How to Calculate Interquartile Range

How to Calculate Interquartile Range, The interquartile range (IQR) is a measure of statistical dispersion, which indicates the spread of the middle 50% of a data set.

It is a useful way to understand the variability within a dataset and is less affected by outliers compared to other measures like the range.

What is the Interquartile Range?

The IQR is defined as the difference between the third quartile (Q3) and the first quartile (Q1):

IQR = Q3 – Q1

• Q1 (First Quartile): The median of the lower half of the data (25th percentile).
• Q3 (Third Quartile): The median of the upper half of the data (75th percentile).

Why Use the IQR?

• It provides a measure of the spread of the middle 50% of the data.
• It helps identify outliers (observations that fall significantly outside the range).
• It is robust against skewed data and outliers.

Step-by-Step Guide to Calculating the IQR

Step 1: Arrange the Data in Ascending Order

Sort your dataset from the smallest to the largest value.

Example:

Data set: 7, 15, 36, 39, 42, 43, 46, 49, 52, 56

Sorted data: 7, 15, 36, 39, 42, 43, 46, 49, 52, 56

Step 2: Find the Median (Q2)

• If the number of data points (n) is odd, the median is the middle value.
• If n is even, the median is the average of the two middle values.

Example:

n = 10 (even)

Median = (42 + 43) / 2 = 42.5

Step 3: Divide the Data into Lower and Upper Halves

• Lower half: data points before the median.
• Upper half: data points after the median.

Example:

Lower half: 7, 15, 36, 39, 42

Upper half: 43, 46, 49, 52, 56

Step 4: Find Q1 and Q3

• Q1 (First Quartile): Median of the lower half. For the lower half: 7, 15, 36, 39, 42 Number of points = 5 (odd) Median of lower half: 36
• Q3 (Third Quartile): Median of the upper half. For the upper half: 43, 46, 49, 52, 56 Median of upper half: 49

Step 5: Calculate the IQR

IQR = Q3 – Q1 = 49 – 36 = 13

Result: The interquartile range of the dataset is 13.

Special Cases and Considerations

• Data with an even number of points: When dividing the data into halves, if the number of data points is even, the median is not included in either half, and the halves are split evenly.
• Calculating quartiles using different methods: Some statistical software or textbooks use different conventions (e.g., including the median in both halves or using interpolation). Be consistent with your method.
• Handling outliers: The IQR is often used to detect outliers. Typically, any data point below ( Q1 – 1.5 * IQR ) or above ( Q3 + 1.5 * IQR ) is considered an outlier.

Summary

Calculating the interquartile range involves:

1. Sorting the data.
2. Finding the median (Q2).
3. Dividing the data into lower and upper halves.
4. Computing Q1 and Q3 as the medians of these halves.
5. Subtracting Q1 from Q3 to get the IQR.

Understanding and calculating the IQR helps in summarizing the spread of data, identifying outliers, and supporting data analysis in various fields such as finance, research, and quality control.

Interquartile Range (IQR) on Calculator