Should I use n2 or n + 12 for the median?

For grouped data on a cumulative frequency graph, n2 is standard.

S3–S4StatisticsHigherGrades 6–8

Cumulative Frequency & Box Plots

Q: Do I join the points with straight lines or a curve?

Either is usually accepted; a smooth curve is conventional.

Q: Where do the minimum and maximum come from?

They are given, or taken from the ends of the data range.

When data come in grouped classes, you cannot list the values to find the median. A cumulative frequency graph solves that: it plots running totals, and the median and quartiles can be read straight off it. A box plot then summarises the whole distribution in five numbers, ready to compare with another group.

50 min AQA GCSE Maths

The big picture

Cumulative frequency is a running total. Plotting it against the $\textbf{upper boundary}$ of each class gives an S-shaped curve. Reading across at half the total gives the median; at a quarter and three-quarters, the quartiles.

The interquartile range (IQR) — upper quartile minus lower quartile — measures the spread of the middle half of the data, so unlike the range it is not distorted by extreme values. Box plots show the median and IQR at a glance.

What you'll learn

Construct a cumulative frequency table and graph
Estimate the median, quartiles and interquartile range
Draw and interpret box plots
Compare two distributions using median and IQR

Building the graph

The times taken by 80 people to finish a puzzle (minutes): $0 < t \le 10$ : 6; $10 < t \le 20$ : 14; $20 < t \le 30$ : 25; $30 < t \le 40$ : 20; $40 < t \le 50$ : 11; $50 < t \le 60$ : 4.

Running totals: 6, 20, 45, 65, 76, 80.

Plot $(10, 6)$ , $(20, 20)$ , $(30, 45)$ , $(40, 65)$ , $(50, 76)$ , $(60, 80)$ , starting from $(0, 0)$ , and join with a smooth curve.

Tip — Plot at the upper class boundary, not the midpoint. By $t = 20$ , all 20 of the first two classes have finished.

Median and quartiles

With $n = 80$ : median at the 40th value, lower quartile at the 20th, upper quartile at the 60th.

Read across from each cumulative frequency to the curve, then down to the time axis.

Example 1Estimate the median, quartiles and IQR of the puzzle times.

1LQ (20th): reads at 20 minutes.

2Median (40th): between 20 and 45, about 28 minutes.

3UQ (60th): between 45 and 65, about 37.5 minutes.

4IQR

= 37.5 - 20

AnswerMedian ≈ 28 min, LQ = 20 min, UQ ≈ 37.5 min, IQR ≈ 17.5 min

Your turnSame method, new numbers — try it before moving on.

A data set has lower quartile 22 and upper quartile 41. What is the interquartile range?

Your answer

Show worked solution

Solution

1IQR $= 41 - 22 = 19$ .

Answer: $19$

Example 2Estimate how many people took more than 45 minutes.

1Read up from 45 minutes: about 70 people had finished.

80 - 70

AnswerAbout 10 people

Why this works

Readings from a hand-drawn curve are estimates, so mark schemes accept a small range. Draw guide lines on the graph so the examiner can follow your method.

Box plots

A box plot uses five values: minimum, lower quartile, median, upper quartile and maximum.

The box runs from LQ to UQ with a line at the median; whiskers extend to the minimum and maximum.

For the puzzle, a box plot might run from 2 to 58 minutes, with the box from 20 to 37.5 and the median at 28.

Your turnSame method, new numbers — try it before moving on.

A cumulative frequency graph shows 80 people. At what cumulative frequency would you read across to estimate the median?

Your answer

Show worked solution

Solution

1For a cumulative frequency graph, use half of the total.
2 $80 \div 2 = 40$ .

Answer: $40$

Comparing distributions

Compare an average (the median) and a spread (the IQR), and interpret both in context.

Example: Group B had median 24 min and IQR 11 min. "Group B were faster on average, as their median was lower (24 min vs 28 min). Their times were also more consistent, as their IQR was smaller (11 min vs 17.5 min)."

Tip — Write one sentence about average and one about spread, each quoting the numbers and linking to the context.

Think like an examiner

Plot points at upper class boundaries.
Show reading lines on the graph.
Compare distributions in context using median and IQR.

Remember these

\text{median at } \tfrac{n}{2}

For large n on a cumulative frequency graph.

\text{LQ at } \tfrac{n}{4},\ \text{UQ at } \tfrac{3n}{4}

Quartile positions.

\text{IQR} = \text{UQ} - \text{LQ}

Interquartile range.

Watch out for these

Plotting cumulative frequency at class midpoints.

Plot at the upper boundary of each class.

Giving the cumulative frequency (40) as the median.

The median is the value on the horizontal axis.

Comparing ranges instead of IQRs.

IQR is less affected by extreme values.

"Group B’s median is lower" with no context.

Say what a lower median means, e.g. faster times.

Stretch yourself

The masses of 120 apples (grams) have cumulative frequencies: $\le 100$ : 8, $\le 120$ : 30, $\le 140$ : 72, $\le 160$ : 104, $\le 180$ : 116, $\le 200$ : 120. Assuming values are spread evenly within classes, estimate the median and IQR.

Hint — Median at the 60th apple, quartiles at the 30th and 90th.

Reveal solution

SolutionMedian (60th): between 30 and 72, so

120 + \dfrac{60 - 30}{42} \times 20 = 134.3

g. LQ (30th): exactly 120 g. UQ (90th): between 72 and 104, so

140 + \dfrac{90 - 72}{32} \times 20 = 151.25

g. IQR

= 151.25 - 120 \approx 31

\blacksquare

Questions students ask

Should I use

\tfrac{n}{2}

\tfrac{n + 1}{2}

for the median?

For grouped data on a cumulative frequency graph,

\tfrac{n}{2}

is standard.

Do I join the points with straight lines or a curve?

Either is usually accepted; a smooth curve is conventional.

Where do the minimum and maximum come from?

They are given, or taken from the ends of the data range.

Key takeaways

Cumulative frequency is a running total.
Plot against upper class boundaries.
Median at $\tfrac{n}{2}$ , quartiles at $\tfrac{n}{4}$ and $\tfrac{3n}{4}$ .
Box plots show the five-number summary.
Compare median and IQR in context.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 63 marks

Times for 60 people: $\begin{array}{c|c} \text{Time (min)} & \text{Cumulative frequency} \\ \hline \le 10 & 5 \\ \le 20 & 17 \\ \le 30 & 38 \\ \le 40 & 52 \\ \le 50 & 60 \end{array}$

(a)How many people took more than 30 minutes?[1]
Show mark scheme (a)
- B1 $22$
Answer: $22$
(b)Estimate the median time.[2]
Show mark scheme (b)
- M1Uses the 30th value, in the $20$ – $30$ class
- A1 $26$ minutes (accept 25–27)
Answer: $26$ minutes

Question 2Grade 72 marks

Test marks: Class A has median 54 and interquartile range 18. Class B has median 61 and interquartile range 9.

Compare the marks of the two classes.[2]
Show mark scheme
- B1Class B scored higher on average (median 61 > 54)
- B1Class B's marks were more consistent (IQR 9 < 18)

Question 3Grade 82 marks

A box plot for the scores of 200 people shows: minimum 12, lower quartile 20, median 26, upper quartile 34, maximum 47.

Estimate how many people scored between 20 and 34.[2]
Show mark scheme
- B1Recognises this is the middle half of the data
- B1 $100$
Answer: $100$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

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