order value is the minimum value.

Minimum value = 4

First quartile of the data

The median of the lower half of the given set of data is said to be the first quartile of the data. It is almost covered 25% of the data. Q₁ is usually used to denote the first quartile. And the equation used for the first quartile is,

Q₁ = (n+1/4)^th 

Example 1

Calculate the first quartile Q₁ of the given data, 24, 13, 21, 18, 12, 34, 15?

Solution 

Step 1: Arrange the data in ascending order.

12, 13, 15, 18, 21, 24, 34

Step 2: Choose the lower half of the ascending order data.

12, 13, 15

Step 3: Choose the central value of the lower half of the median of the lower half that is selected, which is the first quartile.

First quartile = Q₁ = 13

Example 2

Calculate the first quartile Q₁ of the given data, 44, 24, 13, 21, 48, 18, 12, 56, 34, 15, 54?

Solution 

Step 1: Arrange the data in ascending order.

12, 13, 15, 18, 21, 24, 34, 44, 48, 54, 56

Step 2: use formula

Q₁ = (n + 1/4)^th 

Step 3: Put the values in the given formula as n = 11

First quartile = Q₁ = (11 + 1/4)^th 

                                 = (12/4) ^th

First quartile = Q₁ = 3^th

Q₁ is the 3rd term of the arranged data which is 15.

Median of the data

The middle value of the arranged set of data according to ascending order is known as the median of the given data. Almost 50% of the data is used for this. It is usually also referred to as the Q₂ or second quartile of the data. 

When there are no middle values as the terms are even, then take two values and find their mean and the mean value must be the median of the given set of data. To find most perfectly use the equation of the second quartile. 

Median = second quartile = Q₂ = (n + 1/2)^th 

Example 1

Calculate the second quartile Q₂ of the given data, 44, 24, 13, 21, 48, 18, 12, 56, 34, 15, 54?

Solution 

Step 1: Arrange the data in ascending order.

12, 13, 15, 18, 21, 24, 34, 44, 48, 54, 56

Step 2: Since there are 11 numbers so the middle value must be the sixth term of the arranged data.

Median = second quartile = Q₂ = 24

Example 2

Calculate the second quartile Q₂ of the given data, 44, 24, 13, 21, 48, 18, 12, 34, 15, 54?

Solution 

Step 1: Arrange the data in ascending order.

12, 13, 15, 18, 21, 24, 34, 44, 48, 54

Step 2: The term in the given set is even so we have to take two middle values.

21, 24

Step 3: Add the taken two middle terms and divide them by two.

Median = second quartile = Q₂ = 21 + 24/2

Median = second quartile = Q₂ = 45/2 = 22.5

Third quartile of the data

The median of the upper half of the given set of data is said to be the third quartile of the data. It is almost covered 75% of the data. Q₃ is usually used to denote the third quartile. And the equation used for the first quartile is,

Q₃ = (3(n+1)/4)^th 

Example 1

Calculate the second quartile Q₂ of the given data, 44, 24, 13, 21, 48, 18, 12, 56, 34, 15, 54?

Solution 

Step 1: Arrange the data in ascending order.

12, 13, 15, 18, 21, 24, 34, 44, 48, 54, 56

Step 2: Take the terms after median.

34, 44, 48, 54, 56

Step 3: Select the middle value of the upper half of the median of the upper half which is the third quartile.

Third quartile = Q₃ = 48

Example 2

Find the third quartile of the data 44, 33, 21, 48, 12, 34, 45, 3, 11, 54, 60?

Solution 

Step 1: Arrange the data in ascending order.

44, 33, 21, 48, 12, 34, 45, 3, 11, 54, 60

Step 2: Write the formula

Q₃ = (3(n+1)/4)^th 

Step 3: Put the values in the given formula as n = 11

Third quartile = Q₃ = (3(11 + 1)/4) ^th

Third quartile = Q₃ = 3(12/4) ^th

Third quartile = Q₃ = 3*3^th

Third quartile = Q₃ = 9^th

Q₃ is the 9th term of the arranged data which is 11.

Maximum value present in the given data

When the set of numbers is given select the largest one from the set and give it a name as the maximum value of the set. It is the highest value present in the given set of data. 

Example

Find the largest value present in the given set of the data, 4, 33, 21, 48, 12, 34, 15?

Solution 

Step 1: Arrange the data in ascending order.

48, 34, 33, 21, 15, 12, 4

Step 2: The very first value of the descending order value is the minimum value.

Maximum value = 48

Summary

The five-number summary is used to make a list of terms in statistics. All the five terms from the set of the numbers are calculated by using this summary. You can solve any problem related to this summary by following the above-solved examples.