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# Interquartile Range in Statistics

Published by at September 2nd, 2021 , Revised On February 8, 2023

This article highlights the definition, examples, and calculation of the interquartile range. But before that, let’s start with an explanation of what quartiles are.

## What are Quartiles?

The values dividing a list of numerical data into three quarters are called quartiles. The lower quartile, often known as the first quartile, is halfway between the dataset’s smallest value and the median. The median is in the second quartile, Q2. The upper or third quartile, abbreviated as Q3, is the midpoint of the distribution, located between the median and the highest number.

The four groups generated by the quartiles can now be shown on a map. The smallest number up to Q1 is in the first category; Q1 to the median is in the second category; the median to Q3 is in the third category; and Q3 to the highest data point in the total collection is in the fourth category.

Each quartile has a quarter of the total number of observations. In general, the data is organized in the following order: smallest to largest:

The lowest 25% of numbers are in the first quartile.

Between 25.1 percent and 50 percent in the second quartile (up to the median)

50.1 percent to 75 percent in the third quartile (above the median)

The top 25% of numbers are in the fourth quartile.

## What is Range in Statistics?

The range is the shortest of all the measures of dispersion in statistics. It is the difference between the distribution’s two extreme outcomes. In other terms, the range is the difference between the distribution’s maximum and least observation. It is characterized by

Xmax – Xmin= range.

Where Xmax is the largest observation of the variable value and Xmin is the smallest observation.

Now, let’s discuss what interquartile range is in statistics.

## What is Interquartile Range?

The interquartile range is defined as the difference (substraction) between the upper and lower quartiles.. The interquartile range formula is shown below.

Upper Quartile – Lower Quartile = Q3 – Q1 Interquartile range

The first quartile of the series in Q1, and the third quartile is Q3. The existence of interquartile range and median for the data set is depicted in the graph below.

35, 37, 42, 42, 43, 45, 46, 48, 52, 70

Lower Half           Upper half

Median 43+45/2= 44

Q1= 42                  Q3=48

Interquartile Range= Q3-Q1= 48-42= 6

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The metrics of dispersion are defined as the semi-interquartile range. Half of the interquartile range is also known as the semi interquartile range. It is calculated as half of the difference between the 75th and 25th percentiles (Q3) (Q1). Semi-interquartile range is defined as one-half of the difference between the first and third quartile.

The formula for Semi Interquartile Range is:

(Q3– Q1) / 2

## How to Find Out the Interquartile Range?

Following is the procedure to calculate interquartile range:

Sort the integers in the given list in ascending or descending order.

Then add up the values you have been given. If the number is odd, the center value is the median; otherwise, take the mean of the two center values. This is referred to as the Q2 value. The median is the average of the middle two values if there are even many values.

The median divides the given numbers into two equal portions. Q1 and Q3 pieces are what they are called. Q1 is the median of data values below the median. Q3 is the median of data values above the median value. Finally, the median values of Q1 and Q3 can be subtracted.

The interquartile range is the outcome of this calculation.

Here is an example for a better understanding:

Question:

Find out the interquartile range value for the first ten odd numbers.

The solution here is:

The first ten odd numbers are:

1, 3, 5, 7, 9, 11, 13, 15, 17, 19

The number of values here is 10

Because 10 is an even number, here the median would be a mean of 9 and 11

Q2= 9+11/2= 20/2

=10

Next up, we will get two parts: the lower half for finding Q1 and the upper half for finding Q3.

Q1: 1, 3, 5, 7, 9

Number of values= 5

Center value= 5

For Q3: 11, 13, 15, 17, and 19

Number of values= 5

Center value= 15

Q1-Q3= 15-5

The interquartile range= 10

Quartiles are the numbers that divide a list of numerical data into three quarters.

The interquartile range is defined as the difference between the upper and lower quartiles.

The range is the shortest of all the measures of dispersion in statistics. It is the difference between the distribution’s two extreme outcomes.

The metrics of dispersion are defined as the semi-interquartile range. Half of the interquartile range is also known as the semi interquartile range. It is calculated as half of the difference between the 75th and 25th percentiles (Q3) (Q1). The semi-interquartile range accounts for half of the difference between the first and third quartiles.

Sort the integers in the given list in ascending or descending order.
Then add up the values you have been given. If the number is odd, the center value is the median; otherwise, take the mean of the two center values. This is referred to as the Q2 value. The median is the average of the middle two values if there are even many values. Ingram is a dissertation specialist. He has a master's degree in data sciences. His research work aims to compare the various types of research methods used among academicians and researchers.