2.02f Measures of average and spread

3 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.

4 At a certain stage of a football league season, the numbers of goals scored by a sample of 20 teams in the league were as follows. \(\begin{array} { l l l l l l l l l l l l l l l l l l l l } 22 & 23 & 23 & 23 & 26 & 28 & 28 & 30 & 31 & 33 & 33 & 34 & 35 & 35 & 36 & 36 & 37 & 46 & 49 & 49 \end{array}\)

1. Calculate the sample mean and sample variance, \(s ^ { 2 }\), of these data.
2. The three teams with the most goals appear to be well ahead of the other teams. Determine whether or not any of these three pieces of data may be considered outliers.

2 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.

3 The stem and leaf diagram illustrates the heights in metres of 25 young oak trees. Key: 4 |2 represents 4.2

1. State the type of skewness of the distribution.
2. Use your calculator to find the mean and standard deviation of these data.
3. Determine whether there are any outliers.

2 In a traffic survey, the number of people in each car passing the survey point is recorded. The results are given in the following frequency table.

1. Write down the median and mode of these data.
2. Draw a vertical line diagram for these data.
3. State the type of skewness of the distribution.

3 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{b6d84f99-ee39-49c7-a5e8-05838efeef5a-2_804_1372_483_436} Data sourced from the 2001 Census, www.sta is \href{http://ics.gov.uk}{ics.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.

   Age	20	30	40	50	65	100
   Cumulative frequency (thousands)	660	1240	1810	\(a\)	2490	2770

   (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)	1120	650	770	590	680	610

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.

1 The maximum temperatures \(x\) degrees Celsius recorded during each month of 2005 in Cambridge are given in the table below. These data are summarised by \(n = 12 , \Sigma x = 180.6 , \Sigma x ^ { 2 } = 3107.56\).

1. Calculate the mean and standard deviation of the data.
2. Determine whether there are any outliers.
3. The formula \(y = 1.8 x + 32\) is used to convert degrees Celsius to degrees Fahrenheit. Find the mean and standard deviation of the 2005 maximum temperatures in degrees Fahrenheit.
4. In New York, the monthly maximum temperatures are recorded in degrees Fahrenheit. In 2005 the mean was 63.7 and the standard deviation was 16.0 . Briefly compare the maximum monthly temperatures in Cambridge and New York in 2005. The total numbers of hours of sunshine recorded in Cambridge during the month of January for each of the last 48 years are summarised below.

   Hours \(h\)	\(70 \leqslant h < 100\)	\(100 \leqslant h < 110\)	\(110 \leqslant h < 120\)	\(120 \leqslant h < 150\)	\(150 \leqslant h < 170\)	\(170 \leqslant h < 190\)
   Number of years	6	8	10	11	10	3

5. Draw a cumulative frequency graph for these data.
6. Use your graph to estimate the 90th percentile.

2 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.

1. On the insert, draw a cumulative frequency diagram to illustrate the data.
2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
3. State the type of skewness of the distribution of the data.

1 The amounts of electricity, \(x \mathrm { kWh }\) (kilowatt hours), used by 40 households in a three-month period are summarised as follows. $$n = 40 \quad \sum x = 59972 \quad \sum x ^ { 2 } = 96767028$$

1. Calculate the mean and standard deviation of \(x\).
2. The formula \(y = 0.163 x + 14.5\) gives the cost in pounds of the electricity used by each household. Use your answers to part (i) to deduce the mean and standard deviation of the costs of the electricity used by these 40 households.

1 The hourly wages, \(\pounds x\), of a random sample of 60 employees working for a company are summarised as follows. $$n = 60 \quad \sum x = 759.00 \quad \sum x ^ { 2 } = 11736.59$$

1. Calculate the mean and standard deviation of \(x\).
2. The workers are offered a wage increase of \(2 \%\). Use your answers to part (i) to deduce the new mean and standard deviation of the hourly wages after this increase.
3. As an alternative the workers are offered a wage increase of 25 p per hour. Write down the new mean and standard deviation of the hourly wages after this 25p increase.

3 The numbers of eggs laid by a sample of 70 female herring gulls are shown in the table.

1. Find the mean and standard deviation of the number of eggs laid per gull.
2. The sample did not include female herring gulls that laid no eggs. How would the mean and standard deviation change if these gulls were included?

6 A retail analyst records the numbers of loaves of bread of a particular type bought by a sample of shoppers in a supermarket.

1. Calculate the mean and standard deviation of the numbers of loaves bought per person.
2. Each loaf costs \(\pounds 1.04\). Calculate the mean and standard deviation of the amount spent on loaves per person.

2 Dwayne is a car salesman. The numbers of cars, \(x\), sold by Dwayne each month during the year 2008 are summarised by $$n = 12 , \quad \Sigma x = 126 , \quad \Sigma x ^ { 2 } = 1582 .$$

1. Calculate the mean and standard deviation of the monthly numbers of cars sold.
2. Dwayne earns \(\pounds 500\) each month plus \(\pounds 100\) commission for each car sold. Show that the mean of Dwayne's monthly earnings is \(\pounds 1550\). Find the standard deviation of Dwayne's monthly earnings.
3. Marlene is a car saleswoman and is paid in the same way as Dwayne. During 2008 her monthly earnings have mean \(\pounds 1625\) and standard deviation \(\pounds 280\). Briefly compare the monthly numbers of cars sold by Marlene and Dwayne during 2008.

6 In a survey, a sample of 44 fields is selected. Their areas ( \(x\) hectares) are summarised in the grouped frequency table.

1. Calculate an estimate of the sample mean and the sample standard deviation.
2. Determine whether there could be any outliers at the upper end of the distribution.

2 The marks \(x\) scored by a sample of 56 students in an examination are summarised by $$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$

1. Calculate the mean and standard deviation of the marks.
2. The highest mark scored by any of the 56 students in the examination was 93. Show that this result may be considered to be an outlier.
3. The formula \(y = 1.2 x - 10\) is used to scale the marks. Find the mean and standard deviation of the scaled marks.

3 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{dfb0acd8-d84b-4291-a811-a68f4942794b-2_1266_1546_487_335}

1. Use the diagram to estimate the median and interquartile range of the data.
2. Use your answers to part (i) to estimate the number of outliers in the sample.
3. Should these outliers be excluded from any further analysis? Briefly explain your answer.
4. Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected. \(12 \%\) of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
5. Find the probability that
   (A) exactly 2 of these 17 babies require special care,
   (B) more than 2 of the 17 babies require special care.
6. On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.

3 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{79f1015b-7c3d-4576-8d5b-e9fc89d8a49e-2_1266_1546_487_335}

1. Use the diagram to estimate the median and interquartile range of the data.
2. Use your answers to part (i) to estimate the number of outliers in the sample.
3. Should these outliers be excluded from any further analysis? Briefly explain your answer.
4. Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected. \(12 \%\) of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
5. Find the probability that
   (A) exactly 2 of these 17 babies require special care,
   (B) more than 2 of the 17 babies require special care.
6. On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.

1 The stem and leaf diagram illustrates the heights in metres of 25 young oak trees. Key: 4 |2 represents 4.2

1. State the type of skewness of the distribution.
2. Use your calculator to find the mean and standard deviation of these data.
3. Determine whether there are any outliers.

2 The mean daily maximum temperatures at a research station over a 12 -month period, measured to the nearest degree Celsius, are given below.

1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of \(0,10 , \ldots\).
2. Write down the median of these data.
3. The mean of these data is 24.3. Would the mean or the median be a better measure of central tendency of the data? Briefly explain your answer.

3 The stem and leaf diagram shows the weights, rounded to the nearest 10 grams, of 25 female iguanas. Key: 11 | 2 represents 1120 grams

1. Find the mode and the median of the data.
2. Identify the type of skewness of the distribution.

4 A camera records the speeds in miles per hour of 15 vehicles on a motorway. The speeds are given below. $$\begin{array} { l l l l l l l l l l l l l l l } 73 & 67 & 75 & 64 & 52 & 63 & 75 & 81 & 77 & 72 & 68 & 74 & 79 & 72 & 71 \end{array}$$

1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of \(50,60 , \ldots\).
2. Write down the median and midrange of the data.
3. Which of the median and midrange would you recommend to measure the central tendency of the data? Briefly explain your answer.

5 In a traffic survey, the number of people in each car passing the survey point is recorded. The results are given in the following frequency table.

1. Write down the median and mode of these data.
2. Draw a vertical line diagram for these data.
3. State the type of skewness of the distribution.

6 A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a \(\pounds 10\) prize, 20 of them have a \(\pounds 100\) prize, one of them has a \(\pounds 5000\) prize and all of the rest have no prize. This information is summarised in the frequency table below.

1. Find the mean and standard deviation of the prize money per ticket.
2. I buy two of these tickets at random. Find the probability that I win either two \(\pounds 10\) prizes or two \(\pounds 100\) prizes.

7 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{aabf9d8b-5f91-4a3b-bcf8-e46cb45127c4-4_805_1372_392_401} Data sourced from he 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.

   Age	20	30	40	50	65	100
   Cumulative frequency (thousands)	660	1240	1810	\(a\)	2490	2770

   (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)	1120	650	770	590	680	610

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.
   [0pt] [4]

7 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{93bbc0cf-d3ad-4bc2-a6c6-36a3b8e394a9-4_805_1372_392_401} Data sourced from he 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.

   Age	20	30	40	50	65	100
   Cumulative frequency (thousands)	660	1240	1810	\(a\)	2490	2770

   (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)	1120	650	770	590	680	610

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.
   [0pt] [4]