Draw histogram then find median/quartiles from cumulative frequency

Questions that ask students to draw a histogram from a frequency table with unequal class widths, then calculate quartiles, median, or percentiles by constructing and using a cumulative frequency graph or table.

9 questions · Moderate -0.5

2.02b Histogram: area represents frequency
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OCR MEI S1 Q4
8 marks Moderate -0.8
4 The frequency table below shows the distance travelled by 1200 visitors to a particular UK tourist destination in August 2008.
Distance \(( d\) miles \()\)\(0 \leqslant d < 50\)\(50 \leqslant d < 100\)\(100 \leqslant d < 200\)\(200 \leqslant d < 400\)
Frequency360400307133
  1. Draw a histogram on graph paper to illustrate these data.
  2. Calculate an estimate of the median distance.
Edexcel S1 2004 January Q5
18 marks Moderate -0.3
5. The values of daily sales, to the nearest \(\pounds\), taken at a newsagents last year are summarised in the table below.
SalesNumber of days
\(1 - 200\)166
\(201 - 400\)100
\(401 - 700\)59
\(701 - 1000\)30
\(1001 - 1500\)5
  1. Draw a histogram to represent these data.
  2. Use interpolation to estimate the median and inter-quartile range of daily sales.
  3. Estimate the mean and the standard deviation of these data. The newsagent wants to compare last year's sales with other years.
  4. State whether the newsagent should use the median and the inter-quartile range or the mean and the standard deviation to compare daily sales. Give a reason for your answer.
    (2)
OCR MEI S1 2009 June Q5
8 marks Moderate -0.8
5 The frequency table below shows the distance travelled by 1200 visitors to a particular UK tourist destination in August 2008.
Distance \(( d\) miles \()\)\(0 \leqslant d < 50\)\(50 \leqslant d < 100\)\(100 \leqslant d < 200\)\(200 \leqslant d < 400\)
Frequency360400307133
  1. Draw a histogram on graph paper to illustrate these data.
  2. Calculate an estimate of the median distance.
Edexcel S1 2022 June Q3
14 marks Moderate -0.3
  1. Gill buys a bag of logs to use in her stove. The lengths, \(l \mathrm {~cm}\), of the 88 logs in the bag are summarised in the table below.
Length \(( \boldsymbol { l } )\)Frequency \(( \boldsymbol { f } )\)
\(15 < l \leqslant 20\)19
\(20 < l \leqslant 25\)35
\(25 < l \leqslant 27\)16
\(27 < l \leqslant 30\)15
\(30 < l \leqslant 40\)3
A histogram is drawn to represent these data.
The bar representing logs with length \(27 < l \leqslant 30\) has a width of 1.5 cm and a height of 4 cm .
  1. Calculate the width and height of the bar representing log lengths of \(20 < l \leqslant 25\)
  2. Use linear interpolation to estimate the median of \(l\) The maximum length of log Gill can use in her stove is 26 cm .
    Gill estimates, using linear interpolation, that \(x\) logs from the bag will fit into her stove.
  3. Show that \(x = 62\) Gill randomly selects 4 logs from the bag.
  4. Using \(x = 62\), find the probability that all 4 logs will fit into her stove. The weights, \(W\) grams, of the logs in the bag are coded using \(y = 0.5 w - 255\) and summarised by $$n = 88 \quad \sum y = 924 \quad \sum y ^ { 2 } = 12862$$
  5. Calculate
    1. the mean of \(W\)
    2. the variance of \(W\)
Edexcel S1 2001 January Q5
17 marks Moderate -0.3
5. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 200 motorists were delayed by roadworks on a stretch of motorway.
Delay (mins)Number of motorists
\(4 - 6\)15
\(7 - 8\)28
949
1053
\(11 - 12\)30
\(13 - 15\)15
\(16 - 20\)10
  1. Using graph paper represent these data by a histogram.
  2. Give a reason to justify the use of a histogram to represent these data.
  3. Use interpolation to estimate the median of this distribution.
  4. Calculate an estimate of the mean and an estimate of the standard deviation of these data. One coefficient of skewness is given by $$\frac { 3 ( \text { mean - median } ) } { \text { standard deviation } } .$$
  5. Evaluate this coefficient for the above data.
  6. Explain why the normal distribution may not be suitable to model the number of minutes that motorists are delayed by these roadworks.
Edexcel S1 2002 June Q6
14 marks Moderate -0.3
6. The labelling on bags of garden compost indicates that the bags weigh 20 kg . The weights of a random sample of 50 bags are summarised in the table below.
Weight in kgFrequency
14.6-14.81
14.8-18.00
18.0-18.55
18.5-20.06
20.0-20.222
20.2-20.415
20.4-21.01
  1. On graph paper, draw a histogram of these data.
  2. Using the coding \(y = 10\) (weight in \(\mathrm { kg } - 14\) ), find an estimate for the mean and standard deviation of the weight of a bag of compost.
    [0pt] [Use \(\Sigma f y ^ { 2 } = 171\) 503.75]
  3. Using linear interpolation, estimate the median. The company that produces the bags of compost wants to improve the accuracy of the labelling. The company decides to put the average weight in kg on each bag.
  4. Write down which of these averages you would recommend the company to use. Give a reason for your answer.
Edexcel S1 2010 June Q5
14 marks Moderate -0.8
5. A teacher selects a random sample of 56 students and records, to the nearest hour, the time spent watching television in a particular week.
Hours\(1 - 10\)\(11 - 20\)\(21 - 25\)\(26 - 30\)\(31 - 40\)\(41 - 59\)
Frequency615111383
Mid-point5.515.52850
  1. Find the mid-points of the 21-25 hour and 31-40 hour groups. A histogram was drawn to represent these data. The \(11 - 20\) group was represented by a bar of width 4 cm and height 6 cm .
  2. Find the width and height of the 26-30 group.
  3. Estimate the mean and standard deviation of the time spent watching television by these students.
  4. Use linear interpolation to estimate the median length of time spent watching television by these students. The teacher estimated the lower quartile and the upper quartile of the time spent watching television to be 15.8 and 29.3 respectively.
  5. State, giving a reason, the skewness of these data.
Edexcel S1 Q6
13 marks Moderate -0.3
6. A cinema recorded the number of people at each showing of each film during a one-week period. The results are summarised in the table below.
Number of peopleNumber of showings
1-4036
41-6020
61-8033
81-10024
101-15036
151-20039
201-30052
  1. Draw a histogram on graph paper to illustrate these data.
  2. Calculate estimates of the median and quartiles of these data.
  3. Use your answers to part (b) to show that the data is positively skewed.
Edexcel S1 Q5
Moderate -0.3
5. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 200 motorists were delayed by roadworks on a stretch of motorway.
Delay (mins)Number of motorists
\(4 - 6\)15
\(7 - 8\)28
949
1053
\(11 - 12\)30
\(13 - 15\)15
\(16 - 20\)10
  1. Using graph paper represent these data by a histogram.
  2. Give a reason to justify the use of a histogram to represent these data.
  3. Use interpolation to estimate the median of this distribution.
  4. Calculate an estimate of the mean and an estimate of the standard deviation of these data. One coefficient of skewness is given by $$\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } } .$$
  5. Evaluate this coefficient for the above data.
  6. Explain why the normal distribution may not be suitable to model the number of minutes that motorists are delayed by these roadworks.