How to Calculate A Five-Number Summary for Data Analysis?

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Data summarization is a quick and easy way to explain all of the values in a data set using only a few statistical values. The mean and standard deviation are used to summarize data with a Gaussian distribution, but they could be meaningless or even false if your data set does not have a Gaussian distribution.

In this post, we will teach you how to use a five number summary to describe the distribution of a data sample without assuming a complex data distribution. You can use this statistical summary for any type of data analysis.

After finishing this guide, you will be able to implement data summarization techniques, such as estimating the mean and standard deviation which is only applicable to the Gaussian distribution. Along with that, you will be able to use five-number summary to identify a data sample.

What is Data Summarization?

Data summarization techniques allow you to explain the distribution of data using just a few primary measurements.

To calculate standard deviation and mean for data with a Gaussian distribution is the most general example of data summarization. You can understand and recreate the distribution of the data using just these two parameters. The summary of the given data may wrap as little as tens of individual findings or as much as billions.

The issue is that it is difficult to measure the mean and standard deviation of data that does not have a Gaussian distribution. These amounts can be calculated technically, but they do not summarize the data distribution. In reality, they can be very misleading.

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In the case of data that does not have a Gaussian distribution, the five number summary should be used to summarize the data set.

What is Five-Number Summary?

The five-number summary is a non-parametric data summarization method. It is also known as the 5-number summary because it includes a total of five statistical terms which we will discuss later.

Since it was suggested by John Tukey, it is often referred to as the Tukey 5-number summary. It can be used to characterize the distribution of test samples for any kind of data (sample or population).

The 5-number summary contains just the right amount of information as a regular summary for general usage. Here are the five terms of five number summary.

Maximum Number

Maximum number is a number with the greatest value in the data set. It is the biggest number in a given set of data.

Minimum Number

Minimum number is a number with the least value in the data set. It is the smallest number in a given set of data.

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First Quartile

The 1st quartile is calculated by taking the median of the lower half of the given set of data. It tells us that 25% of the numbers in the data set lie below the first quartile and about 75% of the numbers lie above it. It is represented by Q1.

Median

Median is a statistical value that represents the most middle value of a data set. In other words, median separates the lower half of a data set form the upper half of the data set.

Third Quartile

The 3rd quartile is calculated by taking the median of the upper half of the given set of data. It tells us that 75% of the numbers in the data set lie below the third quartile and about 25% lie above it. It is represented by Q3.

How to Calculate Five Number Summary for Data Analysis?

Calculating 5-number summary is easy if we compare the process of calculations to the whole set of data we usually have with us. As discussed above, we have to calculate five statistical terms to get five number summary of our data.

In this section, we will use an example to demonstrate the method. Each term will be calculated separately for ease of understanding. Before diving in the calculations, let us give you a tip to find five number summary. You can use an online 5 number summary calculator by Allmath.com to get the summary of your data set instantly.

Example

Use the following data set.

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2, 4, 7, 3, 5, 1, 9

Step 1: The first step will always be arranging the set of data. Arrange the values in ascending order.

1, 2, 3, 4, 5, 7, 9

Step 2: Find the minimum and maximum number. In this case, the data set is arranged in ascending order, you can simply pick the first value as the minimum and the last value as maximum.

Maximum Number = 9

Minimum Number = 1

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Step 3: Find the median. Start removing elements one by one from both sides of the data set. The remaining value at the end will be the median. If the data set contains an even number of values, then add the last two remaining values and divide them by 2 to get the median.

1, 2, 3, 4, 5, 7, 9

Median = 4

Note: Numbers on the left side of the median are considered as the upper half and the numbers on the right side of the median are considered as the lower half.

Step 4: Find the first quartile by calculating the median of the upper half.

Upper half = 1, 2, 3

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1, 2, 3

First Quartile = 2

Step 5: Find the third quartile by calculating the median of the lower half.

Lower half = 5, 7, 9

Third Quartile = 7

Step 6: Write down all values get the five number summary in one place.

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Maximum = 9

Minimum = 1

Median = 4

First Quartile = 2

Third Quartile = 7

Are quartile and percentile the same?

A quartile is an observable value at a point that helps divide an ordered data sample into four equal-sized bits. The median, or second quartile, divides the ordered data set into two halves, and the first and third quartiles divide each half into thirds.

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A percentile is an observed value at a point that helps in the division of an organized data sample into 100 equal-sized parts. Quartiles are often presented as percentages.

The quartile and percentile values are also representations of rank statistics that can be measured on any data set. They are used to easily summarize how much of the distribution’s data is behind or ahead of a given observed value.

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