Interquartile Range (IQR) Formula

The Formula

\text{IQR} = Q_3 - Q_1

When to use: IQR focuses on where most of the data lives, ignoring the extremes. If regular range is how far the outliers stretched, IQR is how wide the main crowd is. More resistant to outliers than range.

Quick Example

Q_1 = 70, Q_3 = 85. IQR = 85 - 70 = 15 The middle 50% of scores spans 15 points.

Notation

IQR stands for Interquartile Range. Q_1 is the 25th percentile and Q_3 is the 75th percentile. The 1.5 \times IQR rule defines the outlier fences.

What This Formula Means

The interquartile range (IQR) is the range of the middle 50% of data, calculated as Q_3 - Q_1. It measures spread while ignoring the top and bottom 25% of values, making it resistant to outliers.

IQR focuses on where most of the data lives, ignoring the extremes. If regular range is how far the outliers stretched, IQR is how wide the main crowd is. More resistant to outliers than range.

Formal View

The interquartile range is IQR = Q_3 - Q_1. Outliers are defined as observations below Q_1 - 1.5 \cdot IQR or above Q_3 + 1.5 \cdot IQR.

Worked Examples

Example 1

easy
Given the data set: 4, 7, 9, 12, 15, 18, 22, 25, 30, find the interquartile range (IQR).

Solution

  1. 1
    Step 1: Sort the data (already sorted). Find Q_2: 9 values, median is the 5th value = 15.
  2. 2
    Step 2: Lower half {4, 7, 9, 12}: Q_1 = \frac{7+9}{2} = 8. Upper half {18, 22, 25, 30}: Q_3 = \frac{22+25}{2} = 23.5.
  3. 3
    Step 3: IQR = Q_3 - Q_1 = 23.5 - 8 = 15.5.

Answer

IQR = 15.5.
The interquartile range (IQR) measures the spread of the middle 50% of the data. It is calculated as Q_3 - Q_1. Unlike the range, the IQR is not affected by extreme values (outliers), making it a robust measure of spread.

Example 2

medium
Test scores: 55, 60, 65, 70, 72, 75, 78, 80, 85, 90, 95. Find the IQR and use it to determine the boundaries for outliers using the 1.5 \times IQR rule.

Common Mistakes

  • Confusing with range
  • Calculating from min/max instead of Q_1/Q_3
  • Forgetting it represents 50% of data

Why This Formula Matters

IQR is a robust measure of spread. It's used to identify outliers and compare group consistency.

Frequently Asked Questions

What is the Interquartile Range (IQR) formula?

The interquartile range (IQR) is the range of the middle 50% of data, calculated as Q_3 - Q_1. It measures spread while ignoring the top and bottom 25% of values, making it resistant to outliers.

How do you use the Interquartile Range (IQR) formula?

IQR focuses on where most of the data lives, ignoring the extremes. If regular range is how far the outliers stretched, IQR is how wide the main crowd is. More resistant to outliers than range.

What do the symbols mean in the Interquartile Range (IQR) formula?

IQR stands for Interquartile Range. Q_1 is the 25th percentile and Q_3 is the 75th percentile. The 1.5 \times IQR rule defines the outlier fences.

Why is the Interquartile Range (IQR) formula important in Statistics?

IQR is a robust measure of spread. It's used to identify outliers and compare group consistency.

What do students get wrong about Interquartile Range (IQR)?

Students sometimes compute IQR as max minus min (that is the range). IQR specifically uses Q3 − Q1, not the overall extremes.

What should I learn before the Interquartile Range (IQR) formula?

Before studying the Interquartile Range (IQR) formula, you should understand: stat quartiles, stat range.

← Back to Interquartile Range (IQR) See All Examples Practice Problems

Related Formulas

Range Formula