The Interquartile Range (IQR) is a measure of statistical dispersion that describes the spread of the middle 50% of data. It is calculated as:
\(\text{IQR} = Q_3 - Q_1\)
where \(Q_1\) is the 25th percentile and \(Q_3\) is the 75th percentile.
Usefulness of IQR
- Robust to outliers – Unlike the range or standard deviation, the IQR is not influenced by extreme values.
- Identifies variability of the central portion of the data.
- Used to detect outliers – Often, values below \(Q_1 - 1.5 \times \text{IQR}\) or above \(Q_3 + 1.5 \times \text{IQR}\) are considered outliers.
- Helps in comparing distributions – Especially when using box plots.
Example: Exam Scores
Suppose the exam scores (out of 100) of 15 students are:
\(55, 60, 65, 70, 72, 75, 78, 80, 82, 85, 88, 90, 95, 98, 99\)
Step 1: Find \(Q_1\) and \(Q_3\) - \(Q_1\) = median of the first half (excluding overall median if odd) = 70 - \(Q_3\) = median of the second half = 90
Step 2: Calculate IQR
\(\text{IQR} = 90 - 70 = 20\)
Step 3: Interpret - The middle 50% of students scored between 70 and 90. - The spread of these central scores is 20 points.
Step 4: Detect outliers Lower fence: \(70 - 1.5 \times 20 = 40\) Upper fence: \(90 + 1.5 \times 20 = 120\) No scores below 40 or above 120 → No outliers.
Why IQR is useful here
- If one student scored 20 instead of 55, the range would change drastically, but the IQR would remain the same.
- IQR gives a clearer picture of the typical spread unaffected by extremes.