2.02f Measures of average and spread

6 The weights, \(x\) kilograms, of 144 people were recorded. The results are summarised in the cumulative frequency table below.

1. On graph paper, draw a cumulative frequency graph to represent these results.
2. 64 people weigh more than \(c \mathrm {~kg}\). Use your graph to find the value of \(c\).
3. Calculate estimates of the mean and standard deviation of the weights.

7 A typing test is taken by 111 people. The numbers of typing errors they make in the test are summarised in the table below.

1. Draw a histogram on graph paper to represent this information.
2. Calculate an estimate of the mean number of typing errors for these 111 people.
3. State which class contains the lower quartile and which class contains the upper quartile. Hence find the least possible value of the interquartile range.

6 The times taken by 57 athletes to run 100 metres are summarised in the following cumulative frequency table.

1. State how many athletes ran 100 metres in a time between 10.5 and 11.0 seconds.
2. Draw a histogram on graph paper to represent the times taken by these athletes to run 100 metres.
3. Calculate estimates of the mean and variance of the times taken by these athletes.

2 The table summarises the lengths in centimetres of 104 dragonflies.

1. State which class contains the upper quartile.
2. Draw a histogram, on graph paper, to represent the data.

7 The amounts spent by 160 shoppers at a supermarket are summarised in the following table.

1. Draw a cumulative frequency graph of this distribution.
2. Estimate the median and the interquartile range of the amount spent.
3. Estimate the number of shoppers who spent more than \(\\) 115$.
4. Calculate an estimate of the mean amount spent.

5 The following are the maximum daily wind speeds in kilometres per hour for the first two weeks in April for two towns, Bronlea and Rogate.

1. Draw a back-to-back stem-and-leaf diagram to represent this information.
2. Write down the median of the maximum wind speeds for Bronlea and find the interquartile range for Rogate.
3. Use your diagram to make one comparison between the maximum wind speeds in the two towns.

2 A group of children played a computer game which measured their time in seconds to perform a certain task. A summary of the times taken by girls and boys in the group is shown below.

1. On graph paper, draw two box-and-whisker plots in a single diagram to illustrate the times taken by girls and boys to perform this task.
2. State two comparisons of the times taken by girls and boys.

1 Rani and Diksha go shopping for clothes.

1. Rani buys 4 identical vests, 3 identical sweaters and 1 coat. Each vest costs \(\\) 5.50\( and the coat costs \)\\( 90\). The mean cost of Rani's 8 items is \(\\) 29\(. Find the cost of a sweater.
2. Diksha buys 1 hat and 4 identical shirts. The mean cost of Diksha's 5 items is \)\\( 26\) and the standard deviation is \(\\) 0\(. Explain how you can tell that Diksha spends \)\\( 104\) on shirts.

2 Anabel measured the lengths, in centimetres, of 200 caterpillars. Her results are illustrated in the cumulative frequency graph below. \includegraphics[max width=\textwidth, alt={}, center]{184a04ac-4396-4a0f-8fa8-ab11a4b6df39-03_1173_1195_356_466}

1. Estimate the median and the interquartile range of the lengths.
2. Estimate how many caterpillars had a length of between 2 and 3.5 cm .
3. 6\% of caterpillars were of length \(l\) centimetres or more. Estimate \(l\).

7 The following histogram represents the lengths of worms in a garden. \includegraphics[max width=\textwidth, alt={}, center]{67412184-38f6-4b37-afe3-4a149a2e0586-10_789_1195_301_466}

1. Calculate the frequencies represented by each of the four histogram columns.
2. On the grid on the next page, draw a cumulative frequency graph to represent the lengths of worms in the garden. \includegraphics[max width=\textwidth, alt={}, center]{67412184-38f6-4b37-afe3-4a149a2e0586-11_1111_1409_251_408}
3. Use your graph to estimate the median and interquartile range of the lengths of worms in the garden.
4. Calculate an estimate of the mean length of worms in the garden.
   {www.cie.org.uk} after the live examination series. }

2 In a survey 55 students were asked to record, to the nearest kilometre, the total number of kilometres they travelled to school in a particular week. The results are shown below.

1. On the grid, draw a box-and-whisker plot to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{246c92f4-7603-43ff-8533-042a4be99a69-04_512_1596_900_262} An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
2. Show that there are no outliers.

1 Each of a group of 10 boys estimates the length of a piece of string. The estimates, in centimetres, are as follows. $$\begin{array} { l l l l l l l l l l } 37 & 40 & 45 & 38 & 36 & 38 & 42 & 38 & 40 & 39 \end{array}$$

1. Find the mode.
2. Find the median and the interquartile range.

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.

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}

1 The masses in kilograms of 50 children having a medical check-up were recorded correct to the nearest kilogram. The results are shown in the table.

1. Find which class interval contains the lower quartile.
2. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{dd75fa20-fead-48d6-aff4-c5e733769f9f-02_1397_1397_1187_415}

4 Farfield Travel and Lacket Travel are two travel companies which arrange tours abroad. The numbers of holidays arranged in a certain week are recorded in the table below, together with the means and standard deviations of the prices.

1. Calculate the mean price of all 51 holidays.
2. The prices of individual holidays with Farfield Travel are denoted by \(\\) x _ { F }\( and the prices of individual holidays with Lacket Travel are denoted by \)\\( x _ { L }\). By first finding \(\Sigma x _ { F } ^ { 2 }\) and \(\Sigma x _ { L } ^ { 2 }\), find the standard deviation of the prices of all 51 holidays.

4 The Mathematics and English A-level marks of 1400 pupils all taking the same examinations are shown in the cumulative frequency graphs below. Both examinations are marked out of 100 . \includegraphics[max width=\textwidth, alt={}, center]{be6c6525-a20c-42d0-8fef-1cd254baaa76-06_1682_1246_404_445} Use suitable data from these graphs to compare the central tendency and spread of the marks in Mathematics and English.

6

1. Give one advantage and one disadvantage of using a box-and-whisker plot to represent a set of data.
2. The times in minutes taken to run a marathon were recorded for a group of 13 marathon runners and were found to be as follows. $$\begin{array} { l l l l l l l l l l l l l } 180 & 275 & 235 & 242 & 311 & 194 & 246 & 229 & 238 & 768 & 332 & 227 & 228 \end{array}$$ State which of the mean, mode or median is most suitable as a measure of central tendency for these times. Explain why the other measures are less suitable.
3. Another group of 33 people ran the same marathon and their times in minutes were as follows.

   190	203	215	246	249	253	255	254	258	260	261
   263	267	269	274	276	280	288	283	287	294	300
   307	318	327	331	336	345	351	353	360	368	375

   1. On the grid below, draw a box-and-whisker plot to illustrate the times for these 33 people. \includegraphics[max width=\textwidth, alt={}, center]{f4d040a2-6a04-49ce-98ac-8ba5c515f905-09_611_1202_1270_555}
   2. Find the interquartile range of these times.

7 The times in minutes taken by 13 pupils at each of two schools in a cross-country race are recorded in the table below.

1. Draw a back-to-back stem-and-leaf diagram to illustrate these times with Thaters School on the left.
2. Find the interquartile range of the times for pupils at Thaters School.
   The times taken by pupils at Whitefay Park School are denoted by \(x\) minutes.
3. Find the value of \(\Sigma ( x - 60 ) ^ { 2 }\).
4. It is given that \(\Sigma ( x - 60 ) = 46\). Use this result, together with your answer to part (iii), to find the variance of \(x\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 For 10 values of \(x\) the mean is 86.2 and \(\Sigma ( x - a ) = 362\). Find the value of

1. \(\Sigma x\),
2. the constant \(a\).

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.

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.

5 The weights, in kg, of the 11 members of the Dolphins swimming team and the 11 members of the Sharks swimming team are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Dolphins on the left-hand side of the diagram and Sharks on the right-hand side.
2. Find the median and interquartile range for the Dolphins.

7 The weights in kilograms of two groups of 17-year-old males from country \(P\) and country \(Q\) are displayed in the following back-to-back stem-and-leaf diagram. In the third row of the diagram, ... \(4 | 7 | 1 \ldots\) denotes weights of 74 kg for a male in country \(P\) and 71 kg for a male in country \(Q\).

1. Find the median and quartile weights for country \(Q\).
2. You are given that the lower quartile, median and upper quartile for country \(P\) are 84,94 and 98 kg respectively. On a single diagram on graph paper, draw two box-and-whisker plots of the data.
3. Make two comments on the weights of the two groups.

1 A computer can generate random numbers which are either 0 or 2 . On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers.

4 The ages, \(x\) years, of 18 people attending an evening class are summarised by the following totals: \(\Sigma x = 745 , \Sigma x ^ { 2 } = 33951\).

1. Calculate the mean and standard deviation of the ages of this group of people.
2. One person leaves the group and the mean age of the remaining 17 people is exactly 41 years. Find the age of the person who left and the standard deviation of the ages of the remaining 17 people.

4 A group of 10 married couples and 3 single men found that the mean age \(\bar { x } _ { w }\) of the 10 women was 41.2 years and the standard deviation of the women's ages was 15.1 years. For the 13 men, the mean age \(\bar { x } _ { m }\) was 46.3 years and the standard deviation was 12.7 years.

1. Find the mean age of the whole group of 23 people.
2. The individual women's ages are denoted by \(x _ { w }\) and the individual men's ages by \(x _ { m }\). By first finding \(\Sigma x _ { w } ^ { 2 }\) and \(\Sigma x _ { m } ^ { 2 }\), find the standard deviation for the whole group.