Calculate frequency density from frequency

Questions where students must calculate frequency density values given frequencies and class widths, typically to complete a table before drawing a histogram.

18 questions

CAIE S1 2004 November Q2
2 The lengths of cars travelling on a car ferry are noted. The data are summarised in the following table.
Length of car \(( x\) metres \()\)FrequencyFrequency density
\(2.80 \leqslant x < 3.00\)1785
\(3.00 \leqslant x < 3.10\)24240
\(3.10 \leqslant x < 3.20\)19190
\(3.20 \leqslant x < 3.40\)8\(a\)
  1. Find the value of \(a\).
  2. Draw a histogram on graph paper to represent the data.
  3. Find the probability that a randomly chosen car on the ferry is less than 3.20 m in length.
CAIE S1 2015 November Q3
3 Robert has a part-time job delivering newspapers. On a number of days he noted the time, correct to the nearest minute, that it took him to do his job. Robert used his results to draw up the following table; two of the values in the table are denoted by \(a\) and \(b\).
Time \(( t\) minutes \()\)\(60 - 62\)\(63 - 64\)\(65 - 67\)\(68 - 71\)
Frequency (number of days)396\(b\)
Frequency density1\(a\)21.5
  1. Find the values of \(a\) and \(b\).
  2. On graph paper, draw a histogram to represent Robert's times.
CAIE S1 Specimen Q3
3 Robert has a part-time job delivering newspapers. On a number of days he noted the time, correct to the nearest minute, that it took him to do his job. Robert used his results to draw up the following table; two of the values in the table are denoted by \(a\) and \(b\).
Time \(( t\) minutes \()\)\(60 - 62\)\(63 - 64\)\(65 - 67\)\(68 - 71\)
Frequency (number of days)396\(b\)
Frequency density1\(a\)21.5
  1. Find the values of \(a\) and \(b\).
  2. Draw a histogram to represent Robert's times.
    \includegraphics[max width=\textwidth, alt={}]{34ae4f06-d485-4138-82d8-902b70f08995-04_206_100_1516_441}"□\(\_\_\_\_\)□□ □\includegraphics[max width=\textwidth, alt={}]{34ae4f06-d485-4138-82d8-902b70f08995-04_204_28_1518_1197}\(\_\_\_\_\)□
OCR S1 2009 June Q5
5 The diameters of 100 pebbles were measured. The measurements rounded to the nearest millimetre, \(x\), are summarised in the table.
\(x\)\(10 \leqslant x \leqslant 19\)\(20 \leqslant x \leqslant 24\)\(25 \leqslant x \leqslant 29\)\(30 \leqslant x \leqslant 49\)
Number of stones25222924
These data are to be presented on a statistical diagram.
  1. For a histogram, find the frequency density of the \(10 \leqslant x \leqslant 19\) class.
  2. For a cumulative frequency graph, state the coordinates of the first two points that should be plotted.
  3. Why is it not possible to draw an exact box-and-whisker plot to illustrate the data?
OCR S1 2012 January Q5
5 At a certain resort the number of hours of sunshine, measured to the nearest hour, was recorded on each of 21 days. The results are summarised in the table.
Hours of sunshine0\(1 - 3\)\(4 - 6\)\(7 - 9\)\(10 - 15\)
Number of days06942
The diagram shows part of a histogram to illustrate the data. The scale on the frequency density axis is 2 cm to 1 unit.
\includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-3_944_1778_699_148}
  1. (a) Calculate the frequency density of the \(1 - 3\) class.
    (b) Fred wishes to draw the block for the 10 - 15 class on the same diagram. Calculate the height, in centimetres, of this block.
  2. A cumulative frequency graph is to be drawn. Write down the coordinates of the first two points that should be plotted. You are not asked to draw the graph.
  3. (a) Calculate estimates of the mean and standard deviation of the number of hours of sunshine.
    (b) Explain why your answers are only estimates.
OCR S1 2011 June Q4
4 The table shows information about the time, \(t\) minutes correct to the nearest minute, taken by 50 people to complete a race.
Time (minutes)\(t \leqslant 27\)\(28 \leqslant t \leqslant 30\)\(31 \leqslant t \leqslant 35\)\(36 \leqslant t \leqslant 45\)\(46 \leqslant t \leqslant 60\)\(t \geqslant 61\)
Number of people04281440
  1. In a histogram illustrating the data, the height of the block for the \(31 \leqslant t \leqslant 35\) class is 5.6 cm . Find the height of the block for the \(28 \leqslant t \leqslant 30\) class. (There is no need to draw the histogram.)
  2. The data in the table are used to estimate the median time. State, with a reason, whether the estimated median time is more than 33 minutes, less than 33 minutes or equal to 33 minutes.
  3. Calculate estimates of the mean and standard deviation of the data.
  4. It was found that the winner's time had been incorrectly recorded and that it was actually less than 27 minutes 30 seconds. State whether each of the following will increase, decrease or remain the same:
    (a) the mean,
    (b) the standard deviation,
    (c) the median,
    (d) the interquartile range.
OCR MEI S1 2016 June Q6
6 An online store has a total of 930 different types of women's running shoe on sale. The prices in pounds of the types of women's running shoe are summarised in the table below.
Price \(( \pounds x )\)\(10 \leqslant x \leqslant 40\)\(40 < x \leqslant 50\)\(50 < x \leqslant 60\)\(60 < x \leqslant 80\)\(80 < x \leqslant 200\)
Frequency147109182317175
  1. Calculate estimates of the mean and standard deviation of the shoe prices.
  2. Calculate an estimate of the percentage of types of shoe that cost at least \(\pounds 100\).
  3. Draw a histogram to illustrate the data. The corresponding histogram below shows the prices in pounds of the 990 types of men's running shoe on sale at the same online store.
    \includegraphics[max width=\textwidth, alt={}, center]{aff0c5b2-011b-49a0-bf05-6d905f890eba-4_643_1192_340_440}
  4. State the type of skewness shown by the histogram for men's running shoes.
  5. Martin is investigating the percentage of types of shoe on sale at the store that cost more than \(\pounds 100\). He believes that this percentage is greater for men's shoes than for women's shoes. Estimate the percentage for men's shoes and comment on whether you can be certain which percentage is higher.
  6. You are given that the mean and standard deviation of the prices of men's running shoes are \(\pounds 68.83\) and \(\pounds 42.93\) respectively. Compare the central tendency and variation of the prices of men's and women's running shoes at the store.
OCR MEI AS Paper 2 2018 June Q2
2 Doug has a list of times taken by competitors in a 'fun run'. He has grouped the data and calculated the frequency densities in order to draw a histogram to represent the information. Some of the data are presented in Fig. 2. \begin{table}[h]
Time in minutes\(15 -\)\(20 -\)\(25 -\)\(35 -\)\(45 - 60\)
Number of runners12235971
Frequency density2.45.97.11.4
\captionsetup{labelformat=empty} \caption{Fig. 2}
\end{table}
  1. Write down the missing values in the copy of Fig. 2 in the Printed Answer Booklet.
  2. Doug labels the horizontal axis on the histogram 'time in minutes' and the vertical axis 'number of minutes per runner'. State which one of these labels is incorrect and write down a correct version.
    \(3 \quad P\) and \(Q\) are consecutive odd positive integers such that \(P > Q\).
    Prove that \(P ^ { 2 } - Q ^ { 2 }\) is a multiple of 8 .
Edexcel S1 2024 June Q3
  1. The lengths, \(x \mathrm {~mm}\), of 50 pebbles are summarised in the table below.
LengthFrequency
\(20 \leqslant x < 30\)2
\(30 \leqslant x < 32\)16
\(32 \leqslant x < 36\)20
\(36 \leqslant x < 40\)8
\(40 \leqslant x < 45\)3
\(45 \leqslant x < 50\)1
A histogram is drawn to represent these data.
The bar representing the class \(32 \leqslant x < 36\) is 2.5 cm wide and 7.5 cm tall.
  1. Calculate the width and the height of the bar representing the class \(30 \leqslant x < 32\)
  2. Using linear interpolation, estimate the median of \(x\) The weight, \(w\) grams, of each of the 50 pebbles is coded using \(10 y = w - 20\) These coded data are summarised by $$\sum y = 104 \quad \sum y ^ { 2 } = 233.54$$
  3. Show that the mean of \(w\) is 40.8
  4. Calculate the standard deviation of \(w\) The weight of a pebble recorded as 40.8 grams is added to the sample.
  5. Without carrying out any further calculations, state, giving a reason, what effect this would have on the value of
    1. the mean of \(w\)
    2. the standard deviation of \(w\)
Edexcel S1 2022 October Q1
  1. The stem lengths of a sample of 120 tulips are recorded in the grouped frequency table below.
Stem length (cm)Frequency
\(40 \leqslant x < 42\)12
\(42 \leqslant x < 45\)18
\(45 \leqslant x < 50\)23
\(50 \leqslant x < 55\)35
\(55 \leqslant x < 58\)24
\(58 \leqslant x < 60\)8
A histogram is drawn to represent these data.
The area of the bar representing the \(40 \leqslant x < 42\) class is \(16.5 \mathrm {~cm} ^ { 2 }\)
  1. Calculate the exact area of the bar representing the \(42 \leqslant x < 45\) class. The height of the tallest bar in the histogram is 10 cm .
  2. Find the exact height of the second tallest bar.
    \(Q _ { 1 }\) for these data is 45 cm .
  3. Use linear interpolation to find an estimate for
    1. \(Q _ { 2 }\)
    2. the interquartile range. One measure of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
  4. By calculating this measure, describe the skewness of these data.
Edexcel S1 2005 June Q2
2. The following table summarises the distances, to the nearest km , that 134 examiners travelled to attend a meeting in London.
Distance (km)Number of examiners
41-454
46-5019
51-6053
61-7037
71-9015
91-1506
  1. Give a reason to justify the use of a histogram to represent these data.
  2. Calculate the frequency densities needed to draw a histogram for these data.
    (DO NOT DRAW THE HISTOGRAM)
  3. Use interpolation to estimate the median \(Q _ { 2 }\), the lower quartile \(Q _ { 1 }\), and the upper quartile \(Q _ { 3 }\) of these data. The mid-point of each class is represented by \(x\) and the corresponding frequency by \(f\). Calculations then give the following values $$\Sigma f _ { x } = 8379.5 \quad \text { and } \quad \Sigma f _ { x ^ { 2 } } = 557489.75$$
  4. Calculate an estimate of the mean and an estimate of the standard deviation for these data. One coefficient of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
  5. Evaluate this coefficient and comment on the skewness of these data.
  6. Give another justification of your comment in part (e).
Edexcel S1 2009 June Q3
3. The variable \(x\) was measured to the nearest whole number. Forty observations are given in the table below.
\(x\)\(10 - 15\)\(16 - 18\)\(19 -\)
Frequency15916
A histogram was drawn and the bar representing the \(10 - 15\) class has a width of 2 cm and a height of 5 cm . For the \(16 - 18\) class find
  1. the width,
  2. the height
    of the bar representing this class.
Edexcel S1 Q1
  1. A histogram is to be drawn to represent the following grouped continuous data:
Group\(0 - 10\)\(10 - 20\)\(20 - 25\)\(25 - 30\)\(30 - 50\)\(50 - 100\)
Frequency\(2 x\)\(3 x\)\(5 x\)\(6 x\)\(2 x\)\(x\)
The ' \(10 - 20\) ' bar has height 6 cm and width 4 cm . Calculate
  1. the height of the ' \(20 - 25\) ' bar,
  2. the total area under the histogram.
Edexcel S1 Q5
5. The number of patients attending a hospital trauma clinic each day was recorded over several months, giving the data in the table below.
Number of patients\(10 - 19\)\(20 - 29\)\(30 - 34\)\(35 - 39\)\(40 - 44\)\(45 - 49\)\(50 - 69\)
Frequency218243027145
These data are represented by a histogram.
Given that the bar representing the 20-29 group is 2 cm wide and 7.2 cm high,
  1. calculate the dimensions of the bars representing the groups
    1. 30-34
    2. 50-69
  2. Use linear interpolation to estimate the median and quartiles of these data. The lowest and highest numbers of patients recorded were 14 and 67 respectively.
  3. Represent these data with a boxplot drawn on graph paper and describe the skewness of the distribution.
SPS SPS SM Statistics 2023 January Q1
1.
  1. Joseph drew a histogram to show information about one Local Authority. He used data from the "Age structure by LA 2011" tab in the large data set. The table shows an extract from the data that he used.
    Age group0 to 4
    Frequency2143
    Joseph used a scale of \(1 \mathrm {~cm} = 1000\) units on the frequency density axis. Calculate the height of the histogram block for the 0 to 4 class.
  2. Magdalene wishes to draw a statistical diagram to illustrate some of the data from the "Method of travel by LA 2011" tab in the large data set. State why she cannot draw a histogram.
SPS SPS SM Statistics 2025 January Q4
4. The table shows information about the time, \(t\) minutes correct to the nearest minute, taken by 50 people to complete a race.
Time (minutes)\(t \leqslant 27\)\(28 \leqslant t \leqslant 30\)\(31 \leqslant t \leqslant 35\)\(36 \leqslant t \leqslant 45\)\(46 \leqslant t \leqslant 60\)\(t \geqslant 61\)
Number of people04281440
  1. In a histogram illustrating the data, the height of the block for the \(31 \leqslant t \leqslant 35\) class is 5.6 cm . Find the height of the block for the \(28 \leqslant t \leqslant 30\) class. (There is no need to draw the histogram.)
  2. The data in the table are used to estimate the median time. State, with a reason, whether the estimated median time is more than 33 minutes, less than 33 minutes or equal to 33 minutes.
  3. Calculate estimates of the mean and standard deviation of the data.
    [0pt] [BLANK PAGE]
AQA AS Paper 2 2023 June Q13
13 The table below shows the frequencies for a set of data from a continuous variable \(X\)
\(\boldsymbol { x }\)Frequency
\(11 < x \leq 21\)7
\(21 < x \leq 24\)9
\(24 < x \leq 42\)36
\(42 < x \leq 50\)18
A histogram is drawn to represent this data.
Find the frequency density of the bar in the histogram representing the class \(24 < x \leq 42\) Circle your answer. 2183670
Edexcel Paper 3 Specimen Q1
  1. The number of hours of sunshine each day, \(y\), for the month of July at Heathrow are summarised in the table below.
Hours\(0 \leqslant y < 5\)\(5 \leqslant y < 8\)\(8 \leqslant y < 11\)\(11 \leqslant y < 12\)\(12 \leqslant y < 14\)
Frequency126832
A histogram was drawn to represent these data. The \(8 \leqslant y < 11\) group was represented by a bar of width 1.5 cm and height 8 cm .
  1. Find the width and the height of the \(0 \leqslant y < 5\) group.
  2. Use your calculator to estimate the mean and the standard deviation of the number of hours of sunshine each day, for the month of July at Heathrow.
    Give your answers to 3 significant figures. The mean and standard deviation for the number of hours of daily sunshine for the same month in Hurn are 5.98 hours and 4.12 hours respectably.
    Thomas believes that the further south you are the more consistent should be the number of hours of daily sunshine.
  3. State, giving a reason, whether or not the calculations in part (b) support Thomas' belief.
  4. Estimate the number of days in July at Heathrow where the number of hours of sunshine is more than 1 standard deviation above the mean. Helen models the number of hours of sunshine each day, for the month of July at Heathrow by \(\mathrm { N } \left( 6.6,3.7 ^ { 2 } \right)\).
  5. Use Helen's model to predict the number of days in July at Heathrow when the number of hours of sunshine is more than 1 standard deviation above the mean.
  6. Use your answers to part (d) and part (e) to comment on the suitability of Helen's model.