Estimate mean and standard deviation from histogram

Questions that provide a histogram (rather than a frequency table) and require the student to first extract frequency data from the histogram before calculating estimates of mean and/or standard deviation.

9 questions

CAIE S1 2013 November Q4
4 The following histogram summarises the times, in minutes, taken by 190 people to complete a race.
\includegraphics[max width=\textwidth, alt={}, center]{df246a50-157b-49f7-bba0-f9b86960b8b9-2_1210_1125_1251_513}
  1. Show that 75 people took between 200 and 250 minutes to complete the race.
  2. Calculate estimates of the mean and standard deviation of the times of the 190 people.
  3. Explain why your answers to part (ii) are estimates.
OCR MEI S1 2008 June Q7
7 The histogram shows the age distribution of people living in Inner London in 2001.
\includegraphics[max width=\textwidth, alt={}, center]{be764df3-ff20-415d-9c5c-10edabf350de-5_814_1383_349_379} Data sourced from the 2001 Census, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}
  1. State the type of skewness shown by the distribution.
  2. Use the histogram to estimate the number of people aged under 25.
  3. The table below shows the cumulative frequency distribution.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (A) Use the histogram to find the value of \(a\).
    (B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.
    Age ( \(x\) years)\(0 \leqslant x < 20\)\(20 \leqslant x < 30\)\(30 \leqslant x < 40\)\(40 \leqslant x < 50\)\(50 \leqslant x < 65\)\(65 \leqslant x < 100\)
    Frequency (thousands)1120650770590680610
  4. Illustrate these data by means of a histogram.
  5. Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
  6. The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
    The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.
OCR MEI S1 Q3
3 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears.
\includegraphics[max width=\textwidth, alt={}, center]{56f1bd5c-4b45-4e36-a324-e7e0edbb5bdd-2_825_1634_467_295}
  1. Calculate the number of pears which are between 90 and 100 mm long.
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
  3. Calculate an estimate of the standard deviation.
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
  5. Name the type of skewness of the distribution.
  6. Illustrate the data using a cumulative frequency diagram.
OCR MEI S1 Q5
5 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm , of these pears.
\includegraphics[max width=\textwidth, alt={}, center]{056d3e9a-088d-4c97-9546-7cecb59b8727-3_815_1628_505_304}
  1. Calculate the number of pears which are between 90 and 100 mm long.
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
  3. Calculate an estimate of the standard deviation.
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
  5. Name the type of skewness of the distribution.
  6. Illustrate the data using a cumulative frequency diagram.
OCR MEI S1 Q5
5 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm , of these pears.
\includegraphics[max width=\textwidth, alt={}, center]{99c502aa-2c9f-461d-9dc0-ed55e3df32a2-3_815_1628_505_304}
  1. Calculate the number of pears which are between 90 and 100 mm long.
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
  3. Calculate an estimate of the standard deviation.
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
  5. Name the type of skewness of the distribution.
  6. Illustrate the data using a cumulative frequency diagram.
OCR MEI S1 2010 January Q7
7 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears.
\includegraphics[max width=\textwidth, alt={}, center]{2f39c509-5429-4193-9526-15fb45b18a38-4_837_1651_466_246}
  1. Calculate the number of pears which are between 90 and 100 mm long.
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
  3. Calculate an estimate of the standard deviation.
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
  5. Name the type of skewness of the distribution.
  6. Illustrate the data using a cumulative frequency diagram.
OCR PURE Q10
10 The masses of a random sample of 120 boulders in a certain area were recorded. The results are summarized in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{e42b1a99-c3ca-4ce1-becd-cd346aec757e-08_734_1693_342_178}
  1. Calculate the number of boulders with masses between 60 and 65 kg .
    1. Use midpoints to find estimates of the mean and standard deviation of the masses of the boulders in the sample.
    2. Explain why your answers are only estimates.
  3. Use your answers to part (b)(i) to determine an estimate of the number of outliers, if any, in the distribution.
  4. Give one advantage of using a histogram rather than a pie chart in this context.
Edexcel S1 2023 June Q1
  1. The histogram shows the distances, in km , that 274 people travel to work.
    \includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-02_1272_1582_296_175}
Given that 60 of these people travel between 10 km and 20 km to work, estimate
  1. the number of people who travel between 22 km and 45 km to work,
  2. the median distance travelled to work by these 274 people,
  3. the mean distance travelled to work by these 274 people.
SPS SPS SM Statistics 2024 April Q1
1. The masses of a random sample of 120 boulders in a certain area were recorded. The results are summarized in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{d59e9fea-31cb-4b6d-b1d6-f09f912b5b37-04_773_1765_402_148}
  1. Calculate the number of boulders with masses between 60 and 65 kg .
    1. Use midpoints to find estimates of the mean and standard deviation of the masses of the boulders in the sample.
    2. Explain why your answers are only estimates.
  3. Use your answers to part (b)(i) to determine an estimate of the number of outliers, if any, in the distribution.
  4. Give one advantage of using a histogram rather than a pie chart in this context.
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