Five Number Summary

The Five Number Summary is a statistical method used to describe a dataset using five important values. It provides a quick overview of the distribution, center, and spread of the data.

The five values included in a Five Number Summary are

• Minimum Value – The smallest value in the dataset.
• First Quartile (Q1) – The middle value of the lower half of the dataset.
• Median (Q2) – The middle value of the entire dataset.
• Third Quartile (Q3) – The middle value of the upper half of the dataset.
• Maximum Value – The largest value in the dataset.

Steps to Calculate the Five-Number Summary

1. Arrange the data in ascending order.

2. Find the Minimum Value: The smallest value in the dataset.

3. Find the Maximum Value: The largest value in the dataset.

4. Find the Median (Q2)

• Position of Median = (n + 1) / 2
• The value at this position is the median.

5. Find the First Quartile (Q1)

• Position of Q1 = (n + 1) / 4
• The value at this position is the first quartile.

6. Find the Third Quartile (Q3)

• Position of Q3 = 3(n + 1) / 4
• The value at this position is the third quartile.
• n = Total number of observations in the dataset.

This gives the five values: Minimum Value, Q1, Median (Q2), Q3, and Maximum Value.

Uses of Five-Number Summary

• Provides a concise summary of a dataset.
• Helps understand the distribution and spread of data.
• Assists in identifying outliers.
• Forms the basis of a box-and-whisker plot (box plot).
• Makes comparison between datasets easier.

Solved Examples

Question 1: What is the minimum value in the given data: 10, 20, 5, 15, 25, 30?.

Solution:

Step 1: Arrange the data in ascending order:

5, 8, 10, 15, 20, 25, 30

Step 2: The first value in the sorted data is the minimum value.

Minimum Value = 5

Question 2: What is the maximum value in the given data: 10, 20, 5, 15, 25, 30, ?.

Solution:

Step 1: Arrange the data in ascending order:

5, 8, 10, 15, 20, 25, 30

Step 2: The last value in the sorted data is the maximum value.

Maximum Value = 30

Question 3: What is the median value in the given data: 10, 20, 5, 15, 25, 30, 8

Solution:

Step 1: Arrange the data in ascending order:

5, 8, 10, 15, 20, 25, 30

Step 2: Find the median using the formula:

Median Position = (n + 1) / 2

Here, n = 7

Median Position = (7 + 1) / 2 = 8 / 2 = 4

So, the median is the 4th term in the sorted data.

Median Value = 15

Question 4: Find the 5-number summary for the given data: 10, 20, 5, 15, 25, 30, 8

Solution:

Step 1: Arrange the data in ascending order:

5, 8, 10, 15, 20, 25, 30

Step 2: Find the Minimum, Median, and Maximum values.

• Minimum Value = 5
• Median Value = 15
• Maximum Value = 30

Step 3: Find the First Quartile (Q1).

Q1 Position = (n + 1) / 4

= (7 + 1) / 4

= 8 / 4

= 2

So, Q1 is the 2nd term in the sorted data.

Q1 = 8

Step 4: Find the Third Quartile (Q3).

Q3 Position = 3(n + 1) / 4

= 3(7 + 1) / 4

= 3 × 8 / 4

= 24 / 4

= 6

So, Q3 is the 6th term in the sorted data.

Q3 = 25

Question 5: Find out the 5-number summary for the data 1, 10, 5, 15, 2, 12, 4, 14.

Solution:

Step 1: Arrange the data in ascending order:

1, 2, 4, 5, 10, 12, 14, 15

Step 2: Find the Minimum and Maximum values.

• Minimum Value = 1
• Maximum Value = 15

Step 3: Find the Median (Q2).

Median Position = (n + 1) / 2

= (8 + 1) / 2

= 9 / 2

= 4.5

Since the position is 4.5, take the average of the 4th and 5th terms.

Median = (5 + 10) / 2

= 15 / 2

= 7.5

Median (Q2) = 7.5

Step 4: Find the First Quartile (Q1).

Q1 Position = (n + 1) / 4

= (8 + 1) / 4

= 9 / 4

= 2.25

Take the average of the 2nd and 3rd terms.

Q1 = (2 + 4) / 2

= 6 / 2

= 3

Q1 = 3

Step 5: Find the Third Quartile (Q3).

Q3 Position = 3(n + 1) / 4

= 3(8 + 1) / 4

= 27 / 4

= 6.75

Take the average of the 6th and 7th terms.

Q3 = (12 + 14) / 2

= 26 / 2

= 13

Q3 = 13