2.02a Interpret single variable data: tables and diagrams

209 questions

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CAIE S1 2019 November Q5
9 marks Easy -1.8
5 Last Saturday, 200 drivers entering a car park were asked the time, in minutes, that it had taken them to travel from home to the car park. The results are summarised in the following cumulative frequency table.
Time \(( t\) minutes \()\)\(t \leqslant 10\)\(t \leqslant 20\)\(t \leqslant 30\)\(t \leqslant 50\)\(t \leqslant 70\)\(t \leqslant 90\)
Cumulative frequency1650106146176200
  1. On the grid, draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{06f6c8dd-170c-4e94-a960-0c649a7363a1-08_1198_1399_735_415}
  2. Use your graph to estimate the median of the data.
  3. For 80 of the drivers, the time taken was at least \(T\) minutes. Use your graph to estimate the value of \(T\).
  4. Calculate an estimate of the mean time taken by all 200 drivers to travel to the car park.
CAIE S1 2010 November Q5
8 marks Moderate -0.8
5 The following histogram illustrates the distribution of times, in minutes, that some students spent taking a shower. \includegraphics[max width=\textwidth, alt={}, center]{ec425eaf-8afc-4671-9ef3-ba2477b884ef-3_1031_1326_372_406}
  1. Copy and complete the following frequency table for the data.
    Time \(( t\) minutes \()\)\(2 < t \leqslant 4\)\(4 < t \leqslant 6\)\(6 < t \leqslant 7\)\(7 < t \leqslant 8\)\(8 < t \leqslant 10\)\(10 < t \leqslant 16\)
    Frequency
  2. Calculate an estimate of the mean time to take a shower.
  3. Two of these students are chosen at random. Find the probability that exactly one takes between 7 and 10 minutes to take a shower.
CAIE S1 2011 November Q4
6 marks Easy -1.8
4 The marks of the pupils in a certain class in a History examination are as follows. $$\begin{array} { l l l l l l l l l l l l l } 28 & 33 & 55 & 38 & 42 & 39 & 27 & 48 & 51 & 37 & 57 & 49 & 33 \end{array}$$ The marks of the pupils in a Physics examination are summarised as follows.
Lower quartile: 28 , Median: 39, Upper quartile: 67.
The lowest mark was 17 and the highest mark was 74 .
  1. Draw box-and-whisker plots in a single diagram on graph paper to illustrate the marks for History and Physics.
  2. State one difference, which can be seen from the diagram, between the marks for History and Physics.
CAIE S1 2011 November Q4
8 marks Easy -1.3
4 The weights of 220 sausages are summarised in the following table.
Weight (grams)\(< 20\)\(< 30\)\(< 40\)\(< 45\)\(< 50\)\(< 60\)\(< 70\)
Cumulative frequency02050100160210220
  1. State which interval the median weight lies in.
  2. Find the smallest possible value and the largest possible value for the interquartile range.
  3. State how many sausages weighed between 50 g and 60 g .
  4. On graph paper, draw a histogram to represent the weights of the sausages.
CAIE S1 2011 November Q5
9 marks Moderate -0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{b72ace6b-d3d4-401d-bffe-403c9127f2a8-3_1157_1001_258_573} The cumulative frequency graph shows the annual salaries, in thousands of euros, of a random sample of 500 adults with jobs, in France. It has been plotted using grouped data. You may assume that the lowest salary is 5000 euros and the highest salary is 80000 euros.
  1. On graph paper, draw a box-and-whisker plot to illustrate these salaries.
  2. Comment on the salaries of the people in this sample.
  3. An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
    1. How high must a salary be in order to be classified as an outlier?
    2. Show that none of the salaries is low enough to be classified as an outlier.
CAIE S1 2012 November Q4
7 marks Moderate -0.8
4 Prices in dollars of 11 caravans in a showroom are as follows. \(\begin{array} { l l l l l l l l l l l } 16800 & 18500 & 17700 & 14300 & 15500 & 15300 & 16100 & 16800 & 17300 & 15400 & 16400 \end{array}\)
  1. Represent these prices by a stem-and-leaf diagram.
  2. Write down the lower quartile of the prices of the caravans in the showroom.
  3. 3 different caravans in the showroom are chosen at random and their prices are noted. Find the probability that 2 of these prices are more than the median and 1 is less than the lower quartile.
OCR S1 2005 January Q2
6 marks Easy -1.8
2 The back-to-back stem-and-leaf diagram below shows the number of hours of television watched per week by each of 15 boys and 15 girls. $$\begin{aligned} & \text { Boys Girls } \\ & \left. \begin{array} { r r r r r r r r | r r r r r r r r r r r r r } & 677664 & 4 & 3 & 0 & 0 & 5 & 5 & 6 & 677888 \end{array} \right\} \end{aligned}$$ Key: 4 | 2 | 2 means a boy who watched 24 hours and a girl who watched 22 hours of television per week.
  1. Find the median and the quartiles of the results for the boys.
  2. Give a reason why the median might be preferred to the mean in using an average to compare the two data sets.
  3. State one advantage, and one disadvantage, of using stem-and-leaf diagrams rather than box-andwhisker plots to represent the data.
OCR S1 Specimen Q6
11 marks Moderate -0.5
6 \includegraphics[max width=\textwidth, alt={}, center]{2fb25fc5-0445-44fa-a23e-647d14b1a376-3_803_1180_1018_413} The diagram shows the cumulative frequency graphs for the marks scored by the candidates in an examination. The 2000 candidates each took two papers; the upper curve shows the distribution of marks on paper 1 and the lower curve shows the distribution on paper 2. The maximum mark on each paper was 100.
  1. Use the diagram to estimate the median mark for each of paper 1 and paper 2.
  2. State with a reason which of the two papers you think was the easier one.
  3. To achieve grade A on paper 1 candidates had to score 66 marks out of 100. What mark on paper 2 gives equal proportions of candidates achieving grade A on the two papers? What is this proportion?
  4. The candidates' marks for the two papers could also be illustrated by means of a pair of box-and whisker plots. Give two brief comments comparing the usefulness of cumulative frequency graphs and box-and-whisker plots for representing the data.
OCR MEI S1 2005 January Q7
12 marks Easy -1.2
7 The cumulative frequency graph below illustrates the distances that 176 children live from their primary school. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Distance from school} \includegraphics[alt={},max width=\textwidth]{b35b2b3b-0d26-4a35-b4d2-110bf270d5dc-4_1073_1571_580_340}
\end{figure}
  1. Use the graph to estimate, to the nearest 10 metres,
    (A) the median distance from school,
    (B) the lower quartile, upper quartile and interquartile range.
  2. Draw a box and whisker plot to illustrate the data. The graph on page 4 used the following grouped data.
    Distance (metres)20040060080010001200
    Cumulative frequency2064118150169176
  3. Copy and complete the grouped frequency table below describing the same data.
    Distance ( \(d\) metres)Frequency
    \(0 < d \leqslant 200\)20
    \(200 < d \leqslant 400\)
  4. Hence estimate the mean distance these children live from school. It is subsequently found that none of the 176 children lives within 100 metres of the school.
  5. Calculate the revised estimate of the mean distance.
  6. Describe what change needs to be made to the cumulative frequency graph.
OCR MEI S1 2006 January Q1
6 marks Easy -1.8
1 The times taken, in minutes, by 80 people to complete a crossword puzzle are summarised by the box and whisker plot below. \includegraphics[max width=\textwidth, alt={}, center]{acb05873-e441-4b95-9732-6ebd5ae79fa6-2_147_848_507_612}
  1. Write down the range and the interquartile range of the times.
  2. Determine whether any of the times can be regarded as outliers.
  3. Describe the shape of the distribution of the times.
OCR MEI S1 2007 January Q2
7 marks Easy -1.8
2 The numbers of absentees per day from Mrs Smith's reception class over a period of 50 days are summarised below.
Number of absentees0123456\(> 6\)
Frequency8151183410
  1. Illustrate these data by means of a vertical line chart.
  2. Calculate the mean and root mean square deviation of these data.
  3. There are 30 children in Mrs Smith's class altogether. Find the mean and root mean square deviation of the number of children who are present during the 50 days.
OCR MEI S1 2006 June Q1
8 marks Easy -1.8
1 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.
Number correct1234567
Frequency1233475
  1. Draw a vertical line chart to illustrate the data.
  2. State the type of skewness shown by your diagram.
  3. Calculate the mean and the mean squared deviation of the data.
  4. How many correct answers would George need to average over the next 6 days if he is to achieve an average of 5 correct answers for all 31 days of January?
OCR MEI S1 2008 June Q7
20 marks Moderate -0.8
7 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{be764df3-ff20-415d-9c5c-10edabf350de-5_814_1383_349_379} Data sourced from the 2001 Census, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}
  1. State the type of skewness shown by the distribution.
  2. Use the histogram to estimate the number of people aged under 25.
  3. The table below shows the cumulative frequency distribution.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (A) Use the histogram to find the value of \(a\).
    (B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.
    Age ( \(x\) years)\(0 \leqslant x < 20\)\(20 \leqslant x < 30\)\(30 \leqslant x < 40\)\(40 \leqslant x < 50\)\(50 \leqslant x < 65\)\(65 \leqslant x < 100\)
    Frequency (thousands)1120650770590680610
  4. Illustrate these data by means of a histogram.
  5. Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
  6. The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
    The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.
OCR MEI S1 Q1
8 marks Easy -1.2
1 At a tourist information office the numbers of people seeking information each hour over the course of a 12-hour day are shown below. $$\begin{array} { l l l l l l l l l l l l } 6 & 25 & 38 & 39 & 31 & 18 & 35 & 31 & 33 & 15 & 21 & 28 \end{array}$$
  1. Construct a sorted stem and leaf diagram to represent these data.
  2. State the type of skewness suggested by your stem and leaf diagram.
  3. For these data find the median, the mean and the mode. Comment on the usefulness of the mode in this case.
OCR MEI S1 Q3
8 marks Easy -1.8
3 The ages, \(x\) years, of the senior members of a running club are summarised in the table below.
Age \(( x )\)\(20 \leqslant x < 30\)\(30 \leqslant x < 40\)\(40 \leqslant x < 50\)\(50 \leqslant x < 60\)\(60 \leqslant x < 70\)\(70 \leqslant x < 80\)\(80 \leqslant x < 90\)
Frequency10304223951
  1. Draw a cumulative frequency diagram to illustrate the data.
  2. Use your diagram to estimate the median and interquartile range of the data.
OCR MEI S1 Q1
18 marks Moderate -0.3
1 The heights \(x \mathrm {~cm}\) of 100 boys in Year 7 at a school are summarised in the table below.
Height\(125 \leqslant x \leqslant 140\)\(140 < x \leqslant 145\)\(145 < x \leqslant 150\)\(150 < x \leqslant 160\)\(160 < x \leqslant 170\)
Frequency252924184
  1. Estimate the number of boys who have heights of at least 155 cm .
  2. Calculate an estimate of the median height of the 100 boys.
  3. Draw a histogram to illustrate the data. The histogram below shows the heights of 100 girls in Year 7 at the same school. \includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-1_868_1361_1015_381}
  4. How many more girls than boys had heights exceeding 160 cm ?
  5. Calculate an estimate of the mean height of the 100 girls.
OCR MEI S1 Q3
19 marks Moderate -0.3
3 The birth weights of 200 lambs from crossbred sheep are illustrated by the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-3_919_1144_430_476}
  1. Estimate the percentage of lambs with birth weight over 6 kg .
  2. Estimate the median and interquartile range of the data.
  3. Use your answers to part (ii) to show that there are very few, if any, outliers. Comment briefly on whether any outliers should be disregarded in analysing these data. The box and whisker plot shows the birth weights of 100 lambs from Welsh Mountain sheep. \includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-3_321_1610_1818_293}
  4. Use appropriate measures to compare briefly the central tendencies and variations of the weights of the two types of lamb.
  5. The weight of the largest Welsh Mountain lamb was originally recorded as 6.5 kg , but then corrected. If this error had not been corrected, how would this have affected your answers to part (iv)? Briefly explain your answer.
  6. One lamb of each type is selected at random. Estimate the probability that the birth weight of both lambs is at least 3.9 kg .
OCR MEI S1 Q3
18 marks Moderate -0.8
3 The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.
Heating quality \(( x )\)\(9.1 \leqslant x \leqslant 9.3\)\(9.3 < x \leqslant 9.5\)\(9.5 < x \leqslant 9.7\)\(9.7 < x \leqslant 9.9\)\(9.9 < x \leqslant 10.1\)
Frequency5715167
  1. Draw a cumulative frequency diagram to illustrate these data.
  2. Use the diagram to estimate the median and interquartile range of the data.
  3. Show that there are no outliers in the sample.
  4. Three of these 50 sacks are selected at random. Find the probability that
    (A) in all three, the heating quality \(x\) is more than 9.5 , \(( B )\) in at least two, the heating quality \(x\) is more than 9.5.
OCR MEI S1 Q3
16 marks Moderate -0.3
3 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{dfb0acd8-d84b-4291-a811-a68f4942794b-2_1266_1546_487_335}
  1. Use the diagram to estimate the median and interquartile range of the data.
  2. Use your answers to part (i) to estimate the number of outliers in the sample.
  3. Should these outliers be excluded from any further analysis? Briefly explain your answer.
  4. Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected. \(12 \%\) of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
  5. Find the probability that
    (A) exactly 2 of these 17 babies require special care,
    (B) more than 2 of the 17 babies require special care.
  6. On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.
OCR MEI S1 Q3
18 marks Moderate -0.3
3 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{79f1015b-7c3d-4576-8d5b-e9fc89d8a49e-2_1266_1546_487_335}
  1. Use the diagram to estimate the median and interquartile range of the data.
  2. Use your answers to part (i) to estimate the number of outliers in the sample.
  3. Should these outliers be excluded from any further analysis? Briefly explain your answer.
  4. Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected. \(12 \%\) of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
  5. Find the probability that
    (A) exactly 2 of these 17 babies require special care,
    (B) more than 2 of the 17 babies require special care.
  6. On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.
OCR MEI S1 Q2
7 marks Easy -1.8
2 The mean daily maximum temperatures at a research station over a 12 -month period, measured to the nearest degree Celsius, are given below.
JanFebMarAprMayJunJulAugSepOctNovDec
8152529313134363426158
  1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of \(0,10 , \ldots\).
  2. Write down the median of these data.
  3. The mean of these data is 24.3. Would the mean or the median be a better measure of central tendency of the data? Briefly explain your answer.
OCR MEI S1 Q3
3 marks Easy -1.2
3 The stem and leaf diagram shows the weights, rounded to the nearest 10 grams, of 25 female iguanas.
839
935666899
100223469
112478
12345
132
Key: 11 | 2 represents 1120 grams
  1. Find the mode and the median of the data.
  2. Identify the type of skewness of the distribution.
OCR MEI S1 Q4
8 marks Easy -1.2
4 A camera records the speeds in miles per hour of 15 vehicles on a motorway. The speeds are given below. $$\begin{array} { l l l l l l l l l l l l l l l } 73 & 67 & 75 & 64 & 52 & 63 & 75 & 81 & 77 & 72 & 68 & 74 & 79 & 72 & 71 \end{array}$$
  1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of \(50,60 , \ldots\).
  2. Write down the median and midrange of the data.
  3. Which of the median and midrange would you recommend to measure the central tendency of the data? Briefly explain your answer.
OCR MEI S1 Q5
5 marks Easy -1.8
5 In a traffic survey, the number of people in each car passing the survey point is recorded. The results are given in the following frequency table.
Number of people1234
Frequency5031165
  1. Write down the median and mode of these data.
  2. Draw a vertical line diagram for these data.
  3. State the type of skewness of the distribution.
OCR MEI S1 Q7
20 marks Moderate -0.8
7 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{aabf9d8b-5f91-4a3b-bcf8-e46cb45127c4-4_805_1372_392_401} Data sourced from he 2001 Census, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}
  1. State the type of skewness shown by the distribution.
  2. Use the histogram to estimate the number of people aged under 25.
  3. The table below shows the cumulative frequency distribution.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (A) Use the histogram to find the value of \(a\).
    (B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.
    Age ( \(x\) years)\(0 \leqslant x < 20\)\(20 \leqslant x < 30\)\(30 \leqslant x < 40\)\(40 \leqslant x < 50\)\(50 \leqslant x < 65\)\(65 \leqslant x < 100\)
    Frequency (thousands)1120650770590680610
  4. Illustrate these data by means of a histogram.
  5. Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
  6. The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
    The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.
    [0pt] [4]