Interquartile Range

The numbers that split a list of numerical data into three halves are called quartiles. 

The first, second, and third quartiles are indicated by Q1, Q2, and Q3, respectively. Q2 is the median of the supplied data in this case.
Statistics Variation
The range is the difference between the distribution’s max and least values.
Range = Xmax – Xmin is the formula that defines it.
Where Xmax is the greatest observation of the variable values and Xmin is the lowest observation.

Interquartile Range (IQR):
The interquartile range (IQR) is a measurement of the dispersion of your data in the middle half. It refers to the range for the middle half of your sample.
One of numerous measurements of variability is the interquartile range. Read my post Measures of Variability to learn more about the others and how the IQR compares.
Consider splitting your data into quarters to show the interquartile range. 

Quarters are referred to as quartiles by statisticians, and they are labelled Q1, Q2, Q3, and Q4 in order of low to high. 

The smallest quarter of values in your collection is covered by the lowest quartile (Q1). The highest quarter of data is represented by the upper quartile (Q4). 

 To put it another way, the interquartile range encompasses the 50% of data points that fall between Q1 and Q4. In the graph below, the IQR is the red rectangle that contains Q2 and Q3 (not labelled).

Hand Calculation of the Interquartile Range (IQR):
The interquartile range is calculated by subtracting the third quartile value from the first quartile value.
Q3 – Q1 = IQR
The interquartile range is defined as the area between the 75th and 25th percentiles (75 – 25 = 50% of the data).
We need to get the values for Q3 and Q1 using the IQR formula. To do so, just sort your data from lowest to highest value and divide the total into four equal parts.

Use of Excel to Find the Interquartile Range:
As part of their descriptive statistics, all statistical software systems will indicate the interquartile range. Because most readers have access to Excel, I’ll teach you how to discover it using that programme.
Download the Excel file to follow along: IQR. This dataset is identical to the one I used in the last demonstration. The interquartile range calculations for locating outliers and the IQR normalcy test discussed later in this post are also included in this file.
You’ll need to utilise the QUARTILE.EXC function in Excel, 

which takes the following parameters: QUARTILE.EXC (array, quart).

Formula for the interquartile range:
The Interquartile Range (IQR) formula is a measure of a data set’s middle 50%. The Interquartile Range is the lowest of all statistical measures of dispersion. The interquartile range is the difference between the upper and lower quartiles.

IQR Formula: Upper Quartile – Lower Quartile = Interquartile Range
Q2=Q3–Q1, where IQR denotes the interquartile range. (Q2 = IQR)
(1/4) = Q1
term ([(n + 1)]th term)
(3/4) Q3=
[(n + 1)] [(n + 2)] [(n + 3)]
th semester)
n denotes the number of data points.

The steps below will assist us in determining the IQR:
1. Arrange the data points in ascending order as a basic technique.
2.Q2 is the data’s median. The middle term is (n+1)/2 if the number of data points is odd, while the median is the mean of the two middle points if the number of data points is even.
3.Q1 is the median of the data points to the left of step 2’s median.
4. The median of the data points to the right of the median determined in step 2 is.Q3.
3. Interquartile range =Q2=Q3–Q1

Examples:

Example 1: Calculate the range of the following set of data using the interquartile range formula: 3, 16, 10, 4, 4, 11; 4, 17, 7, 14, 18, 12.
Sol:
Given:
There are 12 terms total.(4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11)
3, 4, 4, 4, 7, 10, 11, 12, 14, 16, 17, 18 (ordered set)
When the collection is divided into quartiles,

 Each quarter will have three terms:( 3, 4, 4), (4,7, 10), (11, 12, 14),(16, 17, 18).
First Quartile, Q1= (4 + 4)/ 2 = 4

 Third Quartile, Q3= (14 + 16)/2 = 15
Using the Formula for the Interquartile Range, Q2=Q3–Q1

15 – 4 = 11
As a result, the supplied set’s interquartile range = 11.

Example 2: Calculate the interquartile range value for the first 10 odd integers
sol: To get the IQR of the first ten odd numbers, do the following:
The first ten odd numbers are as follows: 1, 3, 5, 7, 9, 11, 13, 15, 17, 18, 

n = 10
Because 10 is an even number, we use the median formula to calculate the median as the average of the 5th and 6th terms.

Q2 = (9+11)/2 = 10 is the answer.
The first section of Q1 is now 1, 3, 5, 7, and 9.
The amount of data points here is 5 

Q1= median of (1,3,5,7,9) = 5 

Q3part is (11, 13, 15, 17, 19)
The amount of data points in this case = 5

 Q3= median of (11,13,15,18,19) =15
IQR =Q3–Q1

 Q3–Q1 equals 

15–5 = 10 (using the interquartile range calculation).
The interquartile range for the first ten odd numbers in the given set is 10.

Conclusion:

The interquartile range (IQR) is a measurement of the dispersion of your data in the middle half. It refers to the half of your sample’s range.
Use the IQR to determine where the majority of your results lie in terms of variability. Larger values imply that your data’s central component has spread out more.