Draw histogram then estimate mean/standard deviation

Questions that ask students to draw a histogram from a frequency table with unequal class widths, then calculate estimates of the mean and/or standard deviation using midpoints and frequencies.

11 questions · Moderate -0.7

2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation
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CAIE S1 2021 June Q5
8 marks Moderate -0.3
5 The times taken by 200 players to solve a computer puzzle are summarised in the following table.
Time \(( t\) seconds \()\)\(0 \leqslant t < 10\)\(10 \leqslant t < 20\)\(20 \leqslant t < 40\)\(40 \leqslant t < 60\)\(60 \leqslant t < 100\)
Number of players1654783220
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{1a27e2ca-9be5-48a0-a1aa-01844573f4d4-08_1397_1198_808_516}
  2. Calculate an estimate of the mean time taken by these 200 players.
  3. Find the greatest possible value of the interquartile range of these times.
CAIE S1 2021 November Q3
6 marks Moderate -0.8
3 The times taken, in minutes, by 360 employees at a large company to travel from home to work are summarised in the following table.
Time, \(t\) minutes\(0 \leqslant t < 5\)\(5 \leqslant t < 10\)\(10 \leqslant t < 20\)\(20 \leqslant t < 30\)\(30 \leqslant t < 50\)
Frequency231021357624
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{217c5a58-2966-4b86-b3b6-9d1676d2979c-04_1198_1200_836_516}
  2. Calculate an estimate of the mean time taken by an employee to travel to work.
CAIE S1 2023 November Q4
9 marks Moderate -0.8
4 The times, to the nearest minute, of 150 athletes taking part in a charity run are recorded. The results are summarised in the table.
Time in minutes\(101 - 120\)\(121 - 130\)\(131 - 135\)\(136 - 145\)\(146 - 160\)
Frequency1848343218
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-08_1493_1397_936_415}
  2. Calculate estimates for the mean and standard deviation of the times taken by the athletes.
CAIE S1 2008 June Q5
8 marks Moderate -0.8
5 As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest 0.1 hour. The results are displayed in the following table.
Time spent \(( t\) hours \()\)\(0.1 \leqslant t \leqslant 0.5\)\(0.6 \leqslant t \leqslant 1.0\)\(1.1 \leqslant t \leqslant 2.0\)\(2.1 \leqslant t \leqslant 3.0\)\(3.1 \leqslant t \leqslant 4.5\)
Frequency1115183021
  1. Draw, on graph paper, a histogram to illustrate this information.
  2. Calculate an estimate of the mean time spent on their Mathematics homework by members of this year group.
CAIE S1 2018 June Q5
7 marks Easy -1.2
5 The lengths, \(t\) minutes, of 242 phone calls made by a family over a period of 1 week are summarised in the frequency table below.
Length of phone
call \(( t\) minutes \()\)
\(0 < t \leqslant 1\)\(1 < t \leqslant 2\)\(2 < t \leqslant 5\)\(5 < t \leqslant 10\)\(10 < t \leqslant 30\)
Frequency1446102\(a\)40
  1. Find the value of \(a\).
  2. Calculate an estimate of the mean length of these phone calls.
  3. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{a813e127-d116-411c-88ec-2443fdbc9391-07_2002_1513_486_356}
CAIE S1 2009 November Q6
9 marks Moderate -0.8
6 The following table gives the marks, out of 75, in a pure mathematics examination taken by 234 students.
Marks\(1 - 20\)\(21 - 30\)\(31 - 40\)\(41 - 50\)\(51 - 60\)\(61 - 75\)
Frequency403456542921
  1. Draw a histogram on graph paper to represent these results.
  2. Calculate estimates of the mean mark and the standard deviation.
CAIE S1 2015 November Q6
9 marks Moderate -0.8
6 The heights to the nearest metre of 134 office buildings in a certain city are summarised in the table below.
Height (m)\(21 - 40\)\(41 - 45\)\(46 - 50\)\(51 - 60\)\(61 - 80\)
Frequency1815215228
  1. Draw a histogram on graph paper to illustrate the data.
  2. Calculate estimates of the mean and standard deviation of these heights.
CAIE S1 2019 November Q3
6 marks Moderate -0.8
3 The speeds, in \(\mathrm { km } \mathrm { h } ^ { - 1 }\), of 90 cars as they passed a certain marker on a road were recorded, correct to the nearest \(\mathrm { km } \mathrm { h } ^ { - 1 }\). The results are summarised in the following table.
Speed \(\left( \mathrm { km } \mathrm { h } ^ { - 1 } \right)\)\(10 - 29\)\(30 - 39\)\(40 - 49\)\(50 - 59\)\(60 - 89\)
Frequency1024301412
  1. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{5307cf3d-3d3a-441a-83d7-4adad917e168-04_1594_1198_657_516}
  2. Calculate an estimate for the mean speed of these 90 cars as they pass the marker.
OCR MEI S1 Q4
19 marks Moderate -0.3
4 The incomes of a sample of 918 households on an island are given in the table below.
Income
\(( x\) thousand pounds \()\)
\(0 \leqslant x \leqslant 20\)\(20 < x \leqslant 40\)\(40 < x \leqslant 60\)\(60 < x \leqslant 100\)\(100 < x \leqslant 200\)
Frequency23836514212845
  1. Draw a histogram to illustrate the data.
  2. Calculate an estimate of the mean income.
  3. Calculate an estimate of the standard deviation of the incomes.
  4. Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation.
  5. The incomes were converted into another currency using the formula \(y = 1.15 x\). Calculate estimates of the mean and variance of the incomes in the new currency.
OCR MEI S1 2012 June Q6
18 marks Moderate -0.3
6 The engine sizes \(x \mathrm {~cm} ^ { 3 }\) of a sample of 80 cars are summarised in the table below.
Engine size \(x\)\(500 \leqslant x \leqslant 1000\)\(1000 < x \leqslant 1500\)\(1500 < x \leqslant 2000\)\(2000 < x \leqslant 3000\)\(3000 < x \leqslant 5000\)
Frequency72226187
  1. Draw a histogram to illustrate the distribution.
  2. A student claims that the midrange is \(2750 \mathrm {~cm} ^ { 3 }\). Discuss briefly whether he is likely to be correct.
  3. Calculate estimates of the mean and standard deviation of the engine sizes. Explain why your answers are only estimates.
  4. Hence investigate whether there are any outliers in the sample.
  5. A vehicle duty of \(\pounds 1000\) is proposed for all new cars with engine size greater than \(2000 \mathrm {~cm} ^ { 3 }\). Assuming that this sample of cars is representative of all new cars in Britain and that there are 2.5 million new cars registered in Britain each year, calculate an estimate of the total amount of money that this vehicle duty would raise in one year.
  6. Why in practice might your estimate in part (v) turn out to be too high?
Edexcel S1 Q6
13 marks Moderate -0.8
6. The number of people visiting a new art gallery each day is recorded over a three-month period and the results are summarised in the table below.
Number of visitorsNumber of days
400-4593
460-4798
480-49913
500-51912
520-53918
540-55911
560-5999
600-6995
  1. Draw a histogram on graph paper to illustrate these data. In order to calculate summary statistics for the data it is coded using \(y = \frac { x - 509.5 } { 10 }\), where \(x\) is the mid-point of each class.
  2. Find \(\sum\) fy. You may assume that \(\sum f y ^ { 2 } = 2041\).
  3. Using these values for \(\sum f y\) and \(\sum f y ^ { 2 }\), calculate estimates of the mean and standard deviation of the number of visitors per day.
    (6 marks)