2.02f Measures of average and spread

2 The following back-to-back stem-and-leaf diagram shows the reaction times in seconds in an experiment involving two groups of people, \(A\) and \(B\). Key: 5 | 22 | 6 means a reaction time of 0.225 seconds for \(A\) and 0.226 seconds for \(B\)

1. Find the median and the interquartile range for group \(A\).
   The median value for group \(B\) is 0.235 seconds, the lower quartile is 0.217 seconds and the upper quartile is 0.245 seconds.
2. Draw box-and-whisker plots for groups \(A\) and \(B\) on the grid. \includegraphics[max width=\textwidth, alt={}, center]{62812433-baee-490a-bad4-b6b0f917c234-03_805_1495_1729_365}

7 The heights, in cm, of the 11 members of the Anvils athletics team and the 11 members of the Brecons swimming team are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Anvils on the left-hand side of the diagram and Brecons on the right-hand side.
2. Find the median and the interquartile range for the heights of the Anvils.
   The heights of the 11 members of the Anvils are denoted by \(x \mathrm {~cm}\). It is given that \(\Sigma x = 1923\) and \(\Sigma x ^ { 2 } = 337221\). The Anvils are joined by 3 new members whose heights are \(166 \mathrm {~cm} , 172 \mathrm {~cm}\) and 182 cm .
3. Find the standard deviation of the heights of all 14 members of the Anvils.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 Ransha measured the lengths, in centimetres, of 160 palm leaves. His results are illustrated in the cumulative frequency graph below. \includegraphics[max width=\textwidth, alt={}, center]{7ea494c0-5e1a-4da9-a189-30128654fa1d-08_1090_1424_404_356}

1. Estimate how many leaves have a length between 14 and 24 centimetres.
2. \(10 \%\) of the leaves have a length of \(L\) centimetres or more. Estimate the value of \(L\).
3. Estimate the median and the interquartile range of the lengths.
   Sharim measured the lengths, in centimetres, of 160 palm leaves of a different type. He drew a box-and-whisker plot for the data, as shown on the grid below. \includegraphics[max width=\textwidth, alt={}, center]{7ea494c0-5e1a-4da9-a189-30128654fa1d-09_540_1287_1181_424}
4. Compare the central tendency and the spread of the two sets of data.

1 Twelve tourists were asked to estimate the height, in metres, of a new building. Their estimates were as follows. $$\begin{array} { l l l l l l l l l l l l } 50 & 45 & 62 & 30 & 40 & 55 & 110 & 38 & 52 & 60 & 55 & 40 \end{array}$$

1. Find the median and the interquartile range for the data.
2. Give a disadvantage of using the mean as a measure of the central tendency in this case.

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.

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.

5 Last Saturday, 200 drivers entering a car park were asked the time, in minutes, that it had taken them to travel from home to the car park. The results are summarised in the following cumulative frequency table.

1. On the grid, draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{06f6c8dd-170c-4e94-a960-0c649a7363a1-08_1198_1399_735_415}
2. Use your graph to estimate the median of the data.
3. For 80 of the drivers, the time taken was at least \(T\) minutes. Use your graph to estimate the value of \(T\).
4. Calculate an estimate of the mean time taken by all 200 drivers to travel to the car park.

4 The marks of the pupils in a certain class in a History examination are as follows. $$\begin{array} { l l l l l l l l l l l l l } 28 & 33 & 55 & 38 & 42 & 39 & 27 & 48 & 51 & 37 & 57 & 49 & 33 \end{array}$$ The marks of the pupils in a Physics examination are summarised as follows.
Lower quartile: 28 , Median: 39, Upper quartile: 67.
The lowest mark was 17 and the highest mark was 74 .

1. Draw box-and-whisker plots in a single diagram on graph paper to illustrate the marks for History and Physics.
2. State one difference, which can be seen from the diagram, between the marks for History and Physics.

4 The weights of 220 sausages are summarised in the following table.

1. State which interval the median weight lies in.
2. Find the smallest possible value and the largest possible value for the interquartile range.
3. State how many sausages weighed between 50 g and 60 g .
4. On graph paper, draw a histogram to represent the weights of the sausages.

5 \includegraphics[max width=\textwidth, alt={}, center]{b72ace6b-d3d4-401d-bffe-403c9127f2a8-3_1157_1001_258_573} The cumulative frequency graph shows the annual salaries, in thousands of euros, of a random sample of 500 adults with jobs, in France. It has been plotted using grouped data. You may assume that the lowest salary is 5000 euros and the highest salary is 80000 euros.

1. On graph paper, draw a box-and-whisker plot to illustrate these salaries.
2. Comment on the salaries of the people in this sample.
3. 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.
   1. How high must a salary be in order to be classified as an outlier?
   2. Show that none of the salaries is low enough to be classified as an outlier.

4 Prices in dollars of 11 caravans in a showroom are as follows. \(\begin{array} { l l l l l l l l l l l } 16800 & 18500 & 17700 & 14300 & 15500 & 15300 & 16100 & 16800 & 17300 & 15400 & 16400 \end{array}\)

1. Represent these prices by a stem-and-leaf diagram.
2. Write down the lower quartile of the prices of the caravans in the showroom.
3. 3 different caravans in the showroom are chosen at random and their prices are noted. Find the probability that 2 of these prices are more than the median and 1 is less than the lower quartile.

1 The lengths, \(X\) centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows.
3.9
4.8
4.8
4.4

2 The back-to-back stem-and-leaf diagram below shows the number of hours of television watched per week by each of 15 boys and 15 girls. $$\begin{aligned} & \text { Boys Girls } \\ & \left. \begin{array} { r r r r r r r r | r r r r r r r r r r r r r } & 677664 & 4 & 3 & 0 & 0 & 5 & 5 & 6 & 677888 \end{array} \right\} \end{aligned}$$ Key: 4 | 2 | 2 means a boy who watched 24 hours and a girl who watched 22 hours of television per week.

1. Find the median and the quartiles of the results for the boys.
2. Give a reason why the median might be preferred to the mean in using an average to compare the two data sets.
3. State one advantage, and one disadvantage, of using stem-and-leaf diagrams rather than box-andwhisker plots to represent the data.

4 Each of the variables \(W , X , Y\) and \(Z\) takes eight integer values only. The probability distributions are illustrated in the following diagrams. \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_437_394_397_280} \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_433_380_397_685} \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_428_383_402_1082} \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_425_376_402_1482}

1. For which one or more of these variables is
   1. the mean equal to the median,
   2. the mean greater than the median?
   3. Give a reason why none of these diagrams could represent a geometric distribution.
   4. Which one of these diagrams could not represent a binomial distribution? Explain your answer briefly.

8 In the 2001 census, the household size (the number of people living in each household) was recorded. The percentages of households of different sizes were then calculated. The table shows the percentages for two wards, Withington and Old Moat, in Manchester.

1. Calculate the median and interquartile range of the household size for Withington.
2. Making an appropriate assumption for the last class, which should be stated, calculate the mean and standard deviation of the household size for Withington. Give your answers to an appropriate degree of accuracy. The corresponding results for Old Moat are as follows.

   Median	Interquartile range	Mean	Standard deviation
   2	2	2.4	1.5

3. State one advantage of using the median rather than the mean as a measure of the average household size.
4. By comparing the values for Withington with those for Old Moat, explain briefly why the interquartile range may be less suitable than the standard deviation as a measure of the variation in household size.
5. For one of the above wards, the value of Spearman's rank correlation coefficient between household size and percentage is - 1 . Without any calculation, state which ward this is. Explain your answer.

8 The stem-and-leaf diagram shows the age in completed years of the members of a sports club. \section*{Male} \begin{table}[h] \end{table} Key: 1 | 4 | 0 represents a male aged 41 and a female aged 40.

1. Find the median and interquartile range for the males.
2. The median and interquartile range for the females are 27 and 15 respectively. Make two comparisons between the ages of the males and the ages of the females.
3. The mean age of the males is 30.7 and the mean age of the females is 27.5 , each correct to 1 decimal place. Give one advantage of using the median rather than the mean to compare the ages of the males with the ages of the females. A record was kept of the number of hours, \(X\), spent by each member at the club in a year. The results were summarised by $$n = 49 , \quad \Sigma ( x - 200 ) = 245 , \quad \Sigma ( x - 200 ) ^ { 2 } = 9849 .$$
4. Calculate the mean and standard deviation of \(X\).

5 The examination marks obtained by 1200 candidates are illustrated on the cumulative frequency graph, where the data points are joined by a smooth curve. \includegraphics[max width=\textwidth, alt={}, center]{5faf0d93-4037-4958-8665-1008477a79de-4_1344_1335_386_425} Use the curve to estimate

1. the interquartile range of the marks,
2. \(x\), if \(40 \%\) of the candidates scored more than \(x\) marks,
3. the number of candidates who scored more than 68 marks. Five of the candidates are selected at random, with replacement.
4. Estimate the probability that all five scored more than 68 marks. It is subsequently discovered that the candidates' marks in the range 35 to 55 were evenly distributed - that is, roughly equal numbers of candidates scored \(35,36,37 , \ldots , 55\).
5. What does this information suggest about the estimate of the interquartile range found in part (i)?

7 In a UK government survey in 2000, smokers were asked to estimate the time between their waking and their having the first cigarette of the day. For heavy smokers, the results were as follows. Times are given correct to the nearest minute.

1. Assuming that 'At least 60 minutes' means 'At least 60 minutes but less than 240 minutes', calculate estimates for the mean and standard deviation of the time between waking and first cigarette for these smokers.
2. Find an estimate for the interquartile range of the time between waking and first cigarette for these smokers. Give your answer correct to the nearest minute.
3. The meaning of 'At least 60 minutes' is now changed to 'At least 60 minutes but less than 480 minutes'. Without further calculation, state whether this would cause an increase, a decrease or no change in the estimated value of
   1. the mean,
   2. the standard deviation,
   3. the interquartile range.

5 The numbers of births, in thousands, to mothers of different ages in England and Wales, in 1991 and 2001 are illustrated by the cumulative frequency curves. Cumulative frequency (000's) \includegraphics[max width=\textwidth, alt={}, center]{dfad6626-75ca-4dbd-9c45-42f809c163f3-3_949_1338_461_479}

1. In which of these two years were there more births? How many more births were there in this year?
2. The following quantities were estimated from the diagram.

   Year	M edian age (years)	Interquartile range (years)	Proportion of mothers giving birth aged below 25	Proportion of mothers giving birth aged 35 or above
   1991	27.5	7.3	\(33 \%\)	\(9 \%\)
   2001				\(18 \%\)

   1. Find the values missing from the table.
   2. Did the women who gave birth in 2001 tend to be younger or older or about the same age as the women who gave birth in 1991? Using the table and your values from part (a), give two reasons for your answer.

3 The masses, \(m\) grams, of 52 apples of a certain variety were found and summarised as follows. $$n = 52 \quad \Sigma ( m - 150 ) = - 182 \quad \Sigma ( m - 150 ) ^ { 2 } = 1768$$

1. Find the mean and variance of the masses of these 52 apples.
2. Use your answers from part (i) to find the exact value of \(\Sigma m ^ { 2 }\). The masses of the apples are illustrated in the box-and-whisker plot below. \includegraphics[max width=\textwidth, alt={}, center]{b5ce3230-7528-439c-9e85-ef159a49cba3-3_250_1310_662_383}
3. How many apples have masses in the interval \(130 \leqslant m < 140\) ?
4. An 'outlier' is a data item that lies 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. Explain whether any of the masses of these apples are outliers.

1 Janet and John wanted to compare their daily journey times to work, so they each kept a record of their journey times for a few weeks.

1. Janet's daily journey times, \(x\) minutes, for a period of 25 days, were summarised by \(\Sigma x = 2120\) and \(\Sigma x ^ { 2 } = 180044\). Calculate the mean and standard deviation of Janet's journey times.
2. John's journey times had a mean of 79.7 minutes and a standard deviation of 6.22 minutes. Describe briefly, in everyday terms, how Janet and John's journey times compare.

2 A sprinter runs many 100 -metre trials, and the time, \(x\) seconds, for each is recorded. A sample of eight of these times is taken, as follows. $$\begin{array} { l l l l l l l l } 10.53 & 10.61 & 10.04 & 10.49 & 10.63 & 10.55 & 10.47 & 10.63 \end{array}$$

1. Calculate the sample mean, \(\bar { x }\), and sample standard deviation, \(s\), of these times.
2. Show that the time of 10.04 seconds may be regarded as an outlier.
3. Discuss briefly whether or not the time of 10.04 seconds should be discarded.

7 The cumulative frequency graph below illustrates the distances that 176 children live from their primary school. \begin{figure}[h] \end{figure}

1. Use the graph to estimate, to the nearest 10 metres,
   (A) the median distance from school,
   (B) the lower quartile, upper quartile and interquartile range.
2. Draw a box and whisker plot to illustrate the data. The graph on page 4 used the following grouped data.

   Distance (metres)	200	400	600	800	1000	1200
   Cumulative frequency	20	64	118	150	169	176

3. Copy and complete the grouped frequency table below describing the same data.

   Distance ( \(d\) metres)	Frequency
   \(0 < d \leqslant 200\)	20
   \(200 < d \leqslant 400\)

4. Hence estimate the mean distance these children live from school. It is subsequently found that none of the 176 children lives within 100 metres of the school.
5. Calculate the revised estimate of the mean distance.
6. Describe what change needs to be made to the cumulative frequency graph.

4 A company sells sugar in bags which are labelled as containing 450 grams.
Although the mean weight of sugar in a bag is more than 450 grams, there is concern that too many bags are underweight. The company can adjust the mean or the standard deviation of the weight of sugar in a bag.

1. State two adjustments the company could make. The weights, \(x\) grams, of a random sample of 25 bags are now recorded.
2. Given that \(\sum x = 11409\) and \(\sum x ^ { 2 } = 5206937\), calculate the sample mean and sample standard deviation of these weights.

7 At East Cornwall College, the mean GCSE score of each student is calculated. This is done by allocating a number of points to each GCSE grade in the following way.

1. Calculate the mean GCSE score, \(X\), of a student who has the following GCSE grades: $$\mathrm { A } ^ { * } , \mathrm {~A} ^ { * } , \mathrm {~A} , \mathrm {~A} , \mathrm {~A} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm { C } , \mathrm { D } .$$ 60 students study AS Mathematics at the college. The mean GCSE scores of these students are summarised in the table below.

   Mean GCSE score	Number of students
   \(4.5 \leqslant X < 5.5\)	8
   \(5.5 \leqslant X < 6.0\)	14
   \(6.0 \leqslant X < 6.5\)	19
   \(6.5 \leqslant X < 7.0\)	13
   \(7.0 \leqslant X \leqslant 8.0\)	6

2. Draw a histogram to illustrate this information.
3. Calculate estimates of the sample mean and the sample standard deviation. The scoring system for AS grades is shown in the table below.

   AS Grade	A	B	C	D	E	U
   Score	60	50	40	30	20	0

   The Mathematics department at the college predicts each student's AS score, \(Y\), using the formula \(Y = 13 X - 46\), where \(X\) is the student's average GCSE score.
4. What AS grade would the department predict for a student with an average GCSE score of 7.4 ?
5. What do you think the prediction should be for a student with an average GCSE score of 5.5? Give a reason for your answer.
6. Using your answers to part (iii), estimate the sample mean and sample standard deviation of the predicted AS scores of the 60 students in the department.

1 The total annual emissions of carbon dioxide, \(x\) tonnes per person, for 13 European countries are given below. $$\begin{array} { c c c c c c c c c c c c c } 6.2 & 6.7 & 6.8 & 8.1 & 8.1 & 8.5 & 8.6 & 9.0 & 9.9 & 10.1 & 11.0 & 11.8 & 22.8 \end{array}$$

1. Find the mean, median and midrange of these data.
2. Comment on how useful each of these is as a measure of central tendency for these data, giving a brief reason for each of your answers.