GCSE Statistics · Higher · Chapter 5: Grouped data: histograms and cumulative frequency · Lesson 2

Cumulative frequency and percentiles

About 55 minutesCalculator allowed39 marks

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Date

Do Now

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
A histogram shows the times, in minutes, of some runners. The bar for $30 < t \leq 40$ has frequency density $2.8$ . The bar for $40 < t \leq 55$ has frequency density $2.4$ . Which of these two classes holds more runners, and how many more?
[2 marks]
Answer
A2
From chapter 2
A council wants the views of all the adults in a town on a new park. Ben, who runs the survey, plans to choose 30 people at random from the list of members of the town's tennis club. Explain why this is not a suitable sampling frame.
[2 marks]
A3
From chapter 2
Ben chooses a sample of the 600 students in a school. Ben numbers the 600 students and uses a random number generator, ignoring repeats, to choose 30. Does every student have an equal chance of being chosen? Give a reason.
[2 marks]
A4
From chapter 2
A group has 141 day shift, 142 late shift and 233 night shift. A stratified sample of 50 workers is taken. Work out how many of each are in the sample, checking that they add to 50.
[2 marks]
Answer

Example, then your turn

Your teacher works through each example with you. Then try the one beside it.

Example 1

The table shows the heights, $h$ cm, of 50 seedlings. Write down the points to plot for a cumulative frequency graph.

Hint Keep a running total; plot it at the end of its class.

B1
The table shows the times, $t$ seconds, of 80 swimmers. Write down the points you would plot for a cumulative frequency graph.
[2 marks]
Answer

Example 2

The graph shows the journey times, $t$ minutes, of 120 drivers. Estimate the median and the IQR. Compare them with cyclists on the same route: median 26 minutes, IQR 8 minutes.

Hint Median: across from

60

, then down.

B2
The graph shows the masses, $m$ grams, of 80 apples. Estimate the median and the interquartile range.
[3 marks]
Answermedian $\dots\dots$ g, IQR $\dots\dots$ g

Example 3

The graph shows the test marks, $x$ , of 200 students. Estimate the 10th and 90th percentiles and the range between them. What does the 90th percentile tell you?

Hint 10% of 200 is

20

: read across from

20

B3
The graph shows the distances, $d$ km, of 150 delivery vans in one day. Estimate the 20th and 80th percentiles. What does the 80th percentile tell you?
[3 marks]

Example 4

The graph shows the heights, $h$ cm, of 100 plants. Estimate how many of the plants are taller than 25 cm.

Hint The curve counts plants of 25 cm or less.

B4
The graph shows the times, $t$ minutes, of 60 runners in a race. Estimate the percentage who took more than 30 minutes.
[3 marks]
Answer

Practice

C1
Sam says, "The 90th percentile is 54, because 90% of 60 is 54."
Sam has a cumulative frequency graph of the times, in minutes, of 60 cyclists in a race. Explain Sam's mistake, and say how he should find the 90th percentile.
[2 marks]
C3
The cumulative frequency table shows the masses, $m$ kg, of 90 suitcases at an airport. Estimate the number of suitcases with a mass between 15 kg and 25 kg.
[2 marks]
Answer

C4
Freya has a cumulative frequency graph of the masses of 250 lambs. Write down the cumulative frequency she should read across from to estimate (i) the 3rd decile, (ii) the 85th percentile.
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$
C5
On a cumulative frequency graph of the reaction times of 80 drivers, a time of 0.42 seconds reads across to a cumulative frequency of 60. Which percentile is 0.42 seconds? Explain what it means for these drivers.
[2 marks]

Exam-style questions

D1
Jo says, "The median is the class $25 < t \leq 35$ , so the median is 30 seconds, the middle of that class."
The graph shows the times, $t$ seconds, of 80 people in a puzzle. Is Jo correct? You must give a reason.
[2 marks]

D2
The graph shows the waiting times, $t$ minutes, of 100 callers to Centre A.
(a)
Estimate the median and the interquartile range.
[3 marks]
Answermedian $\dots\dots$ min, IQR $\dots\dots$ min
(b)
Estimate the 90th percentile, and interpret your answer for these callers.
[2 marks]
(c)
At Centre B the median waiting time was 8 minutes and the interquartile range was 3 minutes. Compare the waiting times at the two centres.
[2 marks]
Total for D2: 7 marks

Extension

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Two groups of students did the same task. By 10, 20, 30 and 40 minutes, $4$ , $16$ , $34$ and $40$ of the 40 students in Group A had finished, and $6$ , $24$ , $46$ and $60$ of the 60 students in Group B. Estimate the median time for all 100 students together.
[3 marks]
Hint 1 · What to try
Add the two cumulative frequencies at each time.
Hint 2 · The first line
Together: $10$ , $40$ , $80$ , $100$ . The 50th student is between $20$ and $30$ minutes.
Hint 3 · The full method
$40$ students are done by $20$ minutes; $10$ more of the $40$ in the next class: $20 + \frac{10}{40} \times 10$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out cumulative frequencies and the points to plot at the upper class boundaries.
I can read the median, quartiles, deciles and percentiles from a curve, and how many lie above or below a value.
I can say what a percentile means in context, and compare two distributions.

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