Cumulative Frequency and Box Plots

What Is Cumulative Frequency?

Cumulative frequency is a running total of frequencies. For grouped data, it tells you how many data values are less than or equal to the upper boundary of each class. By plotting cumulative frequencies against upper class boundaries, you create a cumulative frequency curve (also called an ogive), which is a powerful tool for estimating medians, quartiles, and percentiles.

Cumulative frequency and box plots appear on both the Core and Extended IGCSE papers. Questions typically carry between four and eight marks and often ask you to draw a curve, read off values, and then draw or interpret a box plot.

Drawing a Cumulative Frequency Table

Start with a grouped frequency table and add a cumulative frequency column.

Example:

Height (h cm)	Frequency	Cumulative Frequency
140 < h ≤ 150	5	5
150 < h ≤ 160	12	17
160 < h ≤ 170	23	40
170 < h ≤ 180	18	58
180 < h ≤ 190	7	65
190 < h ≤ 200	2	67

Each cumulative frequency is the sum of all frequencies up to and including that class.

Drawing the Cumulative Frequency Curve

Plotting Points

Plot each cumulative frequency against the upper class boundary of its class:

(150, 5), (160, 17), (170, 40), (180, 58), (190, 65), (200, 67)

Why the upper boundary? Because the cumulative frequency tells you how many values are “up to” that point. At the upper boundary of the first class (150), all 5 values in that class have been counted.

You can also plot a point at the lower boundary of the first class with cumulative frequency 0: (140, 0). This anchors the start of the curve.

Drawing the Curve

Join the points with a smooth S-shaped curve (not straight lines). The curve should:

Start relatively flat (low frequency at the beginning)
Steepen in the middle (where most data values fall)
Flatten again at the top (as the total is approached)

Common mistake: Joining the points with straight line segments instead of a smooth curve. While some mark schemes accept this, a smooth curve is the expected standard and looks more professional.

Reading the Median and Quartiles

The median and quartiles are found by reading across from the cumulative frequency axis to the curve, then down to the horizontal axis.

For n data values:

Median is at cumulative frequency n/2
Lower quartile (Q1) is at cumulative frequency n/4
Upper quartile (Q3) is at cumulative frequency 3n/4

Worked Example

Using our data (n = 67):

Median position: 67/2 = 33.5 → read across from 33.5 on the cumulative frequency axis to the curve, then down → approximately 168 cm
Q1 position: 67/4 = 16.75 → approximately 159 cm
Q3 position: 3 × 67/4 = 50.25 → approximately 176 cm

The Interquartile Range (IQR)

IQR = Q3 − Q1

In our example: IQR = 176 − 159 = 17 cm

The IQR measures the spread of the middle 50% of the data. A smaller IQR means the data is more tightly clustered around the median.

Drawing a Box Plot

A box plot (or box-and-whisker diagram) is a visual summary of the data using five key values:

Minimum value (or lower boundary of the first class with data)
Lower quartile (Q1)
Median (Q2)
Upper quartile (Q3)
Maximum value (or upper boundary of the last class with data)

How to Draw It

Draw a number line covering the range of the data
Draw a box from Q1 to Q3
Draw a vertical line inside the box at the median
Draw whiskers from the box to the minimum and maximum values

For our example:

Minimum ≈ 140
Q1 ≈ 159
Median ≈ 168
Q3 ≈ 176
Maximum ≈ 200

Note: When drawing box plots from grouped data and cumulative frequency curves, you may use the lowest and highest class boundaries as the minimum and maximum if the actual values are not given.

Reading a Box Plot

Given a box plot, you can determine:

Median: the line inside the box
IQR: the width of the box (Q3 − Q1)
Range: maximum − minimum (the full extent of the whiskers)
Skewness: if the median is closer to Q1, the data is positively skewed; if closer to Q3, negatively skewed

Comparing Two Distributions

A common exam question gives two box plots (or two cumulative frequency curves) and asks you to compare the distributions. You need to make two comparisons:

A comparison of averages (central tendency) — compare the medians
A comparison of spread (dispersion) — compare the IQRs or ranges

How to Write a Comparison

Always refer to the context of the data and use specific values:

Good answer: “Group A has a higher median height (168 cm) compared to Group B (155 cm), suggesting students in Group A are generally taller. Group A also has a larger IQR (17 cm vs 12 cm), indicating more variation in heights.”

Poor answer: “Group A is higher and more spread out.” (Too vague, no values.)

Estimating Percentiles

You can use the cumulative frequency curve to estimate any percentile:

The 10th percentile is at cumulative frequency 0.1n
The 90th percentile is at cumulative frequency 0.9n

This technique is useful for questions asking how many data values fall above or below a particular threshold.

Example: Estimate the number of students taller than 175 cm.

Read across from 175 on the horizontal axis to the curve
Read across to the cumulative frequency axis — say this gives 48
Number taller than 175 = 67 − 48 = 19 students

Common Mistakes to Avoid

Plotting at the midpoint instead of the upper boundary. Cumulative frequencies are always plotted at upper class boundaries.
Drawing a bar chart instead of a smooth curve. A cumulative frequency diagram is a curve, not a histogram.
Using n + 1 for the median position. At IGCSE level, use n/2 for the median position on a cumulative frequency curve. The (n + 1) rule is for listed data.
Not using a ruler for box plots. Box plots should be drawn accurately with straight edges.
Forgetting to give values in comparisons. Always quote specific numbers when comparing distributions.
Reading the wrong axis. Read across from the cumulative frequency axis to find values on the data axis, not the other way around.

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