Cumulative Frequency and Box Plots

Cumulative frequency graphs and box plots are powerful tools for summarising and comparing distributions. These are Higher-tier staples on the Edexcel GCSE, though some Foundation questions also appear.

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Key Vocabulary

Term	Definition
Cumulative frequency	A running total of frequencies
Cumulative frequency graph	An S-shaped curve plotted using upper class boundaries vs cumulative frequency
Box plot (box-and-whisker diagram)	A diagram showing the minimum, $Q_1$Q1​, median, $Q_3$Q3​ and maximum
Median	The middle value (50th percentile); read from the cumulative frequency graph at n/2
Lower quartile ($Q_1$Q1​)	25th percentile; read at n/4
Upper quartile ($Q_3$Q3​)	75th percentile; read at 3n/4
Interquartile range (IQR)	$Q_3 - Q_1$Q3​−Q1​

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Box Plot Anatomy

graph LR
    A["Minimum"] --- B["Q1"]
    B --- C["Median"]
    C --- D["Q3"]
    D --- E["Maximum"]

    style A fill:#7f8c8d,color:#fff
    style B fill:#3498db,color:#fff
    style C fill:#e74c3c,color:#fff
    style D fill:#3498db,color:#fff
    style E fill:#7f8c8d,color:#fff

• Whiskers (outside the box): extend from min to $Q_1$Q1​ and from $Q_3$Q3​ to max.
• Box: from $Q_1$Q1​ to $Q_3$Q3​. The width of the box = IQR.
• Line inside the box: the median.

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Cumulative Frequency Tables

Add a cumulative frequency column by keeping a running total of the frequencies.

Worked Example 1 — Building a cumulative frequency table

The table shows the time (t minutes) taken by 80 students to complete a task. Add a cumulative frequency column.

Time (t minutes)	Frequency	Cumulative frequency
$0 < t \leq 10$0<t≤10	6	6
$10 < t \leq 20$10<t≤20	14	6 + 14 = 20
$20 < t \leq 30$20<t≤30	24	20 + 24 = 44
$30 < t \leq 40$30<t≤40	22	44 + 22 = 66
$40 < t \leq 50$40<t≤50	10	66 + 10 = 76
$50 < t \leq 60$50<t≤60	4	76 + 4 = 80

Check: Final CF = 80 = total number of data items ✓

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Drawing a Cumulative Frequency Graph

Worked Example 2

Using the table above, describe how to draw the cumulative frequency curve.

Solution:

1. Plot each point at the upper class boundary on the x-axis and the cumulative frequency on the y-axis.
2. Start the curve at the lower boundary of the first class with CF = 0, i.e. plot (0, 0).
3. Points to plot: (0, 0), (10, 6), (20, 20), (30, 44), (40, 66), (50, 76), (60, 80).
4. Join the points with a smooth S-shaped curve (an ogive).

Critical: Always plot at the upper class boundary, NOT the midpoint.

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Reading Median and Quartiles from the Graph

For n data items on a cumulative frequency curve:

• Median is at cumulative frequency = n ÷ 2
• $Q_1$Q1​ is at cumulative frequency = n ÷ 4
• $Q_3$Q3​ is at cumulative frequency = 3n ÷ 4

Worked Example 3

Using the curve from Worked Example 2 (n = 80), find estimates of the median, $Q_1$Q1​, $Q_3$Q3​ and IQR.

Solution:

• Median at CF = 40 → read across to the curve, then down to x-axis → median ≈ 29 minutes.
• $Q_1$Q1​ at CF = 20 → $Q_1$Q1​ ≈ 20 minutes.
• $Q_3$Q3​ at CF = 60 → $Q_3$Q3​ ≈ 38 minutes.
• IQR = $Q_3 - Q_1$Q3​−Q1​ ≈ 38 − 20 = 18 minutes.

Worked Example 4 — Full CF workflow

The masses (m grams) of 60 apples are:

Mass (m g)	80–100	100–120	120–140	140–160	160–180
Frequency	5	12	23	14	6

(a) Complete a cumulative frequency table. (b) Estimate the median, $Q_1$Q1​ and $Q_3$Q3​. (c) Estimate how many apples have a mass greater than 150 g.

Solution:

(a) CF: 5, 17, 40, 54, 60. Plot at upper boundaries: (100, 5), (120, 17), (140, 40), (160, 54), (180, 60).

(b) n = 60:

• Median at CF = 30: curve crosses 30 at about m ≈ 133 → median ≈ 133 g.
• $Q_1$Q1​ at CF = 15: about m ≈ 118 → $Q_1$Q1​ ≈ 118 g.
• $Q_3$Q3​ at CF = 45: about m ≈ 148 → $Q_3$Q3​ ≈ 148 g.
• IQR ≈ 148 − 118 = 30 g.

(c) CF at m = 150: approximately 47. Number of apples above 150 g = 60 − 47 = 13 apples.

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Box Plots (Box-and-Whisker Diagrams)

A box plot displays the five-number summary: minimum, $Q_1$Q1​, median, $Q_3$Q3​, maximum.

Worked Example 5 — Drawing a box plot

Using the cumulative frequency data (n = 80, min = 0, $Q_1$Q1​ = 20, median = 29, $Q_3$Q3​ = 38, max = 60), draw a box plot.

Solution:

1. Draw a horizontal number line scaled 0 to 60 minutes.
2. Draw a vertical line at $Q_1$Q1​ = 20 and another at $Q_3$Q3​ = 38.
3. Join them to form a box (width = IQR = 18).
4. Draw a vertical line inside the box at the median (29).
5. Extend whiskers from the left of the box to the minimum (0) and from the right of the box to the maximum (60).

Reading a Box Plot

From a box plot you can find:

• Median: the line inside the box.
• IQR: the length of the box ($Q_3 - Q_1$Q3​−Q1​).
• Range: maximum − minimum (full length of the whiskers).
• Skewness: if the median is closer to $Q_1$Q1​, the data is positively skewed; if closer to $Q_3$Q3​, negatively skewed.

Worked Example 6 — Reading a box plot

A box plot of test scores shows: minimum = 20, $Q_1$Q1​ = 42, median = 55, $Q_3$Q3​ = 63, maximum = 85.

(a) Find the IQR. (b) Find the range. (c) Comment on the skewness.

Solution: (a) IQR = 63 − 42 = 21 marks. (b) Range = 85 − 20 = 65 marks. (c) The median (55) is closer to $Q_3$Q3​ (63) than to $Q_1$Q1​ (42). This suggests the lower half of the distribution is more spread out than the upper half — the data is negatively skewed (a tail on the low side).

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Comparing Box Plots [H]

Edexcel exam tip: Always make TWO comparisons: one about an average (median) and one about spread (IQR or range). Use the context (test scores, times, etc.).

Worked Example 7 [H] — Comparing two classes

Two classes took the same test. Their results are shown below.

Statistic	Class A	Class B
Minimum	22	30
$Q_1$Q1​	40	45
Median	55	58
$Q_3$Q3​	68	65
Maximum	85	78

Compare the two classes.

Solution:

• Class B has a higher median (58 > 55), so on average Class B scored higher marks.
• Class A has a larger IQR (68 − 40 = 28 vs 65 − 45 = 20), so Class A's marks were more spread out — Class B's middle 50% are more tightly clustered and therefore more consistent.
• Class A has a larger range (85 − 22 = 63 vs 78 − 30 = 48), confirming wider spread overall. Class A does, however, include the highest individual mark (85).

Worked Example 8 [H] — Comparing boys vs girls

Box plots summarise the heights (cm) of Year 10 boys and girls.

	Boys	Girls
Minimum	150	148
$Q_1$Q1​	160	156
Median	168	162
$Q_3$Q3​	175	168
Maximum	188	178

Compare the distributions.

Solution:

• Median: boys (168) > girls (162), so on average boys are taller.
• IQR: boys (175 − 160 = 15), girls (168 − 156 = 12). Boys' heights are more spread out in the middle 50%; girls are slightly more consistent.
• Range: boys (188 − 150 = 38), girls (178 − 148 = 30). Boys have a wider overall range of heights.

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Using Cumulative Frequency to Estimate How Many Values Fall in a Range

Worked Example 9

From the cumulative frequency graph of Worked Example 2 (n = 80), estimate the number of students who took between 15 and 35 minutes.

Solution:

• Read CF at t = 35: approximately 58.
• Read CF at t = 15: approximately 12.
• Number of students between 15 and 35 = 58 − 12 = 46 students.