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Five Number Summary

Statistics plays a vital role not only in mathematics, but also in various other fields. Statistics may be defined as the study of vast numerical data. It includes different researches, surveys and experiments.

In statistics, the data is to be analyzed by different means. In a statistical process, It is important to observe about the data values - how far is spread?, what are its mean, median and mode?, what are minimum and maximum values?, which value lie in the middle? etc.
There are two types of statistics:
1) Descriptive statistics
2) Inferential statistics.
There is an important concept in descriptive statistics called five-number summary which indicates information about a frequency distribution. In this page, we shall discuss about the five-number summary and its examples.

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Definition

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Five-number summary is type of summary about the statistical data. It is defined in descriptive statistics. It is a theory which provides information about the given set of values.
According to five-number summary, a frequency distribution includes five important percentiles which are as follows:
1) Minimum: The minimum indicates the smallest value in the data set. It is represented by "min".

2) Maximum: The maximum denotes the biggest value lying in the given data set. It is notified by "max".

3) Lower Quartile:  It is a point that separates first 25% (or lowest 25%) of the values from the rest 75% of the values of the given data. First quartile is also known as lower quartile and is denoted by Q$_{1}$.

4) Median: Median is the value that falls exactly in the middle of the data set. In other words, median is a middle most value when the observations are arranged in ascending order. It is sometimes denoted by Q$_{2}$ (second quartile).

5) Upper Quartile:  Upper quartile is the value that separates out biggest 25% values from the rest 75%. It is represented by Q$_{3}$ and is also called as third quartile.
We can say that the five-number summary is the list of above described five number about a given set of data. It can be represented as:

Min, Q$_{1}$, Q$_{2}$, Q$_{3}$, Max

Following box and whisker plot (box plot) describes five-number summary:
Five Number Summary
The five-number summary is a concise information about the distribution of the values among a set of observations. It provides various information such as spread of the data, location of the median, smallest and biggest values, range of the data etc.

Five-number summary is very important in ordinal measurements. It is also useful in ratio and interval measurements. By the use of this summary, it becomes possible to easily compare different sets of data by making comparisons with different five-number summaries. These summaries can also be illustrated graphically with the help of box and whisker plot.

Examples

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The examples of five-number summary are as follows:

Example 1: Find the five-number summary of the following data.
57, 45, 98, 30, 66, 81, 48, 69.

Solution: Arranging the list in ascending order, we get
30, 45, 48, 57, 66, 69, 81, 98
Here, Minimum = 30
and maximum = 98

Median of the data = $\frac{57+ 66}{2}$

$Q_{2}$=$\frac{123}{2}$ = 61.5

Thus, the lower half portion of the data is 30, 45, 48, 57
Lower quartile $Q_{1}$=$\frac{45+ 48}{2}$ = 46.5

Also, the upper half portion of the data is 66, 69, 81, 98
Upper quartile $Q_{3}$=$\frac{69+ 81}{2}$ = 75

So, the five-number summary of the data is :
Min, Q$_{1}$, Q$_{2}$, Q$_{3}$, Max
30, 46.5, 61.5, 75, 98

Example 2: List the five-number summary of the given data.
8, 5, 8, 9, 3, 0, 6, 1, 4, 6, 8, 6, 9.

Solution: Let us first arrange this list in the ascending order as below.
0, 1, 3, 4, 5, 6, 6, 6, 8, 8, 8, 9, 9
Minimum of the data = 0
Maximum of the data = 9

Median of the data = the middle number
= 6    (since number of entries is odd)

Thus, lower half portion of the data is 0, 1, 3, 4, 5, 6
Lower quartile is given by

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

Upper half portion of the data is 6, 8, 8, 8, 9, 9

Upper quartile $Q_{3}$=$\frac{8+ 8}{2}$ = 8

Therefore, the five-number summary would be:

Min, Q$_{1}$, Q$_{2}$, Q$_{3}$, Max
0, 3.5, 6, 8, 9

Example 3: The marks obtained by 8 coaching students in a mathematics exam out of 100 are as follows.
Evaluate the five-number summary:
99, 77, 79,  86, 80,  87, 94, 87

Solution: On arranging the marks in ascending order, we get the following list.
77, 79,  80,  86,  87,  87,  94,  99
Minimum marks = 77
Maximum marks = 99

Median of the given marks is the middle most number or second quartile is given by

$Q_{2}$=$\frac{86+87}{2}$   (since the number of observations is even)

 = $\frac{173}{2}$
= 86.5

In this way, we get the lower half part as 77, 79,  80,  86
First quartile is calculated as

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

And, upper half part will be 87,  87,  94,  99

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

So, the five-number summary is as follow:
Min, Q$_{1}$, Q$_{2}$, Q$_{3}$, Max
77, 79.5, 86.5, 90.5, 99
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