The 5 number Summary is one of the simplest statistic sets that any statistician will have to find. However, despite its simple computation, all of the descriptive statistics it comprises are extremely important in different levels of statistics.

The 5 number summary comprises 5 distinct, independent sample statistics that help students and statisticians know more about that sample observation or the bigger population data it represents. The five statistics are the minimum of a sample, the lower quartile, the Median, the upper quartile and the maximum.

The minimum and the maximum are the least and highest value in the observation, respectively. The Median (M) is simply the middle value of the data set. The first quartile is the Median of the lower half of the dataset after the main Median (M) has been found. The third quartile is the Median of the upper half of the main Median of an observation.

Knowing how to find the five data set is extremely important in situations where more information is needed about the observation or sample data being observed. For example, knowing the first quartile and the third quartile of a sample will help determine the interquartile, range and subsequently the spread of the sample. It can also help determine if the sample range includes outliers that do not necessarily influence the observation and the population data that it represents.

This article will outline how to find all the five datasets and also consider when they are actually implementable.

**The 5 Number Summary and Respective Formulas**

The 5 summary is a set of five different descriptive statistics that gives information about a particular dataset. The five statistics are

The sample Minimum

The lower quartile or first quartile

The Median or Middle quartile or second quartile

The Upper Quartile

The Maximum Quartile

**The Sample Minimum**

The sample minimum is the least value in the sample observation after it has been arranged in a descending to ascending order.

**The Sample Maximum**

The sample maximum is the highest value in the sample observation after it has been arranged in a descending to ascending order.

**First Quartile,**

Statistically, the first quartile is simply the Median of the values below the main Median. It accounts for 25% of the data set and is responsible for 25% of the entire data set, while the other 75% is outside it.

The first quartile can actually be determined by just performing a count and finding the middle number of all the data sets below the main Median. It is basically just looking for the Median of a smaller data set derived from the lower half of the main Median.

**Median**

The Median is one of the most common parameters for determining the distribution of a simple data set. Mathematically the Median is a number or set of numbers that divide the data set into two equal parts, with 50% of the entire data set below it and the other 50% above it.

The Median is known as the second quartile (. and as such is of the family of the quartiles. Like every quartile

**Third Quartile**

The third quartile is the Median of all the data set above the main Median. It has 75% of all the data set above it and 25% above it. It is the Median for the upper part of the data set.

**There relevant formulas for the 5 Number summary are:**

**Median Formula**

The formula for the position of the Median or second quartile is

n/2 when n is odd

n+1/2 when n is even

Where n is the number of the entire dataset

**First Quartile Formula**

The first quartile is mathematically presented as

when n is odd

**or**

when n is even

Where n is the number of the lower dataset before the Main Median

**Third Quartile Formula**

The first quartile is mathematically presented as

and its formula is

when n is odd

**or**

when n is even

Where n is the number of the upper dataset after the Main Median

**How to Calculate All the Five Descriptive Statistics that Make Up the Set of the Five Summary?**

Step 1: Arrange the Data Set in a descending to Ascending order starting with the minimum value and ending with the maximum value.

Step 2: Find the Sample Minimum by choosing the number that starts the arranged data set

Step 3: Find the Sample maximum by choosing the number that ends the arranged data set

Step 4: Find the Median by either tracing the middle number in your arranged data set or as an alternative, use the given formula.

Step 5: Find the first quartile by either tracing the middle number of the dataset before the Median or alternatively use the given formula.

Step 6: Find the third quartile by either tracing the middle number of the dataset after the Median or alternatively use the given formula.

**Find The 5 summary**

**Example 1:**

Find the 5 Number summary for the following dataset.

31, 59, 10, 50, 12, 33, 24, 43, 36, 25, 35, 27, 34, 11, 34,

**Solution**

**Step 1: Arrange the Data Set in a descending to Ascending order starting with the minimum value and ending with the maximum value.**

Arranged data set = 10, 11, 12, 24, 25, 27, 31, 33, 34, 34, 35, 36, 43, 50, 59

**Step 2: Find the Sample Minimum by choosing the number that starts the arranged data set**

From the arranged data set, 10 is the sample minimum

**Step 3: Find the Sample maximum by choosing the number that ends the arranged data set**

59 is the sample maximum

**Step 4: Find the Median by either tracing the middle number in your arranged data set or alternatively use the given formula.**

10, 11, 12, 24, 25, 27, 31, 33, 34, 34, 35, 36, 43, 50, 59

Tracing, the data it is clear that the median is 33.

Alternatively, using the formula,

obviously, 33 is in the 8^{th} position

**Step 5: Find the first quartile by either tracing the middle number of the dataset before the Median or alternatively using the given formula.**

The lower part of the median dataset is

10, 11, 12, 24, 25, 27, 31,

The Q1 is the Median of the above data set and as such is 24

**Step 6: Find the third quartile by either tracing the middle number of the dataset after the Median or alternatively use the given formula.**

The upper part of the median dataset is

34, 34, 35, 36, 43, 50, 59

The Q3 is the Median of the above data set and as such is 36

**With the above, the 5 Number Summary has been fully computed**

While the 5 Number Summary is a very important descriptive statistic set, it has its limitations in terms of applications. All the descriptive statistics can only be computed when the sample observation being considered is a Univariate model.

**Conclusion**

From the above computation of the 5 Number summary, it is obvious that it is a simple descriptive statistic to calculate. However, its usefulness cannot be underestimated. Basically, finding the minimum and the maximum only requires identifying the smallest and biggest value of the data set. The Median, first quartile and third quartile were also very easy to compute, and none required a lot of math work.

It is important to remember that the aim of the 5 number summary is to solve basic statistics that play very important roles in finding more complex information about a sample and its population. Finally, it is important to note that the 5 Number summary is only implementable when the sample observation is a univariate dataset.