Interquartile Range (IQR) Formula

Interquartile range (iqr) is the interquartile range (IQR) is the range of the middle 50% of data, calculated as Q_3 - Q_1.

The Formula

IQR=Q3−Q1\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

Q1=70Q_1 = 70, Q3=85Q_3 = 85. IQR=85−70=15IQR = 85 - 70 = 15 The middle 50% of scores spans 15 points.

Notation

IQR stands for Interquartile Range. Q1Q_1 is the 25th percentile and Q3Q_3 is the 75th percentile. The 1.5×IQR1.5 \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 Q3−Q1Q_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=Q3−Q1IQR = Q_3 - Q_1. Outliers are defined as observations below Q1−1.5⋅IQRQ_1 - 1.5 \cdot IQR or above Q3+1.5⋅IQRQ_3 + 1.5 \cdot IQR.

Worked Examples

Example 1

medium
Find the IQR of 5,8,11,14,17,20,23,265, 8, 11, 14, 17, 20, 23, 26.

Answer

1212

First step

1
Eight values; median =(14+17)/2=15.5=(14+17)/2=15.5.

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Example 2

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A box plot shows Q1=22Q_1=22, Q3=34Q_3=34. The largest value is 8080. Does 8080 qualify as a high outlier by the 1.5⋅IQR1.5\cdot\text{IQR} rule?

Example 3

medium
For 10,12,14,14,15,18,20,2210, 12, 14, 14, 15, 18, 20, 22, find the IQR.

Common Mistakes

  • Confusing with range - The safer move is to ask "Do I need to describe how far the data values extend or vary, rather than where the middle is?" and then state the data source, denominator, or variable before interpreting the result.
  • Calculating from min/max instead of Q1Q_1/Q3Q_3 - The safer move is to ask "Do I need to describe how far the data values extend or vary, rather than where the middle is?" and then state the data source, denominator, or variable before interpreting the result.
  • Forgetting it represents 50% of data - The safer move is to ask "Do I need to describe how far the data values extend or vary, rather than where the middle is?" and then state the data source, denominator, or variable before interpreting the result.
  • Choosing interquartile range (iqr) from a keyword alone - Keywords like spread, variation, consistent are only clues; the data structure must match the concept.

Why This Formula Matters

Interquartile Range (IQR) prevents students from treating equal centers as equal data sets. The spread tells how predictable the values are, whether a summary is stable, and whether a comparison hides important variation.

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 Q3−Q1Q_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. Q1Q_1 is the 25th percentile and Q3Q_3 is the 75th percentile. The 1.5×IQR1.5 \times IQR rule defines the outlier fences.

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

Interquartile Range (IQR) prevents students from treating equal centers as equal data sets. The spread tells how predictable the values are, whether a summary is stable, and whether a comparison hides important variation.

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

Students often know a procedure related to interquartile range (iqr) but skip the recognition step: Do I need to describe how far the data values extend or vary, rather than where the middle is? That leads to a calculation or graph that looks reasonable but answers a different question.

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

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

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Range Formula