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Interquartile Range Formula

The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles. The values that divide each part are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

  • Q1 is the “middle” value in the first half of the rank-ordered data set.
  • Q2 is the median value in the set.
  • Q3 is the “middle” value in the second half of the rank-ordered data set.

 The formula for inter-quartile range is given below

\[\large IQR=Q_{3}-Q_{1}\]

Where,
IQR=Inter-quartile range
Q1 = First quartile
Q3 = Third quartile

Q1 can also be found by using the following formula

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

Q2 can also be found by using the following formula:

\[\large Q_{3}=\left(\frac{n+1}{2}\right)^{th}term\]

Q3 can also be found by using the following formula:

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

In these cases, if the values are not whole number, we have to round them up to the nearest integer.

Solved Examples

Question: Find the inter-quartile range for first ten odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 ?

Solution:

Total number of terms n = 10.

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 = Median of first part = 5
Q3 = Median of second part = 15

Formula for inter-quartile range is given by: IQR = Q3 – Q1
IQR = 15 – 5 = 10

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