Draw histogram then perform calculations

CAIE S1 2021 June Q5

5 The times taken by 200 players to solve a computer puzzle are summarised in the following table.

Time $( t$ seconds $)$	$0 \leqslant t < 10$	$10 \leqslant t < 20$	$20 \leqslant t < 40$	$40 \leqslant t < 60$	$60 \leqslant t < 100$
Number of players	16	54	78	32	20

Draw a histogram to represent this information.
\includegraphics[max width=\textwidth, alt={}, center]{1a27e2ca-9be5-48a0-a1aa-01844573f4d4-08_1397_1198_808_516}
Calculate an estimate of the mean time taken by these 200 players.
Find the greatest possible value of the interquartile range of these times.

CAIE S1 2023 June Q5

5 The populations of 150 villages in the UK, to the nearest hundred, are summarised in the table.

Population	$100 - 800$	$900 - 1200$	$1300 - 2000$	$2100 - 3200$	$3300 - 4800$
Number of villages	8	12	50	48	32

Draw a histogram to represent this information.
\includegraphics[max width=\textwidth, alt={}, center]{a7157882-d87e-4efb-abc5-9c9f58197012-08_1395_1195_1043_516}
Write down the class interval which contains the median for this information.
Find the greatest possible value of the interquartile range for the populations of the 150 villages.

CAIE S1 2024 June Q3

3 The heights, in cm, of 200 adults in Barimba are summarised in the following table.

Height $( h \mathrm {~cm} )$	$130 \leqslant h < 150$	$150 \leqslant h < 160$	$160 \leqslant h < 170$	$170 \leqslant h < 175$	$175 \leqslant h < 195$
Frequency	16	32	76	64	12

Draw a histogram to represent this information.
\includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-04_1397_1495_762_287}
The interquartile range is $R \mathrm {~cm}$. Show that $R$ is not greater than 15 .

CAIE S1 2022 March Q3

3 At a summer camp an arithmetic test is taken by 250 children. The times taken, to the nearest minute, to complete the test were recorded. The results are summarised in the table.

Time taken, in minutes	$1 - 30$	$31 - 45$	$46 - 65$	$66 - 75$	$76 - 100$
Frequency	21	30	68	86	45

Draw a histogram to represent this information.
\includegraphics[max width=\textwidth, alt={}, center]{c1bc5ac2-6b0e-48c7-92e9-9b8b56b57d90-05_1000_1198_785_516}
State which class interval contains the median.
Given that an estimate of the mean time is 61.05 minutes, state what feature of the distribution accounts for the median and the mean being different.

CAIE S1 2023 March Q1

1 Each year the total number of hours, $x$, of sunshine in Kintoo is recorded during the month of June. The results for the last 60 years are summarised in the table.

$x$

$30 \leqslant x < 60$

$60 \leqslant x < 90$

$90 \leqslant x < 110$

$110 \leqslant x < 140$

$140 \leqslant x < 180$

$180 \leqslant x \leqslant 240$

Number

of years

4

8

14

25

7

2

Draw a cumulative frequency graph to illustrate the data.
\includegraphics[max width=\textwidth, alt={}, center]{2577c597-0a04-4909-ad71-1347aacec6d9-02_1395_1397_881_415}
Use your graph to estimate the 70th percentile of the data.
Calculate an estimate for the mean number of hours of sunshine in Kintoo during June over the last 60 years.

CAIE S1 2024 March Q3

3 The times taken, in minutes, by 150 students to complete a puzzle are summarised in the table.

Time taken

$( t$ minutes $)$

$0 \leqslant t < 20$

$20 \leqslant t < 30$

$30 \leqslant t < 35$

$35 \leqslant t < 40$

$40 \leqslant t < 50$

$50 \leqslant t < 70$

Frequency

8

23

35

52

20

12

Draw a histogram to represent this information.
\includegraphics[max width=\textwidth, alt={}, center]{d1a3524c-a3b5-45fe-86a7-5cbda087efcd-06_1193_1489_886_328}
Calculate an estimate for the mean time for these students to complete the puzzle.
In which class interval does the lower quartile of the times lie?

CAIE S1 2020 November Q7

7 A particular piece of music was played by 91 pianists and for each pianist, the number of incorrect notes was recorded. The results are summarised in the table.

Number of incorrect notes	$1 - 5$	$6 - 10$	$11 - 20$	$21 - 40$	$41 - 70$
Frequency	10	5	26	32	18

Draw a histogram to represent this information.
\includegraphics[max width=\textwidth, alt={}, center]{9f0f0e3c-7baf-42eb-a4fb-9ce61922c3cd-10_1488_1493_785_365}
State which class interval contains the lower quartile and which class interval contains the upper quartile. Hence find the greatest possible value of the interquartile range.
Calculate an estimate for the mean number of incorrect notes.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE S1 2021 November Q3

3 The times taken, in minutes, by 360 employees at a large company to travel from home to work are summarised in the following table.

Time, $t$ minutes	$0 \leqslant t < 5$	$5 \leqslant t < 10$	$10 \leqslant t < 20$	$20 \leqslant t < 30$	$30 \leqslant t < 50$
Frequency	23	102	135	76	24

Draw a histogram to represent this information.
\includegraphics[max width=\textwidth, alt={}, center]{217c5a58-2966-4b86-b3b6-9d1676d2979c-04_1198_1200_836_516}
Calculate an estimate of the mean time taken by an employee to travel to work.

CAIE S1 2023 November Q4

4 The times, to the nearest minute, of 150 athletes taking part in a charity run are recorded. The results are summarised in the table.

Time in minutes	$101 - 120$	$121 - 130$	$131 - 135$	$136 - 145$	$146 - 160$
Frequency	18	48	34	32	18

Draw a histogram to represent this information.
\includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-08_1493_1397_936_415}
Calculate estimates for the mean and standard deviation of the times taken by the athletes.

CAIE S1 2008 June Q5

5 As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest 0.1 hour. The results are displayed in the following table.

Time spent $( t$ hours $)$	$0.1 \leqslant t \leqslant 0.5$	$0.6 \leqslant t \leqslant 1.0$	$1.1 \leqslant t \leqslant 2.0$	$2.1 \leqslant t \leqslant 3.0$	$3.1 \leqslant t \leqslant 4.5$
Frequency	11	15	18	30	21

Draw, on graph paper, a histogram to illustrate this information.
Calculate an estimate of the mean time spent on their Mathematics homework by members of this year group.

CAIE S1 2014 June Q7

7 A typing test is taken by 111 people. The numbers of typing errors they make in the test are summarised in the table below.

Number of typing errors	$1 - 5$	$6 - 20$	$21 - 35$	$36 - 60$	$61 - 80$
Frequency	24	9	21	15	42

Draw a histogram on graph paper to represent this information.
Calculate an estimate of the mean number of typing errors for these 111 people.
State which class contains the lower quartile and which class contains the upper quartile. Hence find the least possible value of the interquartile range.

CAIE S1 2015 June Q2

2 The table summarises the lengths in centimetres of 104 dragonflies.

Length $( \mathrm { cm } )$	$2.0 - 3.5$	$3.5 - 4.5$	$4.5 - 5.5$	$5.5 - 7.0$	$7.0 - 9.0$
Frequency	8	25	28	31	12

State which class contains the upper quartile.
Draw a histogram, on graph paper, to represent the data.

CAIE S1 2018 June Q5

5 The lengths, $t$ minutes, of 242 phone calls made by a family over a period of 1 week are summarised in the frequency table below.

Length of phone

call $( t$ minutes $)$

$0 < t \leqslant 1$

$1 < t \leqslant 2$

$2 < t \leqslant 5$

$5 < t \leqslant 10$

$10 < t \leqslant 30$

Frequency

14

46

102

$a$

40

Find the value of $a$.
Calculate an estimate of the mean length of these phone calls.
On the grid, draw a histogram to illustrate the data in the table.
\includegraphics[max width=\textwidth, alt={}, center]{a813e127-d116-411c-88ec-2443fdbc9391-07_2002_1513_486_356}

CAIE S1 2007 November Q5

5 The arrival times of 204 trains were noted and the number of minutes, $t$, that each train was late was recorded. The results are summarised in the table.

Number of minutes late $( t )$	$- 2 \leqslant t < 0$	$0 \leqslant t < 2$	$2 \leqslant t < 4$	$4 \leqslant t < 6$	$6 \leqslant t < 10$
Number of trains	43	51	69	22	19

Explain what $- 2 \leqslant t < 0$ means about the arrival times of trains.
Draw a cumulative frequency graph, and from it estimate the median and the interquartile range of the number of minutes late of these trains.

CAIE S1 2009 November Q6

6 The following table gives the marks, out of 75, in a pure mathematics examination taken by 234 students.

Marks	$1 - 20$	$21 - 30$	$31 - 40$	$41 - 50$	$51 - 60$	$61 - 75$
Frequency	40	34	56	54	29	21

Draw a histogram on graph paper to represent these results.
Calculate estimates of the mean mark and the standard deviation.

CAIE S1 2015 November Q6

6 The heights to the nearest metre of 134 office buildings in a certain city are summarised in the table below.

Height (m)	$21 - 40$	$41 - 45$	$46 - 50$	$51 - 60$	$61 - 80$
Frequency	18	15	21	52	28

Draw a histogram on graph paper to illustrate the data.
Calculate estimates of the mean and standard deviation of these heights.

CAIE S1 2016 November Q5

1 marks

5 The tables summarise the heights, $h \mathrm {~cm}$, of 60 girls and 60 boys.

Height of girls (cm)	$140 < h \leqslant 150$	$150 < h \leqslant 160$	$160 < h \leqslant 170$	$170 < h \leqslant 180$	$180 < h \leqslant 190$
Frequency	12	21	17	10	0

Height of boys $( \mathrm { cm } )$	$140 < h \leqslant 150$	$150 < h \leqslant 160$	$160 < h \leqslant 170$	$170 < h \leqslant 180$	$180 < h \leqslant 190$
Frequency	0	20	23	12	5

On graph paper, using the same set of axes, draw two cumulative frequency graphs to illustrate the data.
On a school trip the students have to enter a cave which is 165 cm high. Use your graph to estimate the percentage of the girls who will be unable to stand upright.
[0pt]
The students are asked to compare the heights of the girls and the boys. State one advantage of using a pair of box-and-whisker plots instead of the cumulative frequency graphs to do this. [1]

CAIE S1 2019 November Q3

3 The speeds, in $\mathrm { km } \mathrm { h } ^ { - 1 }$, of 90 cars as they passed a certain marker on a road were recorded, correct to the nearest $\mathrm { km } \mathrm { h } ^ { - 1 }$. The results are summarised in the following table.

Speed $\left( \mathrm { km } \mathrm { h } ^ { - 1 } \right)$	$10 - 29$	$30 - 39$	$40 - 49$	$50 - 59$	$60 - 89$
Frequency	10	24	30	14	12

On the grid, draw a histogram to illustrate the data in the table.
\includegraphics[max width=\textwidth, alt={}, center]{5307cf3d-3d3a-441a-83d7-4adad917e168-04_1594_1198_657_516}
Calculate an estimate for the mean speed of these 90 cars as they pass the marker.

OCR MEI S1 2005 June Q2

2 Answer part (i) of this question on the insert provided.
A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.

Length

$( x$ miles $)$

$0 < x \leqslant 1$

$1 < x \leqslant 2$

$2 < x \leqslant 3$

$3 < x \leqslant 4$

$4 < x \leqslant 6$

$6 < x \leqslant 10$

Number of

journeys

38

30

21

14

9

8

On the insert, draw a cumulative frequency diagram to illustrate the data.
Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
State the type of skewness of the distribution of the data.

OCR MEI S1 Q4

4 The incomes of a sample of 918 households on an island are given in the table below.

Income

$( x$ thousand pounds $)$

$0 \leqslant x \leqslant 20$

$20 < x \leqslant 40$

$40 < x \leqslant 60$

$60 < x \leqslant 100$

$100 < x \leqslant 200$

Frequency

238

365

142

128

45

Draw a histogram to illustrate the data.
Calculate an estimate of the mean income.
Calculate an estimate of the standard deviation of the incomes.
Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation.
The incomes were converted into another currency using the formula $y = 1.15 x$. Calculate estimates of the mean and variance of the incomes in the new currency.

OCR MEI S1 Q2

2 The lifetimes in hours of 90 components are summarised in the table.

Lifetime $( x$ hours $)$	$0 < x \leqslant 20$	$20 < x \leqslant 30$	$30 < x \leqslant 50$	$50 < x \leqslant 65$	$65 < x \leqslant 100$
Frequency	24	13	14	21	18

Draw a histogram to illustrate these data.
In which class interval does the median lie? Justify your answer.

OCR MEI S1 Q4

4 The frequency table below shows the distance travelled by 1200 visitors to a particular UK tourist destination in August 2008.

Distance $( d$ miles $)$	$0 \leqslant d < 50$	$50 \leqslant d < 100$	$100 \leqslant d < 200$	$200 \leqslant d < 400$
Frequency	360	400	307	133

Draw a histogram on graph paper to illustrate these data.
Calculate an estimate of the median distance.

OCR MEI S1 Q5

5 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.

Time $( t$ minutes $)$	$0 \leqslant t < 5$	$5 \leqslant t < 10$	$10 \leqslant t < 20$	$20 \leqslant t < 30$	$30 \leqslant t < 40$	$40 \leqslant t < 60$
Frequency	34	153	188	73	27	5

Illustrate these data by means of a histogram.
Identify the type of skewness of the distribution.

OCR MEI S1 Q6

6 The weights, $w$ grams, of a random sample of 60 carrots of variety A are summarised in the table below.

Weight	$30 \leqslant w < 50$	$50 \leqslant w < 60$	$60 \leqslant w < 70$	$70 \leqslant w < 80$	$80 \leqslant w < 90$
Frequency	11	10	18	14	7

Draw a histogram to illustrate these data.
Calculate estimates of the mean and standard deviation of $w$.
Use your answers to part (ii) to investigate whether there are any outliers. The weights, $x$ grams, of a random sample of 50 carrots of variety B are summarised as follows. $$n = 50 \quad \sum x = 3624.5 \quad \sum x ^ { 2 } = 265416$$
Calculate the mean and standard deviation of $x$.
Compare the central tendency and variation of the weights of varieties A and B .

OCR MEI S1 Q6

6 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.

Time $( t$ minutes $)$	$0 \leqslant t < 5$	$5 \leqslant t < 10$	$10 \leqslant t < 20$	$20 \leqslant t < 30$	$30 \leqslant t < 40$	$40 \leqslant t < 60$
Frequency	34	153	188	73	27	5

Illustrate these data by means of a histogram.
Identify the type of skewness of the distribution.

Length \(( \mathrm { cm } )\)	\(2.0 - 3.5\)	\(3.5 - 4.5\)	\(4.5 - 5.5\)	\(5.5 - 7.0\)	\(7.0 - 9.0\)
Frequency	8	25	28	31	12

Height of boys \(( \mathrm { cm } )\)	\(140 < h \leqslant 150\)	\(150 < h \leqslant 160\)	\(160 < h \leqslant 170\)	\(170 < h \leqslant 180\)	\(180 < h \leqslant 190\)
Frequency	0	20	23	12	5

Time \(( t\) minutes \()\)	\(0 \leqslant t < 5\)	\(5 \leqslant t < 10\)	\(10 \leqslant t < 20\)	\(20 \leqslant t < 30\)	\(30 \leqslant t < 40\)	\(40 \leqslant t < 60\)
Frequency	34	153	188	73	27	5

Time \(( t\) minutes \()\)	\(0 \leqslant t < 5\)	\(5 \leqslant t < 10\)	\(10 \leqslant t < 20\)	\(20 \leqslant t < 30\)	\(30 \leqslant t < 40\)	\(40 \leqslant t < 60\)
Frequency	34	153	188	73	27	5

Time \(( t\) seconds \()\)	\(0 \leqslant t < 10\)	\(10 \leqslant t < 20\)	\(20 \leqslant t < 40\)	\(40 \leqslant t < 60\)	\(60 \leqslant t < 100\)
Number of players	16	54	78	32	20

Population	\(100 - 800\)	\(900 - 1200\)	\(1300 - 2000\)	\(2100 - 3200\)	\(3300 - 4800\)
Number of villages	8	12	50	48	32

Height \(( h \mathrm {~cm} )\)	\(130 \leqslant h < 150\)	\(150 \leqslant h < 160\)	\(160 \leqslant h < 170\)	\(170 \leqslant h < 175\)	\(175 \leqslant h < 195\)
Frequency	16	32	76	64	12

Time taken, in minutes	\(1 - 30\)	\(31 - 45\)	\(46 - 65\)	\(66 - 75\)	\(76 - 100\)
Frequency	21	30	68	86	45

Number of incorrect notes	\(1 - 5\)	\(6 - 10\)	\(11 - 20\)	\(21 - 40\)	\(41 - 70\)
Frequency	10	5	26	32	18

Time, \(t\) minutes	\(0 \leqslant t < 5\)	\(5 \leqslant t < 10\)	\(10 \leqslant t < 20\)	\(20 \leqslant t < 30\)	\(30 \leqslant t < 50\)
Frequency	23	102	135	76	24

Time in minutes	\(101 - 120\)	\(121 - 130\)	\(131 - 135\)	\(136 - 145\)	\(146 - 160\)
Frequency	18	48	34	32	18

Time spent \(( t\) hours \()\)	\(0.1 \leqslant t \leqslant 0.5\)	\(0.6 \leqslant t \leqslant 1.0\)	\(1.1 \leqslant t \leqslant 2.0\)	\(2.1 \leqslant t \leqslant 3.0\)	\(3.1 \leqslant t \leqslant 4.5\)
Frequency	11	15	18	30	21

Number of typing errors	\(1 - 5\)	\(6 - 20\)	\(21 - 35\)	\(36 - 60\)	\(61 - 80\)
Frequency	24	9	21	15	42

Number of minutes late \(( t )\)	\(- 2 \leqslant t < 0\)	\(0 \leqslant t < 2\)	\(2 \leqslant t < 4\)	\(4 \leqslant t < 6\)	\(6 \leqslant t < 10\)
Number of trains	43	51	69	22	19

Speed \(\left( \mathrm { km } \mathrm { h } ^ { - 1 } \right)\)	\(10 - 29\)	\(30 - 39\)	\(40 - 49\)	\(50 - 59\)	\(60 - 89\)
Frequency	10	24	30	14	12

Marks	\(1 - 20\)	\(21 - 30\)	\(31 - 40\)	\(41 - 50\)	\(51 - 60\)	\(61 - 75\)
Frequency	40	34	56	54	29	21

Height (m)	\(21 - 40\)	\(41 - 45\)	\(46 - 50\)	\(51 - 60\)	\(61 - 80\)
Frequency	18	15	21	52	28

Lifetime \(( x\) hours \()\)	\(0 < x \leqslant 20\)	\(20 < x \leqslant 30\)	\(30 < x \leqslant 50\)	\(50 < x \leqslant 65\)	\(65 < x \leqslant 100\)
Frequency	24	13	14	21	18

Distance \(( d\) miles \()\)	\(0 \leqslant d < 50\)	\(50 \leqslant d < 100\)	\(100 \leqslant d < 200\)	\(200 \leqslant d < 400\)
Frequency	360	400	307	133

Weight	\(30 \leqslant w < 50\)	\(50 \leqslant w < 60\)	\(60 \leqslant w < 70\)	\(70 \leqslant w < 80\)	\(80 \leqslant w < 90\)
Frequency	11	10	18	14	7