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Five Number Summary

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Statistics is the study of data. It deals with collection, interpretation, organization and analysis of data. To summarize a data set  five number summary is used, which tells about the minimum, Q1, median, Q3 and maximum values of the data set.

Mean, median, mode are the three basic and important concepts of data analysis. Mean stand for the average value of the data set and mode stands for the most repeated value of the data set. When the data is written in an ascending order, the value that comes exactly in the middle, that is, the value which divides the data set into two halves is known as the median.

Five Number Summary Definition

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The median divides the data set into two halves. The data to the left of the median, that is, from the minimum value to the median is called the lower half. The next half of the data, from the median to the maximum value of the data set is known as upper half.

• Minimum : The lowest value in the data set.

• Q1 : First quartile is the median of the lower half.

• Median : The middle value of the data set.

• Q3 : Third quartile is the median of the upper half.

• Maximum : The highest value of the data set.

How to Find Five Number Summary

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For a given set of numbers, following steps can be taken to get the five number summary of that data set.

1) Arrange all the numbers in the ascending order.

2) The leftmost element is the minimum and the rightmost is the maximum.

3) The element exactly in the middle of the ordered data is the median. If numbers of observation is even, median is the average       of the two middle elements.

4) The median of the data from the minimum element to the element just left of the median is Q1.

5) The median of the data from the element just right of the median and the maximum element is Q3.

Five Number Summary Example

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Below find some solved examples of five number summary,
Example 1:

List the five-number summary when the marks by 8 schools students in a physics exam are as follows. 99, 77, 79,  86, 80,  87, 94, 87 out of 100.

Solution:

Given 99,77,79,86,80,87,94,87 out of 100  

First arranging the marks.

77, 79,  80,  86,  87,  87,  94,  99

Find the minimum marks

which is 77

Find the maximum marks

which is  99

Find the median of the given marks,

$Q_{2 }$ = $\frac{86+87}{2}$   

= $\frac{173}{2}$

= $86.5$

The lower half part = 77, 79,  80,  86

First quartile is,

$Q_{1}$ = $\frac{79+80}{2}$ = $79.5$

The upper half part =  87,  87,  94,  99

Third quartile

$Q_{3}$ = $\frac{87+94}{2}$ = $90.5$

Hence the five-number summary is in order to,

Min, $Q_{1}$, $Q_{2}$
, $Q_{3}$, Max

77, 79.5, 86.5, 90.5, 99
Example 2:

Evaluate five-number summary for below given data.

8, 5, 8, 9, 3, 0, 6, 1, 4, 6, 8, 6, 9.

Solution:

Given 8, 5, 8, 9, 3, 0, 6, 1, 4, 6, 8, 6, 9

First arrange the given data,

0, 1, 3, 4, 5, 6, 6, 6, 8, 8, 8, 9, 9

The minimum of the data is 0

The maximum of the data is 9

Median of the data is 6,

The lower half part = 0, 1, 3, 4, 5, 6

The Lower quartile is,

$Q_{1}$ = $\frac{3+4}{2}$ = 3.5

The Upper half part = 6, 8, 8, 8, 9, 9

The Upper quartile is

$Q_{3}$ = $\frac{8+8}{2}$ = 8

Hence five-number summary is for given data is 

Min, $Q_{1}$, $Q_{2}$, $Q_{3}$, Max


0, 3.5, 6, 8, 9