Cumulative frequency, quartiles and box plots

Q: What is cumulative frequency and how is it calculated from a grouped frequency table?

Cumulative frequency is a running total of frequencies, found by adding each class frequency to the sum of all preceding frequencies.

Q: Where should points be plotted when drawing a cumulative frequency curve?

Plot points at the upper class boundary of each interval against its cumulative frequency, and anchor the curve at the lower boundary of the first interval with a cumulative frequency of 0.

Q: How do you find the median and quartiles from a cumulative frequency curve?

Locate \(\frac{n}{4}\) for the lower quartile (\(Q_1\)), \(\frac{n}{2}\) for the median (\(Q_2\)), and \(\frac{3n}{4}\) for the upper quartile (\(Q_3\)) on the vertical axis, then read across to the curve and down to the horizontal axis.

Q: What is the Interquartile Range (IQR) and why is it preferred over the range?

\(\text{IQR} = Q_3 - Q_1\); it measures the spread of the middle 50% of data and is not affected by extreme values or outliers.

Q: How do you estimate the number of items greater than a given value \(k\) from a cumulative frequency curve?

Find \(k\) on the horizontal axis, read up to the curve and across to find the cumulative frequency \(cf_k\), then subtract this value from the total frequency: \(n - cf_k\).

Q: What five values make up the five-figure summary used to construct a box plot?

Minimum value, Lower Quartile (\(Q_1\)), Median (\(Q_2\)), Upper Quartile (\(Q_3\)), and Maximum value.

Q: What do the box and whiskers represent on a box plot?

The box is drawn from \(Q_1\) to \(Q_3\) (its width is the IQR) with a vertical line at the median, and the whiskers extend to the minimum and maximum values.

Q: What two key points must be included when comparing two distributions or box plots?

One comparison of an average (usually the median) and one comparison of spread or consistency (the IQR or range), both written using comparative language in context.

Mathematics 2210 · CCEA GCSE · 7 min read · Updated: 31 Aug 2026

Handling Data: Cumulative Frequency, Quartiles, and Box Plots

Welcome to one of the most practical and high-scoring topics in GCSE Mathematics! Have you ever wondered how exam boards decide grade boundaries, or how sports scientists compare the fitness levels of two different football teams? They use tools like cumulative frequency curves and box plots to make sense of large amounts of data. Don't worry if these diagrams look a bit intimidating right now — we will break everything down into simple, step-by-step chunks so you can tackle these exam questions with confidence.

1. Cumulative Frequency Tables and Curves

What Does "Cumulative" Mean?

The word cumulative simply means a running total. Think of it like scoring in a game of basketball: each time a basket is made, the points are added to the running total on the scoreboard.

To find the cumulative frequency (\(cf\)) from a grouped frequency table, you keep adding each class frequency to the sum of the frequencies before it. The final number in your cumulative frequency column must always equal the total number of items in the dataset (\(n = \sum f\)).

Example: Building the Table

Imagine 60 students timed how long it took them to complete a puzzle (in seconds, \(t\)):

Time (\(t\) seconds) | Frequency (\(f\)) | Cumulative Frequency (\(cf\))
\(0 < t \le 10\) | \(5\) | \(5\)
\(10 < t \le 20\) | \(12\) | \(5 + 12 = 17\)
\(20 < t \le 30\) | \(25\) | \(17 + 25 = 42\)
\(30 < t \le 40\) | \(14\) | \(42 + 14 = 56\)
\(40 < t \le 50\) | \(4\) | \(56 + 4 = 60\)

Quick Check: The final cumulative frequency is \(60\), which matches our total number of students!

How to Plot a Cumulative Frequency Curve (Ogive)

Plotting the curve accurately is crucial in your exam. Remember these three golden rules:

Rule 1: Always plot at the UPPER class boundary.
Plot your cumulative frequency against the highest value of each interval on the horizontal (\(x\)) axis. For the table above, you would plot the coordinates: \((10, 5)\), \((20, 17)\), \((30, 42)\), \((40, 56)\), and \((50, 60)\).

Rule 2: Anchor the curve at zero.
Your curve must start at the lower boundary of the very first interval with a cumulative frequency of \(0\). In our example, the lowest interval starts at \(0\), so plot the starting point at \((0, 0)\). If the first class was \(20 < t \le 30\), you would anchor at \((20, 0)\).

Rule 3: Draw a smooth S-shaped curve.
Join your plotted points with a smooth, continuous curve (an ogive) or neat straight-line segments from point to point.

Common Mistake to Avoid: Never plot cumulative frequency at the midpoint of the interval! Plotting at the midpoint is for frequency polygons, not cumulative frequency curves. Cumulative frequency shows how many values are up to and including that boundary, so it must go on the upper boundary.

2. Finding Quartiles and Estimates from the Curve

Once your curve is drawn, you can use it to estimate key statistical values. Quartiles divide your ordered data into four equal quarters (\(25\%\) chunks).

Locating Quartiles on the Vertical Axis

To find quartiles, you always start on the vertical cumulative frequency axis (\(y\)-axis) using total frequency \(n\):

1. Lower Quartile (\(Q_1\) or \(LQ\)):
Position = \(\frac{n}{4}\) (or \(25\%\) of \(n\)).
Go to \(\frac{n}{4}\) on the vertical axis, draw a horizontal line across to hit your curve, and then read directly down to the horizontal axis.

2. Median (\(Q_2\) or \(M\)):
Position = \(\frac{n}{2}\) (or \(50\%\) of \(n\)).
This is the middle value of your data distribution.

3. Upper Quartile (\(Q_3\) or \(UQ\)):
Position = \(\frac{3n}{4}\) (or \(75\%\) of \(n\)).
This marks the top \(25\%\) threshold of your data.

Measuring Spread: Range and Interquartile Range (\(\text{IQR}\))

In statistics, we don't just look at the average; we also look at how spread out the data is.

Range:
\(\text{Range} = \text{Maximum value} - \text{Minimum value}\)
The range measures the overall spread, but it can be heavily distorted by extreme values (outliers).

Interquartile Range (\(\text{IQR}\)):
\(\text{IQR} = Q_3 - Q_1\)
The \(\text{IQR}\) measures the spread of the middle \(50\%\) of the data. Because it ignores the lowest \(25\%\) and highest \(25\%\), it is not affected by freak results or outliers, making it a very reliable measure of consistency.

Crucial Exam Tip: Do not calculate \(\text{IQR}\) by subtracting the cumulative frequency numbers (e.g. \(\frac{3n}{4} - \frac{n}{4}\))! You must read the actual values off the horizontal axis (\(Q_3\) and \(Q_1\)) and subtract those.

Estimating Proportions and Frequencies

Exam questions frequently ask you to estimate how many items fall above or below a specific target value \(k\):

• Estimating "Less than or equal to \(k\)": Find \(k\) on the horizontal axis, move up to the curve, and read across to the cumulative frequency axis. That reading is your answer.

• Estimating "Greater than \(k\)" / "More than \(k\)": Find \(k\) on the horizontal axis, move up to the curve, and read the cumulative frequency \(cf_k\). The number of items above \(k\) is the total minus this reading: \(\text{Number above } k = n - cf_k\).

3. Box Plots (Box-and-Whisker Diagrams)

A box plot is a clean, visual summary that shows the spread and skew of a dataset using exactly five key values, known as the five-figure summary.

The Five-Figure Summary

1. Minimum Value: The lowest value in the dataset.
2. Lower Quartile (\(Q_1\)): The \(25\%\) mark.
3. Median (\(Q_2\)): The middle value (\(50\%\) mark).
4. Upper Quartile (\(Q_3\)): The \(75\%\) mark.
5. Maximum Value: The highest value in the dataset.

Anatomy of a Box Plot

• The Box: Drawn between \(Q_1\) and \(Q_3\). The length of this box represents the Interquartile Range (\(\text{IQR}\)).
• The Median Line: A vertical line drawn inside the box at the exact position of \(Q_2\).
• The Whiskers: Horizontal lines extending outwards from the box to the Minimum and Maximum values.

Visual representation:

|-------[ | ]-------|
Min Q1 Med Q3 Max

Always draw your box plot carefully using a sharp pencil and a ruler above a clear, uniformly spaced number line scale.

4. Comparing Two Distributions

Comparing two box plots or distributions is a classic multi-mark question. To get full marks, examiners look for two distinct comparative statements written in context.

The Golden Rule for Full Marks:

You must compare one measure of average (the Median) AND one measure of spread (the \(\text{IQR}\) or Range), using comparative words like higher, lower, greater, or more consistent, and refer to the real-world context of the problem.

Step-by-Step Template for Comparisons:

Statement 1: Compare the Medians (Average / Central Tendency)
• Formula: "[Group A] has a higher/lower median [context] than [Group B], which means on average [Group A did better / took longer / scored higher]."

Statement 2: Compare the Interquartile Ranges (Spread / Consistency)
• Formula: "[Group A] has a smaller/larger IQR than [Group B], which means [Group A]'s results are more consistent / less varied."

Memory Trick:
• Smaller \(\text{IQR}\) = More Consistent (values are packed closer together).
• Larger \(\text{IQR}\) = Less Consistent / More Spread Out.

Example Exam Comparison:

Scenario: Comparing the test scores of Class 10A and Class 10B.
Class 10A: \(\text{Median} = 65\), \(\text{IQR} = 12\)
Class 10B: \(\text{Median} = 54\), \(\text{IQR} = 22\)

• Correct Answer:
"Class 10A had a higher median test score (\(65\)) than Class 10B (\(54\)), meaning Class 10A performed better on average."
"Class 10A had a smaller interquartile range (\(12\)) than Class 10B (\(22\)), meaning Class 10A's scores were more consistent."

Examiner Warning: Writing just numbers without context or comparative language (e.g. "The median of A is 65 and B is 54") will score zero marks! You must interpret what the numbers actually mean.

5. Quick Summary & Exam Checklist

Before sitting your exam, make sure you can tick off every item on this checklist:

[ ] Plotting Cumulative Frequency: Plotted at the upper class boundary on the \(x\)-axis, anchored at the lower bound of the first class at \(y = 0\), joined with a smooth curve.
[ ] Finding Quartiles: Found at \(\frac{n}{4}\) (Lower Quartile), \(\frac{n}{2}\) (Median), and \(\frac{3n}{4}\) (Upper Quartile) on the vertical \(y\)-axis.
[ ] Calculating \(\text{IQR}\): Calculated as \(Q_3 - Q_1\) using values from the horizontal \(x\)-axis.
[ ] Constructing Box Plots: Accurately marked using the five-figure summary: Minimum, \(Q_1\), Median, \(Q_3\), Maximum.
[ ] Comparing Distributions: Provided two contextual sentences comparing (1) the Medians (average) and (2) the \(\text{IQR}\) (consistency).

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is cumulative frequency and how is it calculated from a grouped frequency table?

Cumulative frequency is a running total of frequencies, found by adding each class frequency to the sum of all preceding frequencies.

Where should points be plotted when drawing a cumulative frequency curve?

Plot points at the upper class boundary of each interval against its cumulative frequency, and anchor the curve at the lower boundary of the first interval with a cumulative frequency of 0.

How do you find the median and quartiles from a cumulative frequency curve?

Locate \(\frac{n}{4}\) for the lower quartile (\(Q_1\)), \(\frac{n}{2}\) for the median (\(Q_2\)), and \(\frac{3n}{4}\) for the upper quartile (\(Q_3\)) on the vertical axis, then read across to the curve and down to the horizontal axis.

What is the Interquartile Range (IQR) and why is it preferred over the range?

\(\text{IQR} = Q_3 - Q_1\); it measures the spread of the middle 50% of data and is not affected by extreme values or outliers.

How do you estimate the number of items greater than a given value \(k\) from a cumulative frequency curve?

Find \(k\) on the horizontal axis, read up to the curve and across to find the cumulative frequency \(cf_k\), then subtract this value from the total frequency: \(n - cf_k\).

What five values make up the five-figure summary used to construct a box plot?

Minimum value, Lower Quartile (\(Q_1\)), Median (\(Q_2\)), Upper Quartile (\(Q_3\)), and Maximum value.

What do the box and whiskers represent on a box plot?

The box is drawn from \(Q_1\) to \(Q_3\) (its width is the IQR) with a vertical line at the median, and the whiskers extend to the minimum and maximum values.

What two key points must be included when comparing two distributions or box plots?

One comparison of an average (usually the median) and one comparison of spread or consistency (the IQR or range), both written using comparative language in context.

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