# Quartile Deviation

## Quartile Deviation Formula

Statistics is an important branch of Mathematics. Quartile deviation is part of Statistics. By statistics, we mean a collection of useful data. The quartile deviation comes in handy to assess the spread of a distribution. This distribution measures the spread from its central tendency (commonly referred to as ‘the mean’). Hence, quartile deviation provides helpful segmentation within which the central 50% of your sample data lies. Statistics is about the collection of data, the frequency, and the distribution of the trends. We can define Quartile deviation as the difference between the first quartile and the third quartile in the frequency distribution table. This difference is known as the interquartile range. When the difference is divided by two, it is known as quartile deviation or semi interquartile range.

### How to Calculate Quartile Deviation

A quartile helps break up the observations and divide into four intervals based upon the values of the data. It also examines how far is the distribution of the observations from the mean.

It is crucial to understand the median to understand quartile. As you already know that standard deviation, mean, mode, and median are essential concepts that are in use in statistics. All these methods of studying the trends help in estimating the flow and the diversions.

It is vital to understand the importance of the median as a measure of central tendency. The median in statistics is the middle value of a set of numbers. It is the point at which exactly half of the data lies, below and above the central value.

### Quartile Deviation – An example

Here is an example to help you understand better.

In a given set of 13 numbers, the median would be the seventh number. The six numbers above are the lowest in the data. The six numbers after the median are the highest number in the given data. So, it is logical to say that the median is not affected by extreme values.

We can safely say that the median is a robust estimator of location but does not divulge much information about data on either side of the high and low values. It is where quartile comes in. Quartiles help to measure the spread of values above and below the mean by dividing the data into four groups.

That is where the quartile steps in. The quartile measures the spread of values above and below the mean by dividing the distribution into four groups. So you have first, second, and third quartiles written as Q1, Q2, and Q3 respectively. Q2 is the median. To find quartiles in a grouped data, we have to arrange the data. The information is always arranged in ascending order.

### Quartile Formula

Let us assume that-

Q3 is the upper quartile in the median of the upper half of the data sample.

Q1is the lower quartile and median of the lower half of the data.

Median is Q2.

Number of items in data is n, the quartiles are given by

Q1= [(n+1)/4]th item

Q2=[(n+1)/2]th item

Q3=[3(n+1)/4]th item

Hence, the formula for quartile can be written as

Qr= l 1 + r( n/4)-c (l2 –l1)

f

Where, Qr is the rth quartile, l1 is the lower limit, l2 is the upper limit, f is the frequency, and c is the cumulative frequency of the class preceding the quartile class.

### About Quartile Deviation

We can define Quartile deviation as half of the distance between the third and the first quartile. It is also known as the Semi Interquartile range. When one takes half of the difference or variance between the 3rd and the 1st quartiles of a simple distribution or frequency distribution it is quartile deviation.

The quartile deviation formula is

Q.D. = Q3-Q1/ 2

Example –

Quartiles are values that divide a list of numbers into quarters. Put the numbers in ascending order, then cut the list in four equal parts. The quartiles are the cuts.

For example- 5, 7, 4, 4, 6, 2, and 8.

Arrange them in order – 2, 4, 4, 5, 6, 7, and 8.

Cut the list into quarters.

Quartile 1 (Q1) = 4 or lower quartile

Quartile 2 (Q2) = which is also the Median = 5

Quartile 3 (Q3) = 7 or lower quartile

1. Why do we calculate the quartile deviation?

The quartile deviation helps to examine the spread of a distribution about a measure of its central tendency, usually the mean or the average. Hence, it is in use to give you an idea about the range within which the central 50% of your sample data lies. Quartiles are in use for reporting on a set of data and for making box and whisker plots. Quartiles are of particular use when the data does not have a symmetrical distribution. HR teams use it to determine what salary range to provide to an employee / new joinee based on his experience and qualifications.

2. What is the role of quartiles in statistics?

Quartiles divide the given data into four groups (after the data is sorted). Each group contains an equal number of values. Quartiles are divided by the 25th, 50th, and the 75th percentile, also called as first, second, and the third quartile. Quartile splits the collected data into quarters so that 25 percent of the estimations are less than the lower quartiles, 50 percent of the estimations are less than the mean, and 75 percent of the estimations are less than the upper quartile. In order to understand it better, let us arrange it in the following way:

First quartile= 25% from smallest to largest numbers

Second quartile = between 25.1% and 50 % i.e. the median

Third quartile = 51 % to 75% i.e. above the median

Fourth quartile = 25% of the largest numbers