Cumulative frequency graph construction then interpretation

A frequency or cumulative frequency table is provided, the question requires drawing/constructing a cumulative frequency graph first, then using it to estimate median, quartiles, or other measures.

7 questions

CAIE S1 2020 March Q7
7 Helen measures the lengths of 150 fish of a certain species in a large pond. These lengths, correct to the nearest centimetre, are summarised in the following table.
Length (cm)\(0 - 9\)\(10 - 14\)\(15 - 19\)\(20 - 30\)
Frequency15486621
  1. Draw a cumulative frequency graph to illustrate the data.
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  2. 40\% of these fish have a length of \(d \mathrm {~cm}\) or more. Use your graph to estimate the value of \(d\).
    The mean length of these 150 fish is 15.295 cm .
  3. Calculate an estimate for the variance of the lengths of the fish.
    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 2022 November Q3
3 The times, \(t\) minutes, taken to complete a walking challenge by 250 members of a club are summarised in the table.
Time taken \(( t\) minutes \()\)\(t \leqslant 20\)\(t \leqslant 30\)\(t \leqslant 35\)\(t \leqslant 40\)\(t \leqslant 50\)\(t \leqslant 60\)
Cumulative frequency3266112178228250
  1. Draw a cumulative frequency graph to illustrate the data.
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  2. Use your graph to estimate the 60th percentile of the data.
    It is given that an estimate for the mean time taken to complete the challenge by these 250 members is 34.4 minutes.
  3. Calculate an estimate for the standard deviation of the times taken to complete the challenge by these 250 members.
CAIE S1 2016 March Q4
4 A survey was made of the journey times of 63 people who cycle to work in a certain town. The results are summarised in the following cumulative frequency table.
Journey time (minutes)\(\leqslant 10\)\(\leqslant 25\)\(\leqslant 45\)\(\leqslant 60\)\(\leqslant 80\)
Cumulative frequency018505963
  1. State how many journey times were between 25 and 45 minutes.
  2. Draw a histogram on graph paper to represent the data.
  3. Calculate an estimate of the mean journey time.
CAIE S1 2012 November Q3
3 The table summarises the times that 112 people took to travel to work on a particular day.
Time to travel to
work \(( t\) minutes \()\)
\(0 < t \leqslant 10\)\(10 < t \leqslant 15\)\(15 < t \leqslant 20\)\(20 < t \leqslant 25\)\(25 < t \leqslant 40\)\(40 < t \leqslant 60\)
Frequency191228221813
  1. State which time interval in the table contains the median and which time interval contains the upper quartile.
  2. On graph paper, draw a histogram to represent the data.
  3. Calculate an estimate of the mean time to travel to work.
CAIE S1 2014 November Q6
6 On a certain day in spring, the heights of 200 daffodils are measured, correct to the nearest centimetre. The frequency distribution is given below.
Height \(( \mathrm { cm } )\)\(4 - 10\)\(11 - 15\)\(16 - 20\)\(21 - 25\)\(26 - 30\)
Frequency2232784028
  1. Draw a cumulative frequency graph to illustrate the data.
  2. \(28 \%\) of these daffodils are of height \(h \mathrm {~cm}\) or more. Estimate \(h\).
  3. You are given that the estimate of the mean height of these daffodils, calculated from the table, is 18.39 cm . Calculate an estimate of the standard deviation of the heights of these daffodils.
OCR MEI S1 Q3
3 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.
  1. On the insert, draw a cumulative frequency diagram to illustrate the data.
  2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
  3. State the type of skewness of the distribution of the data.
Edexcel S1 2011 January Q5
5. On a randomly chosen day, each of the 32 students in a class recorded the time, \(t\) minutes to the nearest minute, they spent on their homework. The data for the class is summarised in the following table.
Time, \(t\)Number of students
10-192
20-294
30-398
40-4911
50-695
70-792
  1. Use interpolation to estimate the value of the median. Given that $$\sum t = 1414 \quad \text { and } \quad \sum t ^ { 2 } = 69378$$
  2. find the mean and the standard deviation of the times spent by the students on their homework.
  3. Comment on the skewness of the distribution of the times spent by the students on their homework. Give a reason for your answer.