Interquartile Range Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Interquartile Range.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: IQR is resistant to outliers—extreme values don't affect it.

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

Sense of Study hint: Find Q1 and Q3 first. Then just subtract: IQR = Q3 - Q1. This tells you the spread of the middle 50% of data.

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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A box plot shows Q_1 = 30 and Q_3 = 50. Calculate the IQR and the lower and upper fences for outlier detection.

Example 2

hard
A data set has Q_1 = 40, median = 55, Q_3 = 70. A new value of 120 is added. Without recalculating quartiles, explain why the IQR may remain unchanged and identify whether 120 is an outlier.
← Back to Interquartile Range Formula Explained Practice Mode

Related Concepts

QuartilesBox Plot

Background Knowledge

These ideas may be useful before you work through the harder examples.

quartiles