Interquartile Range (IQR)

What is the IQR?

The interquartile range = $Q3 - Q1$. It measures the spread of the middle 50% of the data, ignoring the extreme 25% on each end. It's robust to outliers (unlike range or standard deviation).

Step-by-step: {1, 3, 5, 7, 9, 11, 13, 15, 17}

$n = 9$. Median = 9 (5th value).

Q1 = median of {1, 3, 5, 7} = $(3 + 5)/2 = 4$

Q3 = median of {11, 13, 15, 17} = $(13 + 15)/2 = 14$

$IQR = Q3 - Q1 = 14 - 4 = 10$

IQR vs Range vs SD

• Range ($\text{max} - \text{min} = 17 - 1 = 16$): Uses only extremes. Very sensitive to outliers.
• IQR ($= 10$): Uses only the middle 50%. Robust to outliers.
• SD: Uses all data. Moderately sensitive to outliers.

Try it in the visualization

The data is sorted. The middle 50% is highlighted. Q1 and Q3 are marked, and the IQR is shown as the distance between them.