Interquartile Range Formula: Definition, Formula and Solved Examples

The interquartile range is the difference between the third and first quartiles. Quartiles are partitioned values that divide an entire series into four equal parts. So there are three quartiles.

• The first quartile is marked by Q₁, also known as the lower quartile.
• The second quartile is denoted by Q₂, and the third quartile is represented by Q₃, known as the upper quartile.
• Hence, the interquartile range equals the difference between the upper and lower quartiles.
• The interquartile range (IQR) is used to identify and eliminate deviation from a series of data at both ends.
• It is a measure of variability that is calculated by categorizing a data set into quartiles like first, second, third, and fourth.

Key Terms: Interquartile range, Data collection, Range, Statistics, Data set, Median, Data points. Quartile, First Quartile, Mean, Third Quartile, Ordered set

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What is Interquartile Range Formula?

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The Interquartile Range (IQR) formula calculates the middle 50% of a data collection. The Interquartile Range is the smallest of all statistical measures of dispersion.

• The interquartile range is the difference between the upper and lower quartiles.
• This can be used to calculate the difference between the third and first quartiles.
• It calculates variability by splitting an ordered set of data into quartiles.
• The First Quartile or Q₁ gives the middle value in the first half of the data set.
• The second Quartile or Q₂ gives the median of the data set.
• The third Quartile or Q₃ gives the middle value in the second half of the data set.

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Formula of Interquartile Range

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The Interquartile Range (IQR) formula can be represented as,

\(IQR = Q_3 - Q_1\)

Where,

• IQR is the Inter-quartile range
• Q₁ is the First quartile
• Q₃ is the Third quartile

The First Quartile, Q₁ can also be found by using the following formula

\(\large Q_{1}=\left(\frac{n+1}{4}\right)^{th}term\)

The Third Quartile, Q₃ can also be found by using the following formula

\(\large Q_{3}=\left(\frac{3(n+1)}{4}\right)^{th}term\)

In the above cases, if the values are not whole numbers, we must round them to the nearest integer.

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How to Find Interquartile Range?

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Interquartile Range (IQR) of a given set of data can be determined using some simple steps which are mentioned below:

• Step 1: Arrange the given values in ascending order. 
• Step 2: The Second Quartile or Q₂ is the median of the given values. The middle term is (n+1)/2 if the number of values is odd, while the median is the mean of the two middle points if the number of values is even.
• Step 3: First Quartile or Q₁ is the median of the values present to the left of the median calculated in Step 2. 
• Step 4: The third Quartile or Q₃ is the median of the values present to the right of the median calculated in Step 2. 
• Step 5: Now we have all the required values, we can calculate IQR = Q₃ - Q₁.

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Solved Examples

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Ques. Calculate the range of the following set of data: {4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11}, using the interquartile range formula

Ans. Given

• Number of terms = 12
• Set = {4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11} 
• Ordered set = {3, 4, 4, 4, 7, 10, 11, 12, 14, 16, 17, 18}  

Now divide the set into quartiles and each quarter will have 3 terms:

{3, 4, 4}, {4, 7, 10}, {11, 12, 14}, and  {16, 17, 18}

The First Quartile is given by,

Q₁ = (4 + 4)/ 2 = 4

The Third Quartile is given by

Q₃ = (14 + 16)/2 = 15

Using the Interquartile Range Formula, we get

Q₂ = Q₃ – Q₁

⇒ Q₂ = 15 – 4 = 11

Therefore, the Interquartile range of the given set = 11

Ques. Find the interquartile range for the first ten odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.

Ans.  Given, the total number of terms n = 10.

The formula of the median is given by

Median = \(\frac{(\frac{n}{2})^{th}term + (\frac{n}{2}+1)^{th}term}{2}\)

⇒ Median = \(\frac{9 + 11}{2} = 10\)

Therefore, the set of data is divided into two parts: 1, 3, 5, 7, 9 and 11, 13, 15, 17, 19

• Q₁ is the Median of the first part = 5
• Q₃ is the Median of the second part = 15

The formula for the interquartile range is given by

IQR = Q₃ – Q₁

On substituting the values, we get
IQR = 15 – 5 = 10

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Advantages of Interquartile Range

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The advantages of IQR are mentioned below:

• IQR is a more powerful data analysis tool than the mean or median of a data collection.
• The interquartile range is useful for determining and eliminating variance at both ends of a data collection.
• In cases with skewed data distribution, an interquartile range is also an excellent measure of variation.

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Disadvantages of Interquartile Range

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The disadvantages of IQR are mentioned below:

• IQR is less susceptible to a variety of severe scores.
• IQR has a sample consistency that is less potent than the standard deviation.

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Things to Remember

• The Quartiles split a set of data series into four equal sections namely First Quartile, Second Quartile, Third Quartile, Forth Quartile.
• The First Quartile (Q₁) is known as the lower quartile.
• The second Quartile (Q₂) is referred to as Median.
• The third Quartile (Q₃) is called the upper quartile.
• The Interquartile Range (IQR) formula calculates the middle 50% of a data collection.
• The Interquartile Range is the smallest of all statistical measures of dispersion.
• The interquartile range is the difference between the upper and lower quartiles.
• The Interquartile Range (IQR) = Upper Quartile - Lower Quartile 
• The First Quartile, Q₁ = [(n+1)/4] ^th Term
• The Third Quartile, Q₃ = [(n+1)3/4] ^th Term

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