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

$I Q R = Q_{3} - Q_{1}$

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

Q₁can also be found by using the following formula

$Q_{1} = {(\frac{n + 1}{4})}^{t h} t e r m$

Q₃ can also be found by using the following formula:

$Q_{3} = {(\frac{3 (n + 1)}{4})}^{t h} t e r m$

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

Q₂ can also be found by using the following formula:

Q₂= Q₃– Q₁

Which is equivalent to median.

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 =

\begin{array}{l} \frac{(\frac{n}{2})^{t h} t e r m + (\frac{n}{2} + 1)^{t h} t e r m}{2} \end{array}

Median =

\begin{array}{l} \frac{9 + 11}{2} \end{array}

= 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 – Q₁IQR = 15 – 5 = 10

Solved Examples

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