Interquartile Range Formula

The Formula

\text{IQR} = Q3 - Q1

When to use: The IQR ignores the extreme 25% on each end, capturing only the spread of the central bulk of data — making it robust when outliers inflate the regular range.

Quick Example

Data: \{2, 5, 7, 9, 12, 15, 20\}. Q1 = 5, Q3 = 15, \text{IQR} = 15 - 5 = 10.

Notation

\text{IQR} = Q_3 - Q_1; the middle 50\% of the data

What This Formula Means

The interquartile range (IQR) is Q3 - Q1 — the spread of the middle 50% of the data, resistant to outliers.

The IQR ignores the extreme 25% on each end, capturing only the spread of the central bulk of data — making it robust when outliers inflate the regular range.

Formal View

\text{IQR} = Q_3 - Q_1 where Q_1 = Q_{0.25} and Q_3 = Q_{0.75} are the first and third quartiles

Worked Examples

Example 1

easy
Calculate the IQR for: \{15, 22, 28, 35, 42, 50, 58, 65\} and explain what it measures.

Solution

  1. 1
    Find Q_1: lower half is \{15, 22, 28, 35\}; Q_1 = \frac{22+28}{2} = 25
  2. 2
    Find Q_3: upper half is \{42, 50, 58, 65\}; Q_3 = \frac{50+58}{2} = 54
  3. 3
    Calculate IQR: IQR = Q_3 - Q_1 = 54 - 25 = 29
  4. 4
    Interpretation: the middle 50% of values span a range of 29 units

Answer

IQR = 29
The IQR measures the spread of the middle 50% of data. It is resistant to outliers (unlike the range) because it ignores the top and bottom 25% of values. A larger IQR indicates more variability in the central portion of the data.

Example 2

medium
Data set A: \{1, 2, 3, 4, 5, 6, 7, 8, 9, 100\} and Data set B: \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}. Compare the range and IQR for both sets and explain why IQR is preferred as a measure of spread.

Common Mistakes

  • Computing IQR as Q1 - Q3 instead of Q3 - Q1 — IQR must be positive
  • Confusing IQR with range — range uses min and max, IQR uses Q1 and Q3
  • Using IQR alone to describe the data without reporting the median — IQR measures spread but not center

Why This Formula Matters

IQR is the standard spread measure for skewed or outlier-heavy data, and it defines the "fences" beyond which data points are classified as outliers in a box plot.

Frequently Asked Questions

What is the Interquartile Range formula?

The interquartile range (IQR) is Q3 - Q1 — the spread of the middle 50% of the data, resistant to outliers.

How do you use the Interquartile Range formula?

The IQR ignores the extreme 25% on each end, capturing only the spread of the central bulk of data — making it robust when outliers inflate the regular range.

What do the symbols mean in the Interquartile Range formula?

\text{IQR} = Q_3 - Q_1; the middle 50\% of the data

Why is the Interquartile Range formula important in Math?

IQR is the standard spread measure for skewed or outlier-heavy data, and it defines the "fences" beyond which data points are classified as outliers in a box plot.

What do students get wrong about Interquartile Range?

Outliers are typically defined as values more than 1.5 \times \text{IQR} from Q1 or Q3.

What should I learn before the Interquartile Range formula?

Before studying the Interquartile Range formula, you should understand: quartiles.

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Related Concepts

QuartilesBox Plot