Statistics • Topic 3 of 3

Interpreting Box Plots and Histograms

What is a Box Plot (Box-and-Whisker Plot)?

A box plot shows how data is spread out using 5 key numbers:

Minimum    Q1    Median    Q3    Maximum
   │        │       │       │        │
   ├────────┼───────┼───────┼────────┤
   │        │       │       │        │
   └─Whisker┴─Box───┴───────┴─Whisker─┘

The 5-Number Summary:

Term	Meaning	Example (data: 1,3,5,7,9,11,13)
Minimum	Smallest value	1
Q1	Lower quartile (25th percentile)	3
Median	Middle value (50th percentile)	7
Q3	Upper quartile (75th percentile)	11
Maximum	Largest value	13

What is a Histogram?

A histogram is a bar graph that shows frequency (how many) of data in intervals (bins).

Histogram of Test Scores:

Frequency
   10│        ┌─┐
    9│        │ │
    8│        │ │
    7│     ┌─┐│ │
    6│     │ ││ │
    5│     │ ││ │  ┌─┐
    4│  ┌─┐│ ││ │  │ │
    3│  │ ││ ││ │  │ │
    2│  │ ││ ││ │  │ │
    1│  │ ││ ││ │  │ │
    0└──┴─┴┴─┴┴─┴──┴─┴──
       50-59 60-69 70-79 80-89 90-100
                    Score Range

BOX PLOT - STEP BY STEP:

    Data: 4, 5, 7, 8, 9, 11, 12, 15, 18
    
    Step 1: Order data: 4,5,7,8,9,11,12,15,18
    Step 2: Min=4, Max=18
    Step 3: Median = 9 (5th number)
    Step 4: Q1 (median of lower half 4,5,7,8) = (5+7)/2 = 6
    Step 5: Q3 (median of upper half 11,12,15,18) = (12+15)/2 = 13.5
    
    Box plot:
    
    4    6    9    13.5  18
    │    │    │     │     │
    ├────┼────┼─────┼─────┤
    │    │    │     │     │
    └─Whisker─┴─Box─┴──Whisker─┘


BOX PLOT INTERPRETATION:

    ┌─────────────────────────────────────┐
    │  Left whisker: 25% of data          │
    │  Left box:      25% of data          │
    │  Right box:     25% of data          │
    │  Right whisker: 25% of data          │
    └─────────────────────────────────────┘
    
    Longer whisker = more spread out
    Median in center = symmetric data
    Median left of center = skewed right
    Median right of center = skewed left


HISTOGRAM VS BAR GRAPH:

    Bar Graph:                    Histogram:
    (categories, gaps)            (continuous data, no gaps)
    
    Favorite Color                Test Scores
    5│┌─┐                         5│┌─┐
    4││ │┌─┐                      4││ │┌─┐
    3││ ││ │┌─┐                   3││ ││ │┌─┐
    2││ ││ ││ │                    2││ ││ ││ │
    1││ ││ ││ │                    1││ ││ ││ │
     └─┴─┴─┴─┴─                      └─┴─┴─┴─┴─
      R B G Y                        0-9 10-19 20-29

Solved Examples

Worked Example

Create a 5-number summary for: 2, 4, 6, 8, 10, 12, 14, 16

Solution

Order: already ordered
Min = 2, Max = 16
Count = 8 (even), median = (8+10)/2 = 9
Lower half: 2,4,6,8 → Q1 = (4+6)/2 = 5
Upper half: 10,12,14,16 → Q3 = (12+14)/2 = 13
Answer: Min=2, Q1=5, Median=9, Q3=13, Max=16

Worked Example

What percent of data falls between Q1 and Q3 in a box plot?

Solution

Q1 to Q3 is the "box" which contains the middle 50%
Answer: 50%

Worked Example

A histogram shows test scores. The tallest bar is 70-79. What does this mean?

Solution

Tallest bar = most frequent/most common interval
Most students scored in the 70-79 range
Answer: Mode interval is 70-79

Key Points

Box plot shows spread using 5-number summary
Box contains middle 50% of data
Whiskers show remaining 25% each
Histogram groups data into intervals (bins)
Taller bar = more data in that interval
No gaps between bars in histogram (continuous data)

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.A histogram displays ___ data.

Explanation: Grouped data.

Q2.A box plot shows the:

Explanation: Five-number summary.

Q3.The interquartile range (IQR) is:

Explanation: $Q_3-Q_1$.

Q4.In a histogram, between the bars there are:

Explanation: No gaps.

Practice more MCQs → 📝 Assignment · 35 marks