Interquartile Range Formula, Definition, Solved Examples

Interquartile Range Formula: The formula for the Interquartile Range (IQR) is used to determine the spread between the third quartile (Q ₃ ) and the first quartile (Q ₁ ). This calculation evaluates the variability by dividing a dataset into quartiles. Quartiles, identified as Q ₁ , Q ₂ , and Q ₃ , partition the data into four equal parts.

Q ₁ denotes the point that separates the first half of the ordered dataset, while Q ₂ represents the median value. Q ₃ , on the other hand, divides the second half of the ordered data set.

Let's discuss the IQR formula and explore its application through a few solved examples.

What Is the Formula for Interquartile Range

The Interquartile Range (IQR) formula serves as a measure of the middle 50% of a dataset. It represents the smallest dispersion measure in statistics. This range quantifies the difference between the upper and lower quartiles.

Interquartile Range Formula: Interquartile range = Upper Quartile – Lower Quartile

Interquartile Range Formula IQR=Q ₃ ​ −Q ₁ ​

Where: IQR = Interquartile range (also represented as Q ₂ ​ )

Q ₁ = (1/4)[(n + 1)] ^{th term} )

Q ₃ = (3/4)[(n + 1)] ^{th term} )

n = number of data points

Finding the Interquartile Range (IQR)

Here are the steps to calculate the IQR:

• Arrange the data points in ascending order. Q ₂ ​ represents the median of the dataset.
• If the number of data points is odd, the middle term is (n+1)/2.
• If the number of data points is even, the median is the mean of the two middle points.
• Q ₁ ​ is the median of the data points to the left of the median found in step 2.
• Q ₃ ​ is the median of the data points to the right of the median found in step 2.
• IQR can be calculated using the formula: IQR=Q ₂ ​ =Q ₃ ​ −Q _{1 ​}

Interquartile Range Formula Solved Examples

Example 1: Compute the Interquartile Range for the following dataset: {22, 25, 28, 30, 32, 35, 40, 42, 45}

Given: Dataset = {22, 25, 28, 30, 32, 35, 40, 42, 45}

Step 1: Arrange the dataset in ascending order: {22, 25, 28, 30, 32, 35, 40, 42, 45}

Step 2: Determine the quartiles. n=9 (number of terms)

Q 2 ​ (median): Since the number of terms is odd, the median will be the middle term, which is the ( 9 + 1 ) / 2 = 5 (9+1)/2=5th term:

Q 2 ​ =32

Next, to determine Q1 and Q3, the dataset is divided into two segments:

Lower half: {22, 25, 28, 30, 32}

Upper half: {35, 40, 42, 45}

Q 1 ​ (first quartile) is the median of the lower half:

Q 1 ​ =median of {22,25,28,30,32}=28

Q 3 ​ (third quartile) is the median of the upper half:

Q 3 ​ =median of {35,40,42,45}=38.5

Step 3: Calculate the Interquartile Range using the formula:

Interquartile Range Formula IQR=Q 3 ​ −Q 1 ​ =38.5−28=10.5

Thus, the Interquartile Range (IQR) for the given dataset is 10.5.

Example 2: Determine the Interquartile Range for the following set of numbers: {12, 15, 18, 21, 24, 28, 31, 35, 39, 42, 45}

Given: Dataset = {12, 15, 18, 21, 24, 28, 31, 35, 39, 42, 45}

Step 1: Arrange the data in ascending order: {12, 15, 18, 21, 24, 28, 31, 35, 39, 42, 45}

Step 2: Determine the quartiles. n=11 (number of terms)

Q ₂ ​ (median): Since the number of terms is odd, the median will be the middle term, which is the ( 11 + 1 ) / 2 = 6 (11+1)/2=6th term:

Q ₂ ​ =28 Split the dataset into two parts:

Lower half: {12, 15, 18, 21, 24}

Upper half: {31, 35, 39, 42, 45}

Q ₁ ​ (first quartile) is the median of the lower half:

Q ₁ ​ =median of {12,15,18,21,24}=18

Q ₃ ​ (third quartile) is the median of the upper half:

Q ₃ ​ =median of {31,35,39,42,45}=37

Step 3: Calculate the Interquartile Range using the formula:

Interquartile Range Formula IQR=Q ₃ ​ −Q ₁ ​ =37−18=19

T hus, the Interquartile Range (IQR) for the given dataset is 19.

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