Interquartile Range Formula

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

The Formula

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

Notation

IQR=Q3−Q1\text{IQR} = Q_3 - Q_1; the middle 50%50\% of the data

What This Formula Means

The interquartile range (IQR) is Q3−Q1Q3 - 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

IQR=Q3−Q1\text{IQR} = Q_3 - Q_1 where Q1=Q0.25Q_1 = Q_{0.25} and Q3=Q0.75Q_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}\{15, 22, 28, 35, 42, 50, 58, 65\} and explain what it measures.

Answer

IQR=29IQR = 29

First step

1
Find Q1Q_1: lower half is {15,22,28,35}\{15, 22, 28, 35\}; Q1=22+282=25Q_1 = \frac{22+28}{2} = 25

Full solution

  1. 2
    Find Q3Q_3: upper half is {42,50,58,65}\{42, 50, 58, 65\}; Q3=50+582=54Q_3 = \frac{50+58}{2} = 54
  2. 3
    Calculate IQR: IQR=Q3−Q1=54−25=29IQR = Q_3 - Q_1 = 54 - 25 = 29
  3. 4
    Interpretation: the middle 50% of values span a range of 29 units
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}\{1, 2, 3, 4, 5, 6, 7, 8, 9, 100\} and Data set B: {1,2,3,4,5,6,7,8,9,10}\{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.

Example 3

medium
Find the IQR of test scores: 55,60,62,68,72,75,78,80,85,9055,60,62,68,72,75,78,80,85,90.

Common Mistakes

  • Subtracting min from max instead of Q3−Q1Q_3-Q_1 — the IQR uses the quartiles, never the extreme values.
  • Computing quartiles on unsorted data — order the data from smallest to largest before finding Q1Q_1 and Q3Q_3.
  • Confusing Q3−Q1Q_3-Q_1 with Q2Q_2 — the IQR is a width (a difference), not the median itself.

Why This Formula Matters

The IQR is the spread measure that survives outliers, so it pairs with the median to honestly describe skewed or messy data. It is also the engine behind box plots and the 1.5×IQR1.5\times\text{IQR} outlier rule, so getting Q1Q_1 and Q3Q_3 right unlocks both. Recognizing it by "Am I measuring the width of the middle half of sorted data, not the full extent?" — rather than by familiar numbers — is what lets a student tell it apart from range and standard deviation and quartiles in a mixed problem set.

Frequently Asked Questions

What is the Interquartile Range formula?

The interquartile range (IQR) is Q3−Q1Q3 - 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?

IQR=Q3−Q1\text{IQR} = Q_3 - Q_1; the middle 50%50\% of the data

Why is the Interquartile Range formula important in Math?

The IQR is the spread measure that survives outliers, so it pairs with the median to honestly describe skewed or messy data. It is also the engine behind box plots and the 1.5×IQR1.5\times\text{IQR} outlier rule, so getting Q1Q_1 and Q3Q_3 right unlocks both. Recognizing it by "Am I measuring the width of the middle half of sorted data, not the full extent?" — rather than by familiar numbers — is what lets a student tell it apart from range and standard deviation and quartiles in a mixed problem set.

What do students get wrong about Interquartile Range?

The procedure for interquartile range is the easy part; the trap is subtracting min from max instead of Q3−Q1Q_3-Q_1. Asking "Am I measuring the width of the middle half of sorted data, not the full extent?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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