Box Plot Skewness Guide
Box Plots & Skewness Guide
Exercises 21 & 22 • Step-by-Step Five-Number Summary & Distribution Shape
Statistics Reference
The Core Framework
How to compute summaries and determine skewness
1. Five-Number Summary
- • Order data from least to greatest.
- • Minimum & Maximum: Lowest & highest values.
- • Median (\(Q_2\)): Exact middle value.
- • Quartiles (\(Q_1, Q_3\)): Medians of lower & upper halves.
2. Diagnosing Skewness
- • Right-Skewed: Longer right whisker; \(\text{Max} - Q_2 > Q_2 - \text{Min}\).
- • Left-Skewed: Longer left whisker; \(Q_2 - \text{Min} > \text{Max} - Q_2\).
- • Symmetric: Whiskers and box halves approximately equal.
Exercise 21
Analysis & Complete Solution
Result: Right-Skewed (Positively Skewed)
Raw Data: 39, 36, 30, 27, 26, 24, 28, 35, 39, 60, 50, 41, 35, 32, 51 (\(n = 15\))
Sorted Data:
24, 26, 27, 28, 30, 32, 35, 35, 36, 39, 39, 41, 50, 51, 60
(a) Five-Number Summary
Minimum
24
Q1 (25th %)
28
Median (\(Q_2\))
35
Q3 (75th %)
41
Maximum
60
(b) Box-and-Whisker Plot
20 25 30 35 40 45 50 55 60 65 24 28 35 41 60
← Short Left Whisker (4 units) Long Right Whisker (19 units) →
Why is this distribution Right-Skewed?
1. Whisker Length: The right whisker (\(\text{Max} - Q_3 = 60 - 41 = 19\)) is almost 5 times longer than the left whisker (\(Q_1 - \text{Min} = 28 - 24 = 4\)).
2. Span from Median: The distance from Median to Maximum (\(60 - 35 = 25\)) greatly exceeds the distance from Minimum to Median (\(35 - 24 = 11\)). The upper 50% of values are spread far toward high values.
Statistics Mastery Series • Five-Number Summary & Box Plots Page 1 of 2
Exercise 22
Analysis & Complete Solution
Result: Left-Skewed (Negatively Skewed)
Raw Data: 171, 176, 182, 150, 178, 180, 173, 170, 174, 178, 181, 180 (\(n = 12\))
Sorted Data:
150, 170, 171, 173, 174, 176 | 178, 178, 180, 180, 181, 182
Median (\(Q_2\)) Mean of 6th & 7th values:
\(\frac{176 + 178}{2} = \mathbf{177}\)
First Quartile (\(Q_1\)) Median of lower 6 values:
\(\frac{171 + 173}{2} = \mathbf{172}\)
Third Quartile (\(Q_3\)) Median of upper 6 values:
\(\frac{180 + 180}{2} = \mathbf{180}\)
(a) Five-Number Summary
Minimum
150
Q1 (25th %)
172
Median (\(Q_2\))
177
Q3 (75th %)
180
Maximum
182
(b) Box-and-Whisker Plot
145 150 155 160 165 170 175 180 185 150 172 177 180 182
← Extremely Long Left Whisker (22 units) Short Right Whisker (2 units) →
Why is this distribution Left-Skewed?
1. Whisker Disparity: The left whisker (\(Q_1 - \text{Min} = 172 - 150 = 22\)) is 11 times longer than the right whisker (\(\text{Max} - Q_3 = 182 - 180 = 2\)).
2. Span from Median: Distance from Minimum to Median (\(177 - 150 = 27\)) is over 5 times greater than Median to Maximum (\(182 - 177 = 5\)). The low value (\(150\)) drags a long tail out to the left.
Direct Comparison & Quick Diagnostic Rules
Side-by-Side Review
Exercise 21 (Right-Skewed) → Tail Right
- • Left Whisker: \(28 - 24 = 4\)
- • Right Whisker: \(60 - 41 = 19\) (Dominates)
- • Box halves: \(35 - 28 = 7\) vs \(41 - 35 = 6\)
- • Key Driver: High values (\(50, 51, 60\)) stretch the upper tail.
Exercise 22 (Left-Skewed) ← Tail Left
- • Left Whisker: \(172 - 150 = 22\) (Dominates)
- • Right Whisker: \(182 - 180 = 2\)
- • Box halves: \(177 - 172 = 5\) vs \(180 - 177 = 3\)
- • Key Driver: Single low value (\(150\)) stretches the lower tail.
Golden Memory Rule: Skewness follows the longer tail / whisker. Tail to the right = Right-Skewed; tail to the left = Left-Skewed.
Statistics Mastery Series • Five-Number Summary & Box Plots Page 2 of 2
