[Maths Class Notes] on Quartile Deviation Pdf for Exam

Statistics is the branch of mathematics that deals with the compilation and calculations of data. It is the single biggest field of research in Mathematics and Computer science today with Automation being done at a rapid pace and billions of bytes of data available online. One of the more important topics of statistics is known as Quartile deviation. The concept of quartile deviation finds relevance when we have to assess the spread of a distribution. The important thing we have to usually find out using statistical tools is the distribution measures and their spread from its central tendency (known as ‘the mean’). So, quartile deviation provides helpful insight into the segmentation and within which the central 50% of your sample data lies. 

Importance of Quartile Deviation

Statistics is about understanding the art and science of the collection of data, the frequency, and the distribution of the trends. The accepted definition of Quartile deviation is the difference between the first quartile and the third quartile in the frequency distribution table. This difference is known as the interquartile range. The interquartile range is important as this is the spread of the data that is most important and from this point, numerous regressions and deviations can be calculated which are very helpful to assess the characteristics of the data. When the difference is divided by two, it is known as quartile deviation or semi-inter-quartile range.

Here is an example to help you understand better.

If a set of 13 numbers is given to you, then 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 natural to say that the median is not affected by extreme values. This is where the quartile comes in.  These 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(n+1)/4th item

Q2=(n+1)/2(n+1)/2th item 

Q3=3(n+1)/43(n+1)/4th item

Hence, the formula for quartile can be written as 

Qr= [1 + frac {r(n/4)-c(2-1)} {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