Draw cumulative frequency graph from cumulative frequency table

Questions that provide data already in cumulative frequency form and ask students to draw the cumulative frequency graph or curve directly.

11 questions

CAIE S1 2022 June Q1
1 The time taken, \(t\) minutes, to complete a puzzle was recorded for each of 150 students. These times are summarised in the table.
Time taken \(( t\) minutes \()\)\(t \leqslant 25\)\(t \leqslant 50\)\(t \leqslant 75\)\(t \leqslant 100\)\(t \leqslant 150\)\(t \leqslant 200\)
Cumulative frequency164486104132150
  1. Draw a cumulative frequency graph to illustrate the data.
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  2. Use your graph to estimate the 20th percentile of the data.
CAIE S1 2020 November Q6
6 The times, \(t\) minutes, taken by 150 students to complete a particular challenge are summarised in the following cumulative frequency table.
Time taken \(( t\) minutes \()\)\(t \leqslant 20\)\(t \leqslant 30\)\(t \leqslant 40\)\(t \leqslant 60\)\(t \leqslant 100\)
Cumulative frequency1248106134150
  1. Draw a cumulative frequency graph to illustrate the data.
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  2. \(24 \%\) of the students take \(k\) minutes or longer to complete the challenge. Use your graph to estimate the value of \(k\).
  3. Calculate estimates of the mean and the standard deviation of the time taken to complete the challenge.
CAIE S1 2023 November Q4
6 marks
4 The weights, \(x \mathrm {~kg}\), of 120 students in a sports college are recorded. The results are summarised in the following table.
Weight \(( x \mathrm {~kg} )\)\(x \leqslant 40\)\(x \leqslant 60\)\(x \leqslant 65\)\(x \leqslant 70\)\(x \leqslant 85\)\(x \leqslant 100\)
Cumulative frequency0143860106120
  1. Draw a cumulative frequency graph to represent this information.
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  2. It is found that \(35 \%\) of the students weigh more than \(W \mathrm {~kg}\). Use your graph to estimate the value of \(W\).
  3. Calculate estimates for the mean and standard deviation of the weights of the 120 students. [6]
CAIE S1 2024 November Q3
3 The time taken, in minutes, to walk to school was recorded for 200 pupils at a certain school. These times are summarised in the following table.
Time taken
\(( t\) minutes \()\)
\(t \leqslant 15\)\(t \leqslant 25\)\(t \leqslant 30\)\(t \leqslant 40\)\(t \leqslant 50\)\(t \leqslant 70\)
Cumulative
frequency
184688140176200
  1. Draw a cumulative frequency graph to illustrate the data.
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  2. Use your graph to estimate the median and the interquartile range of the data.
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  3. Calculate an estimate for the mean value of the times taken by the 200 pupils to walk to school.
CAIE S1 2004 June Q2
2 In a recent survey, 640 people were asked about the length of time each week that they spent watching television. The median time was found to be 20 hours, and the lower and upper quartiles were 15 hours and 35 hours respectively. The least amount of time that anyone spent was 3 hours, and the greatest amount was 60 hours.
  1. On graph paper, show these results using a fully labelled cumulative frequency graph.
  2. Use your graph to estimate how many people watched more than 50 hours of television each week.
CAIE S1 2011 June Q6
6 There are 5000 schools in a certain country. The cumulative frequency table shows the number of pupils in a school and the corresponding number of schools.
Number of pupils in a school\(\leqslant 100\)\(\leqslant 150\)\(\leqslant 200\)\(\leqslant 250\)\(\leqslant 350\)\(\leqslant 450\)\(\leqslant 600\)
Cumulative frequency20080016002100410047005000
  1. Draw a cumulative frequency graph with a scale of 2 cm to 100 pupils on the horizontal axis and a scale of 2 cm to 1000 schools on the vertical axis. Use your graph to estimate the median number of pupils in a school.
  2. \(80 \%\) of the schools have more than \(n\) pupils. Estimate the value of \(n\) correct to the nearest ten.
  3. Find how many schools have between 201 and 250 (inclusive) pupils.
  4. Calculate an estimate of the mean number of pupils per school.
CAIE S1 2013 June Q6
6 The weights, \(x\) kilograms, of 144 people were recorded. The results are summarised in the cumulative frequency table below.
Weight \(( x\) kilograms \()\)\(x < 40\)\(x < 50\)\(x < 60\)\(x < 65\)\(x < 70\)\(x < 90\)
Cumulative frequency012346492144
  1. On graph paper, draw a cumulative frequency graph to represent these results.
  2. 64 people weigh more than \(c \mathrm {~kg}\). Use your graph to find the value of \(c\).
  3. Calculate estimates of the mean and standard deviation of the weights.
CAIE S1 2015 June Q6
6 Seventy samples of fertiliser were collected and the nitrogen content was measured for each sample. The cumulative frequency distribution is shown in the table below.
Nitrogen content\(\leqslant 3.5\)\(\leqslant 3.8\)\(\leqslant 4.0\)\(\leqslant 4.2\)\(\leqslant 4.5\)\(\leqslant 4.8\)
Cumulative frequency0618416270
  1. On graph paper draw a cumulative frequency graph to represent the data.
  2. Estimate the percentage of samples with a nitrogen content greater than 4.4.
  3. Estimate the median.
  4. Construct the frequency table for these results and draw a histogram on graph paper.
CAIE S1 2018 November Q6
6 The daily rainfall, \(x \mathrm {~mm}\), in a certain village is recorded on 250 consecutive days. The results are summarised in the following cumulative frequency table.
Rainfall, \(x \mathrm {~mm}\)\(x \leqslant 20\)\(x \leqslant 30\)\(x \leqslant 40\)\(x \leqslant 50\)\(x \leqslant 70\)\(x \leqslant 100\)
Cumulative frequency5294142172222250
  1. On the grid, draw a cumulative frequency graph to illustrate the data.
  2. On 100 of the days, the rainfall was \(k \mathrm {~mm}\) or more. Use your graph to estimate the value of \(k\).
  3. Calculate estimates of the mean and standard deviation of the daily rainfall in this village.
CAIE S1 2019 November Q5
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.
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  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.
Edexcel S1 Q6
6. 1000 houses were sold in a small town in a one-year period. The selling prices were as given in the following table:
Selling PriceNumber of HousesSelling PriceNumber of Houses
Up to \(\pounds 50000\)60Up to \(\pounds 150000\)642
Up to \(\pounds 75000\)227Up to \(\pounds 200000\)805
Up to \(\pounds 100000\)305Up to \(\pounds 500000\)849
Up to \(\pounds 125000\)414Up to \(\pounds 750000\)1000
  1. Name (do not draw) a suitable type of graph for illustrating this data.
  2. Use interpolation to find estimates of the median and the quartiles.
  3. Estimate the 37th percentile. Given further that the lowest price was \(\pounds 42000\) and the range of the prices was \(\pounds 690000\),
  4. draw a box plot to represent the data. Show your scale clearly. In another town the median price was \(\pounds 149000\), and the interquartile range was \(\pounds 90000\).
  5. Briefly compare the prices in the two towns using this information.