Box-and-Whisker Plots

What is a Box-and-Whisker Plot?

A box-and-whisker plot (or box plot) shows how data is distributed using five key numbers.

Parts:

• Minimum: smallest value
• Q1 (First Quartile): 25% mark
• Median (Q2): middle value (50% mark)
• Q3 (Third Quartile): 75% mark
• Maximum: largest value

Visual:

    |----[=====|=====]----|
   min  Q1   median  Q3  max
    ←whisker→ ←box→ ←whisker→

The Five-Number Summary

Steps to find it:

1. Order data from least to greatest
2. Find minimum and maximum
3. Find median (middle)
4. Find Q1 (median of lower half)
5. Find Q3 (median of upper half)

Example: Find Five-Number Summary

Data: 3, 7, 8, 5, 12, 14, 21, 13, 18

Step 1: Order the data

• 3, 5, 7, 8, 12, 13, 14, 18, 21

Step 2: Find minimum and maximum

• Minimum: 3
• Maximum: 21

Step 3: Find median (Q2)

• 9 values → middle is 5th value
• Median: 12

Step 4: Find Q1 (median of lower half)

• Lower half: 3, 5, 7, 8
• Median of 4 values: (5 + 7) ÷ 2 = 6
• Q1: 6

Step 5: Find Q3 (median of upper half)

• Upper half: 13, 14, 18, 21
• Median: (14 + 18) ÷ 2 = 16
• Q3: 16

Five-Number Summary: 3, 6, 12, 16, 21

Drawing a Box-and-Whisker Plot

Example: Draw the plot for 3, 6, 12, 16, 21

Step 1: Draw a number line that includes the range

Step 2: Mark all five values above the line

Step 3: Draw a box from Q1 to Q3

Step 4: Draw a line inside the box at the median

Step 5: Draw whiskers from box to min and max

    |----[==|==]----|
    3    6 12 16   21

Understanding Quartiles

Quartiles divide data into four equal parts:

• Q1: 25% of data is below this value
• Median (Q2): 50% of data is below this value
• Q3: 75% of data is below this value

Interquartile Range (IQR): Q3 − Q1

• Shows the spread of the middle 50% of data

Example: Find IQR

Five-number summary: 3, 6, 12, 16, 21

IQR = Q3 − Q1 = 16 − 6 = 10

The middle 50% of data spans 10 units.

Reading Box-and-Whisker Plots

Example: Analyze this plot

    |---[====|====]--------|
   20   30  40  50        80

Five-number summary: 20, 30, 40, 50, 80

Observations:

• Range: 80 − 20 = 60
• IQR: 50 − 30 = 20
• Median: 40
• Right whisker is longer: data is more spread out on the high end

Identifying Outliers

Outlier: A value that is much higher or lower than the rest

Outlier rule:

• Below Q1 − 1.5 × IQR
• Above Q3 + 1.5 × IQR

Example: Check for Outliers

Data: 5, 6, 7, 8, 9, 10, 11, 25

Five-number summary: 5, 6.5, 8.5, 10.5, 25

IQR = 10.5 − 6.5 = 4

Lower boundary: 6.5 − 1.5(4) = 6.5 − 6 = 0.5 Upper boundary: 10.5 + 1.5(4) = 10.5 + 6 = 16.5

Check 25: 25 > 16.5 → Yes, 25 is an outlier!

Mark outliers with * and draw whisker to largest non-outlier:

    |---[==|==]---| *
    5  6.5 8.5 10.5  25
                 11

Comparing Data Sets

Box plots make it easy to compare multiple data sets!

Example: Test Scores

Class A: min=65, Q1=72, median=80, Q3=88, max=95 Class B: min=55, Q1=68, median=75, Q3=82, max=98

Class A:  |---[====|====]--|
         65  72  80  88  95

Class B: |----[====|===]----|
        55   68  75 82    98

Observations:

• Class A has higher median (80 vs 75)
• Class A has less variability (smaller IQR)
• Class B has wider range

Advantages of Box Plots

Shows:

• Center (median)
• Spread (range and IQR)
• Skewness (symmetric vs. skewed)
• Outliers

Good for:

• Large data sets
• Comparing multiple groups
• Seeing distribution shape

Symmetric vs. Skewed Data

Symmetric: Median is centered in box, whiskers similar length

    |----[==|==]----|

Right-skewed: Right whisker longer

    |--[==|==]--------|

Left-skewed: Left whisker longer

    |--------[==|==]--|

Real-World Applications

Sports: Compare player statistics

• Salaries, ages, performance metrics

Education: Analyze test scores

• Compare classes, identify struggling students

Weather: Temperature distributions

• Daily highs/lows across seasons

Business: Sales data

• Identify best/worst performers, outliers

Practice