Five number summary is a pretty simple topic. It gives information about the location (from the median), spread, and range (from sample minimum and maximum) of the observation. It gives a rough idea about what the data set looks like. The number chosen helps us to know the center of data.

In this post, we will learn the definition, rule, and how to calculate the five number summary along with a lot of examples.

**What is the Five Number Summary?**

Five number summary is simply consisted of the smallest data value, the first quartile, median, the third quartile, and the largest data value. It gives information about the location (from the median), spread, and range (from sample minimum and maximum) of the observation. It gives a rough idea of what the data set looks like. In general, we can write the five number summary as,

- Smallest data value.
- First quartile
**Q**_{1} - The median or the second quartile
**Q**_{2} - The third quartile
**Q**_{3} - Largest data value.

## Rules of Five Number Summary

Let us learn the rules of five numbers

**1. Minimum value**

The smallest data value is known as a minimum value of the data. In order to find the minimum value, arrange the given data in ascending order.

**Example**

Find the minimum value of the data 44, 33, 21, 48, 12, 34, 45?

**Solution**

**Step 1:**Arrange the data in ascending order.

12, 21, 33, 34, 44, 45, 48

**Step 2:**the first value of the arranged data is the minimum value

Minimum value = 12

**2. Maximum value**

The largest data value is known as the maximum value of the data. In order to find the maximum value, arrange the given data in ascending order.

**Example**

Find the maximum value of the data 44, 33, 21, 48, 12, 34, 45?

**Solution**

**Step 1:**Arrange the data in ascending order.

12, 21, 33, 34, 44, 45, 48

**Step 2:**the last value of the arranged data is the maximum value

Maximum value = 48

**3. First Quartile**

First quartile is the median of the lower half of the data as 25% of data falls below the first quartile. It is denoted by **Q _{1}**. It can also be found by using formula.

Q_{1} = ((n + 1)/4)^{th}

**Example 1**

Find the first quartile of the data 44, 33, 21, 48, 12, 34, 45?

**Solution**

**Step 1:**Arrange the data in ascending order.

12, 21, 33, 34, 44, 45, 48

**Step 2:**take the lower half of the data

12, 21, 33

**Step 3:**choose the middle value of the lower half of the median of the lower half which is the first quartile.

First quartile = Q_{1} = 21

**Example 2**

Find the first quartile of the data 44, 33, 21, 48, 12, 34, 45?

**Solution**

**Step 1:**Arrange the data in ascending order.

12, 21, 33, 34, 44, 45, 48

**Step 2:**use formula

Q_{1} = ((n + 1)/4)^{th}

**Step 3:**Put the values in the given formula as n = 7

First quartile = Q_{1} = ((7 + 1)/4)^{th}

= (8/4)^{th}

First quartile = Q_{1} = 2^{th}

Q_{1} is the 2th term of the arranged data which is 21.

#### 4.** Median**

Median is the middle most value of the data as 50% of all the data falls below the median. Median is the middle value from smallest to largest. It is also known as the second quartile of the data and is denoted by Q_{2}. If there is no middle value as the number of data set is even then we take two middle values and find the mean of those values the result will be the median. It can also be found by using formula.

Q_{2} = ((n + 1)/2)^{th}

**Example 1**

Find the median of the data 44, 33, 21, 48, 12, 34, 45?

**Solution**

**Step 1:**Arrange the data in ascending order.

12, 21, 33, 34, 44, 45, 48

**Step 2:**the middle value of the arranged data is the median.

Median = Q_{2} = 34

**Example 2**

Find the median of the data 3, 5, 6, 4, 2, 9?

**Solution**

**Step 1:**Arrange the data in ascending order.

2, 3, 4, 5, 6, 9

**Step 2:**the data is even so we take two middle values of the arranged data.

4, 5

**Step 3:**add both the terms and divide by 2 we get the median.

Median = Q_{2} = 4 + 5/2

= 9/2 = 4.5

**5. Third Quartile**

Third quartile is the median of the upper half of the data as 75% of data falls below the third quartile. It is denoted by **Q _{3}**. It can also be found by using formula.

Q_{3} = (3(n + 1)/4)^{th}

**Example 1**

Find the third quartile of the data 44, 33, 21, 48, 12, 34, 45?

**Solution**

**Step 1:**Arrange the data in ascending order.

12, 21, 33, 34, 44, 45, 48

**Step 2:**take the upper half of the data

44, 45, 48

**Step 3:**choose the middle value of the upper half or the median of the upper half which is the third quartile.

Third quartile = Q_{3} = 21

**Example 2**

Find the third quartile of the data 44, 33, 21, 48, 12, 34, 45?

**Solution**

**Step 1:**Arrange the data in ascending order.

12, 21, 33, 34, 44, 45, 48

**Step 2:**use formula

Q_{3} = (3(n + 1)/4)^{th}

**Step 3:**Put the values in the given formula as n = 7

First quartile = Q_{3} = (3(7 + 1)/4)^{th}

= 3(8/4)^{th}

First quartile = Q_{1} = 3*2^{th}

First quartile = Q_{1} = 6^{th}

Q_{3} is the 6th term of the arranged data which is 45.

**How to calculate Five Number Summary?**

In order to calculate five number summary, you have to keep a few steps in mind.

- Put the data in ascending order.
- Find minimum and maximum of your data.
- Find median.
- Place parenthesis around the upper and lower from median.
- Find Q
_{1},Q_{3}. - Write summary

Let us take some examples in order to learn how to calculate five number summary. You can also find it using the online Five Number Summary calculator.

**Read more: Why Daily Total One Contact Lenses Are Better Than Monthly Lenses?**

**Example 1**

Find the five number summary of the data set 18, 7, 12, 5, 15, 9, 17?

**Solution**

**Step 1:**Arrange the data in ascending order.

5, 7, 9, 12, 15, 17, 18

**Step 2:**Find maximum and minimum number.

**Minimum number** in the arranged data = 5

**Maximum number** in the arranged data = 18

**Step 3:**Find median

The middle value of the arranged data is the median.

**Median** = 12

**Step 4:**put parenthesis above and below the median.

**(**5, 7, 9**)**, 12, **(**15, 17, 18**)**

**Step 5:**Find Q_{1},Q_{3}.

**Q _{1}** = median of lower half = 7

**Q _{3}** = median of upper half = 17

**Step 6:**write all values to form a summary.

**Minimum number** = 5

**Maximum number** = 18

**Median** = 12

**Q _{1}** = 7

**Q _{3}** = 17

**Example 2**

Find the five number summary of the data set 33, 23, 16,19, 44, 65, 41?

**Solution**

**Step 1:**Arrange the data in ascending order.

16, 19, 23, 33, 41, 44, 65

**Step 2:**Find maximum and minimum number.

**Minimum number** in the arranged data = 16

**Maximum number** in the arranged data = 65

**Step 3:**Find median

The middle value of the arranged data is the median.

**Median** = 33

**Step 4:**put parenthesis above and below the median.

**(**16, 19, 23**)**, 33, **(**41, 44, 65**)**

**Step 5:**Find Q_{1},Q_{3}.

**Q _{1}** = median of lower half = 19

**Q _{3}** = median of upper half = 44

**Step 6:**write all values to form a summary.

**Minimum number** = 16

**Maximum number** = 65

**Median** = 33

**Q _{1}** = 19

**Q _{3}** = 44

After calculating the 5 number summary, you may want to calculate variance or standard deviation. You can use a standard deviation calculator to find the difference, sum of difference, variance, and standard deviation in one place.

**Summary**

In this post we have learned about the definition and rules to find five number summary with a lot of examples. By practicing these examples, you will easily solve any problem related to five number summary.