# Interquartile Range(IQR) Formula

The Interquartile range formula helps in finding the difference between the third quartile and the first quartile. The Interquartile range formula measures the variability, based on dividing an ordered set of data into quartiles. Quartiles are three values or cuts that divide each respective part as the first, second, and third quartiles, denoted by \(Q_{1}\), \(Q_{2}\), and \(Q_{3}\), respectively.

- \(Q_{1}\) is the cut in the first half of the rank-ordered data set
- \(Q_{2}\) is the median value of the set
- \(Q_{3}\) is the cut in the second half of the rank-ordered data set.

Let's learn about the range formula with a few solved examples.

## What is Interquartile Range Formula?

The Interquartile Range (IQR) formula is a measure of the middle 50% of a data set. The smallest of all the measures of dispersion in statistics is called the Interquartile Range. The difference between the upper and lower quartile is known as the interquartile range.

### IQR Formula

Interquartile range = Upper Quartile – Lower Quartile

**\(Q_{2} = Q_{3} – Q_{1}\)**

where,

IQR = Interquartile range (IQR = \(Q_{2}\))

\(Q_{1}\) = (1/4)[(n + 1)]^{th} term)

\(Q_{3}\) = (3/4)[(n + 1)]^{th} term)

n = number of data points

The following steps help us to find the IQR:

- The simple trick is to arrange the data points in ascending order.
- \(Q_{2}\) is the median of the data. If the number of data points is odd, the middle term is (n+1)/2 and 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 = \(Q_{2} = Q_{3} – Q_{1}\)

## Solved Examples Using Interquartile Range Formula

**Example 1: Using the interquartile range formula, calculate the range of the following set of data: {4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11}**

**Solution:** 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}

Dividing the set into quartiles, each quarter will have 3 terms as: {3, 4, 4}, {4, 7, 10}, {11, 12, 14}, {16, 17, 18}

First Quartile,

\(Q_{1}\) = (4 + 4)/ 2 = 4

Third Quartile,

\(Q_{3}\) = (14 + 16)/2 = 15

Using Interquartile Range Formula,\(Q_{2} = Q_{3} – Q_{1}\)

= 15 - 4

= 11

Therefore, the Interquartile range of the given set = 11

**Example 2: **Determine the interquartile range value for the first ten odd numbers.

**Solution: **

To find IQR of the first 10 odd numbers:

The first ten odd numbers:1, 3, 5, 7, 9, 11, 13, 15, 17, 18

n = 10

Since 10 is even, using the median formula, we find the median as the mean of the 5^{th} and 6^{th} terms.

That is \(Q_{2}\) = (9+11)/2 ⇒ \(Q_{2}\) = 10.

Now \(Q_{1}\) part is {1, 3, 5, 7, 9}

Here the number of data points = 5

\(Q_{1}\) = median of {1,3,5,7,9} = 5

\(Q_{3}\) part is {11, 13, 15, 17, 19}

Here the number of data points = 5

\(Q_{3}\) = median of {11,13,15,18,19} = 15

Using Interquartile range formula, IQR = \( Q_{3} – Q_{1}\)

\(Q_{3} – Q_{1}\) is 15 – 5 = 10

**Answer:** 10 is the interquartile range for the given set of first 10 odd numbers.

**Example 3: Europe has an estimated population of 1,420,062,022 people. Brazil's population is 132,328,035. India has a population of 1,368,737,513. The United States has a population of 329,093,110. What is the range of this set?**

**Solution:** Let's arrange the values in a set: {132,328,035; 329,093,110; 1,368,737,513; 1,420,062,022}

Using the interquartile range formula, IQR = \( Q_{3} – Q_{1}\)

\(Q_{3}\) is the highest number i.e. 1,420,062,022 and \(Q_{1}\) is the lowest number i.e. 132,328,038

IQR = 1,420,062,022 - 132,328,038

IQR = 1,287,733,984

Therefore, the interquartile range of the population of the 4 countries is 1,287,733,984

## FAQs on Interquartile Range Formula

### What is Interquartile Range Formula?

The difference between the upper and lower quartile is known as the interquartile range formula. The Interquartile Range formula finds the difference between the two extreme observations or data points of the distribution.

### What is the Formula to Calculate the IQR in a given Set?

The interquartile range formula helps in finding the difference between the third quartile and the first quartile. IQR **=**\(Q_{3} – Q_{1}\

where,

IQR = Interquartile range

\(Q_{1}\) = First Quartile

\(Q_{3}\)= Third Quartile

### What is the Formula to Find the First Quartile in a given Set?

Once the given set is arranged in ascending order, to find the interquartile range, we need to determine the lower quartile or the first quartile using this formula:

\(Q_{1}\) = (1/4)[(n + 1)]^{th} term, where n is the number of data points

### What is the Formula to Find the Third Quartile in a given Set?

Once the given set is arranged in ascending order, to find the interquartile range, we need to determine the upper quartile or the third quartile using this formula:

\(Q_{3}\) = (3/4)[(n + 1)]^{th} term, where n is the number of data points

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