Draw cumulative frequency graph from frequency table (equal class widths)

Questions providing a frequency table with equal or near-equal class widths where students must calculate cumulative frequencies and draw the graph.

8 questions · Easy -1.2

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CAIE S1 2009 June Q6
14 marks Moderate -0.8
6 During January the numbers of people entering a store during the first hour after opening were as follows.
Time after opening,
\(x\) minutes
Frequency
Cumulative
frequency
\(0 < x \leqslant 10\)210210
\(10 < x \leqslant 20\)134344
\(20 < x \leqslant 30\)78422
\(30 < x \leqslant 40\)72\(a\)
\(40 < x \leqslant 60\)\(b\)540
  1. Find the values of \(a\) and \(b\).
  2. Draw a cumulative frequency graph to represent this information. Take a scale of 2 cm for 10 minutes on the horizontal axis and 2 cm for 50 people on the vertical axis.
  3. Use your graph to estimate the median time after opening that people entered the store.
  4. Calculate estimates of the mean, \(m\) minutes, and standard deviation, \(s\) minutes, of the time after opening that people entered the store.
  5. Use your graph to estimate the number of people entering the store between ( \(m - \frac { 1 } { 2 } s\) ) and \(\left( m + \frac { 1 } { 2 } s \right)\) minutes after opening.
CAIE S1 2007 November Q5
8 marks Easy -1.8
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 trains4351692219
  1. Explain what \(- 2 \leqslant t < 0\) means about the arrival times of trains.
  2. 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 2016 November Q5
8 marks Easy -1.3
5 The tables summarise the heights, \(h \mathrm {~cm}\), of 60 girls and 60 boys.
Height of girls (cm)\(140 < h \leqslant 150\)\(150 < h \leqslant 160\)\(160 < h \leqslant 170\)\(170 < h \leqslant 180\)\(180 < h \leqslant 190\)
Frequency122117100
Height of boys \(( \mathrm { cm } )\)\(140 < h \leqslant 150\)\(150 < h \leqslant 160\)\(160 < h \leqslant 170\)\(170 < h \leqslant 180\)\(180 < h \leqslant 190\)
Frequency02023125
  1. On graph paper, using the same set of axes, draw two cumulative frequency graphs to illustrate the data.
  2. 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]
  3. 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]
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 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 Q2
8 marks Easy -1.8
2 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.
OCR MEI S1 2011 June Q8
18 marks Moderate -0.3
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 \(\leqslant x <\) 9.39.3 \(< x \leqslant\) 9.59.5 \(< x \leqslant\) 9.79.7 \(< x \leqslant\) 9.99.9 \(< x \leqslant\) 10.1
Frequency5715167
  1. Draw a cumulative frequency diagram to illustrate these data. [5]
  2. Use the diagram to estimate the median and interquartile range of the data. [3]
  3. Show that there are no outliers in the sample. [3]
  4. Three of these 50 sacks are selected at random. Find the probability that
    1. in all three, the heating quality \(x\) is more than 9.5, [3]
    2. in at least two, the heating quality \(x\) is more than 9.5. [4]
OCR MEI S1 2014 June Q1
8 marks Easy -1.3
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. [5]
  2. Use your diagram to estimate the median and interquartile range of the data. [3]