2.02f Measures of average and spread

8 The stem-and-leaf diagram shows the heights, in centimetres, of 17 plants, measured correct to the nearest centimetre. Key: 5 | 6 means 56

1. Find the median and inter-quartile range of these heights.
2. Calculate the mean and standard deviation of these heights.
3. State one advantage of using the median rather than the mean as a measure of average for these heights.

1. Joshua is investigating the daily total rainfall in Hurn for May to October 2015

Using the information from the large data set, Joshua wishes to calculate the mean of the daily total rainfall in Hurn for May to October 2015

1. Using your knowledge of the large data set, explain why Joshua needs to clean the data before calculating the mean. Using the information from the large data set, he produces the grouped frequency table below.

   Daily total rainfall ( \(r \mathrm {~mm}\) )	Frequency	Midpoint ( \(\boldsymbol { x } \mathbf { m m }\) )
   \(0 \leqslant r < 0.5\)	121	0.25
   \(0.5 \leqslant r < 1.0\)	10	0.75
   \(1.0 \leqslant r < 5.0\)	24	3.0
   \(5.0 \leqslant r < 10.0\)	12	7.5
   \(10.0 \leqslant r < 30.0\)	17	20.0

   $$\text { You may use } \sum \mathrm { f } x = 539.75 \text { and } \sum \mathrm { f } x ^ { 2 } = 7704.1875$$
2. Use linear interpolation to calculate an estimate for the upper quartile of the daily total rainfall.
3. Calculate an estimate for the standard deviation of the daily total rainfall in Hurn for May to October 2015
4. 1. State the assumption involved with using class midpoints to calculate an estimate of a mean from a grouped frequency table.
   2. Using your knowledge of the large data set, explain why this assumption does not hold in this case.
   3. State, giving a reason, whether you would expect the actual mean daily total rainfall in Hurn for May to October 2015 to be larger than, smaller than or the same as an estimate based on the grouped frequency table.

The histogram summarises the heights of 256 seedlings two weeks after they were planted. \includegraphics[max width=\textwidth, alt={}, center]{08e3b0b0-2155-4b37-83e3-343c317ca10c-06_1242_1810_287_132}

1. Use linear interpolation to estimate the median height of the seedlings.
   (4) Chris decides to model the frequency density for these 256 seedlings by a curve with equation $$y = k x ( 8 - x ) \quad 0 \leqslant x \leqslant 8$$ where \(k\) is a constant.
2. Find the value of \(k\) Using this model,
3. write down the median height of the seedlings.

1. Jiang is studying the variable Daily Mean Pressure from the large data set.

He drew the following box and whisker plot for these data for one of the months for one location using a linear scale but

• he failed to label all the values on the scale
• he gave an incorrect value for the median \includegraphics[max width=\textwidth, alt={}, center]{08e3b0b0-2155-4b37-83e3-343c317ca10c-09_248_1264_573_402}

Daily Mean Pressure (hPa)
Using your knowledge of the large data set, suggest a suitable value for

1. the median,
2. the range.
   (You are not expected to have memorised values from the large data set. The question is simply looking for sensible answers.)
   1. Jiang is studying the variable Daily Mean Pressure from the large data set.
   He drew the following box and whisker plot for these data for one of the months for one nong asing at
   Using your knowledge of the large data set, suggest a suitable value for

1. the median,
   " \includegraphics[max width=\textwidth, alt={}, center]{08e3b0b0-2155-4b37-83e3-343c317ca10c-09_42_31_1213_1304}
   "

The histogram and its frequency polygon below give information about the weights, in grams, of 50 plums. \includegraphics[max width=\textwidth, alt={}, center]{854568d2-b32d-44de-8a9c-26372e509c20-02_908_1307_328_386}

1. Show that an estimate for the mean weight of the 50 plums is 63.72 grams.
2. Calculate an estimate for the standard deviation of the 50 plums. Later it was discovered that the scales used to weigh the plums were broken.
   Each plum actually weighs 5 grams less than originally thought.
3. State the effect this will have on the estimate of the standard deviation in part (b). Give a reason for your answer.

1. A coach recorded the heights of some adult rugby players and found the following summary statistics.

$$\begin{array} { r } \text { Median } = 1.85 \mathrm {~m} \\ \text { Range } = 0.28 \mathrm {~m} \\ \text { Interquartile range } = 0.11 \mathrm {~m} \end{array}$$ The coach also noticed that

• the height of the shortest player is 1.72 m
• \(25 \%\) of the players' heights are below the height of a player whose height is 1.81 m

Draw a box and whisker plot to represent this information on the grid below. \includegraphics[max width=\textwidth, alt={}, center]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-02_342_1096_1027_488}

1. The partially completed table and partially completed histogram give information about the ages of passengers on an airline.

There were no passengers aged 90 or over. \includegraphics[max width=\textwidth, alt={}, center]{6dfefd72-338f-40be-ac37-aef56bfaccaa-04_1173_1792_721_139}

1. Complete the histogram.
2. Use linear interpolation to estimate the median age. An outlier is defined as a value greater than \(Q _ { 3 } + 1.5 \times\) interquartile range.
   Given that \(Q _ { 1 } = 27.3\) and \(Q _ { 3 } = 58.9\)
3. determine, giving a reason, whether or not the oldest passenger could be considered as an outlier.
   (2)

1. Ben is studying the Daily Total Rainfall, \(x \mathrm {~mm}\), in Leeming for 1987

He used all the data from the large data set and summarised the information in the following table.

1. Explain how the data will need to be cleaned before Ben can start to calculate statistics such as the mean and standard deviation. Using all 184 of these values, Ben estimates \(\sum x = 390\) and \(\sum x ^ { 2 } = 4336\)
2. Calculate estimates for
   1. the mean Daily Total Rainfall,
   2. the standard deviation of the Daily Total Rainfall. Ben suggests using the statistic calculated in part (b)(i) to estimate the annual mean Daily Total Rainfall in Leeming for 1987
3. Using your knowledge of the large data set,
   1. give a reason why these data would not be suitable,
   2. state, giving a reason, how you would expect the estimate in part (b)(i) to differ from the actual annual mean Daily Total Rainfall in Leeming for 1987

1. Ming is studying the large data set for Perth in 2015

He intended to use all the data available to find summary statistics for the Daily Mean Air Temperature, \(x { } ^ { \circ } \mathrm { C }\).
Unfortunately, Ming selected an incorrect variable on the spreadsheet.
This incorrect variable gave a mean of 5.3 and a standard deviation of 12.4

1. Using your knowledge of the large data set, suggest which variable Ming selected. The correct values for the Daily Mean Air Temperature are summarised as $$n = 184 \quad \sum x = 2801.2 \quad \sum x ^ { 2 } = 44695.4$$
2. Calculate the mean and standard deviation for these data. One of the months from the large data set for Perth in 2015 has
   • mean \(\bar { X } = 19.4\)
   • standard deviation \(\sigma _ { x } = 2.83\) for Daily Mean Air Temperature.
   • Suggest, giving a reason, a month these data may have come from.

1. Each member of a group of 27 people was timed when completing a puzzle.

The time taken, \(x\) minutes, for each member of the group was recorded.
These times are summarised in the following box and whisker plot. \includegraphics[max width=\textwidth, alt={}, center]{2b63aa7f-bc50-4422-8dc0-e661b521c221-08_353_1436_458_319}

1. Find the range of the times.
2. Find the interquartile range of the times. For these 27 people \(\sum x = 607.5\) and \(\sum x ^ { 2 } = 17623.25\)
3. calculate the mean time taken to complete the puzzle,
4. calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean.
5. State how many outliers Taruni would say there are in these data, giving a reason for your answer. Adam and Beth also completed the puzzle in \(a\) minutes and \(b\) minutes respectively, where \(a > b\).
   When their times are included with the data of the other 27 people
   • the median time increases
   • the mean time does not change
   • Suggest a possible value for \(a\) and a possible value for \(b\), explaining how your values satisfy the above conditions.
   • Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).

9 The table shows information about the number of days absent last year by students in class 2A at a certain school.

1. Calculate an estimate of the mean for these data.
2. Find the median of these data. The headteacher is writing a report on the numbers of absences at her school. She wishes to include a figure for the average number of absences in class 2A. A governor suggests that she should quote the mean. The class teacher suggests that she should quote the median, because it is lower than the mean.
3. Give another reason for using the median rather than the mean for the average number of absences in class 2A.

6 The large data set gives information about life expectancy at birth for males and females in different London boroughs. Fig. 6.1 shows summary statistics for female life expectancy at birth for the years 2012-2014. Fig. 6.2 shows summary statistics for male life expectancy at birth for the years 2012-2014. \section*{Female Life Expectancy at Birth} \begin{table}[h] \end{table} Male Life Expectancy at Birth \begin{table}[h] \end{table}

1. Use the information in Fig. 6.1 and Fig. 6.2 to draw two box plots. Draw one box plot for female life expectancy at birth in London boroughs and one box plot for male life expectancy at birth in London boroughs.
2. Compare and contrast the distribution of male life expectancy at birth with the distribution of female life expectancy at birth in London boroughs in 2012-2014. Lorraine, who lives in Lancashire, says she wishes her daughter (who was born in 2013) had been born in the London borough of Barnet, because her daughter would have had a higher life expectancy.
3. Give two reasons why there is no evidence in the large data set to support Lorraine's comment.
4. Use the mean and standard deviation for the summary statistics given in Fig. 6.1 and Fig. 6.2 to show that there is at least one outlier in each set. The scatter diagram in Fig. 6.3 shows male life expectancy at birth plotted against female life expectancy at birth for London boroughs in 2012-14. The outliers have been removed. Male life expectancy at birth against female life expectancy at birth \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{11e5167f-9f95-4494-9b66-b59fdce8b1ef-5_593_1054_1260_246} \captionsetup{labelformat=empty} \caption{Fig. 6.3}
   \end{figure}
5. Describe the association between male life expectancy at birth and female life expectancy at birth in London boroughs in 2012-14.

5 Ali collected data from a random sample of 200 workers and recorded the number of days they each worked from home in the second week of September 2019. These data are shown in Fig. 5.1. \begin{table}[h] \end{table}

1. Represent the data by a suitable diagram.
2. Calculate
   Ali then collected data from a different random sample of 200 workers for the same week in September 2019. The mean number of days worked from home for this sample was 1.94 and the standard deviation was 1.75.
3. Explain whether there is any evidence to suggest that one or both of the samples must be flawed. Fig. 5.2 shows a cumulative frequency diagram for the ages of the workers in the first sample who worked from home on at least one day. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e0b502a8-c742-4d78-993c-8c0c7329ec9c-04_671_1362_1452_241} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} Ali concludes that \(90 \%\) of the workers in this sample who worked from home on at least one day were under 60 years of age
4. Explain whether Ali's conclusion is correct.

1 A researcher collects data concerning the number of different social media platforms used by school pupils on a typical weekday. The frequency table for the data is shown below. The researcher uses software to represent the results in this diagram. \includegraphics[max width=\textwidth, alt={}, center]{82438df0-6550-4ffd-92d8-3c67bec59a6b-04_961_1195_737_242}

1. Explain why this diagram is inappropriate.
2. Calculate the following for the number of social media platforms used:
   1. the mean,
   2. the standard deviation.

6 An app on my new smartphone records the number of times in a day I use the phone. The data for each day since I bought the phone are shown in the stem and leaf diagram. Key: 3|1 means 31

1. Explain whether these data are a sample or a population.
2. Describe the shape of the distribution.
3. Determine the interquartile range.
4. Use your answer to part (c) to determine whether there are any outliers in the lower tail.

3 A student conducts an investigation into the number of hours spent cooking per week by people who live in village A. The student represents the data in the cumulative frequency diagram below. \section*{Hours spent cooking per week by people who live in village A} \includegraphics[max width=\textwidth, alt={}, center]{ce94c1ea-ffe5-42d0-8f8a-43c47105d6bf-3_796_1494_918_233}

1. How many people were involved in the investigation?
2. Use the copy of the diagram in the Printed Answer Booklet to determine an estimate for the interquartile range. The student conducts a similar investigation into the number of hours spent cooking per week by 200 people who live in village B. The interquartile range is found to be 3.9 hours.
3. Explain whether the evidence suggests that the number of hours spent cooking by people who live in village B is more variable, equally variable or less variable than the number of hours spent cooking by people who live in village A .

10 The pre-release material contains information about the birth rate per 1000 people in different countries of the world. These countries have been classified into different regions. The table shows some data for three of these regions: the mean and standard deviation (sd) of the birth rate per 1000, and the number of countries for which data was used, n. \section*{Birth rate per 1000 by region}

1. Use the information in the table to compare and contrast the birth rate per 1000 in Africa with the birth rate per 1000 in Europe.
2. The birth rate per 1000 in Mauritius, which is in Africa, is recorded as 9.86. Use the information in the table to show that this value is an outlier.
3. Use your knowledge of the pre-release material to explain whether the value for Mauritius should be discarded.
4. The pre-release material identifies 27 countries in Oceania. Suggest a reason why only 21 values were used to calculate the mean and standard deviation.

2 A student measures the upper arm lengths of a sample of 97 women. The results are summarised in the frequency table in Fig. 2.1. \begin{table}[h] \end{table} The student constructs two cumulative frequency diagrams to represent the data using different class intervals. These are shown in Fig. 2.2 opposite One of these diagrams is correct and the other is incorrect.

1. State which diagram is incorrect, justifying your answer.
2. Use the correct diagram in Fig. 2.2 to find an estimate of the median. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c08a2212-3104-425e-8aee-7f2d46f23924-05_2256_1230_191_148} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}

3 A researcher is conducting an investigation into the number of portions of fruit adults consume each day. The researcher decides to ask 50 men and 50 women to complete a simple questionnaire.

1. State the type of sampling procedure the researcher is using.
2. Write down one disadvantage of this sampling procedure. The researcher represents the data in Fig. 3.1. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Number of portions of fruit consumed by adults} \includegraphics[alt={},max width=\textwidth]{c08a2212-3104-425e-8aee-7f2d46f23924-06_531_991_701_248}
   \end{figure} Fig. 3.1
3. Describe the shape of the distribution. The data are summarised in the frequency table in Fig. 3.2. \begin{table}[h]

   Number of portions of fruit	0	1	2	3	4	5
   Number of adults	18	34	26	11	7	4

   \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{table}
4. For the data in Fig. 3.2, use your calculator to find
   Give your answers correct to 2 decimal places. A second researcher chooses a proportional stratified sample of 100 children from years 5 and 6 in a certain primary school. There are 220 children to choose from. In year 5 there are 125 children, of whom 81 are boys.
5. How many year 5 girls should be included in the sample? The second researcher found that the mean number of portions of fruit consumed per day by the children in this sample was 1.61 and the standard deviation was 0.53 .
6. Comment on the amount of fruit consumed per day by the children compared to the amount of fruit consumed per day by the adults.

9 Fig. 9.1 shows box and whisker diagrams which summarise the birth rates per 1000 people for all the countries in three of the regions as given in the pre-release data set.
The diagrams were drawn as part of an investigation comparing birth rates in different regions of the world. Africa (Sub-Saharan) \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_104_991_557_730} East and South East Asia \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_109_757_744_671} Caribbean \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_99_369_982_730} \begin{figure}[h] \end{figure}

1. Discuss the distributions of birth rates in these regions of the world. Make three different statements. You should refer to both information from the box and whisker diagrams and your knowledge of the large data set.
2. The birth rates for all the countries in Australasia are shown below.

   Country	Birth rate per 1000
   Australia	12.19
   New Zealand	13.4
   Papua New Guinea	24.89

   1. Explain why the calculation below is not a correct method for finding the birth rate per 1000 for Australasia as a whole. $$\frac { 12.19 + 13.4 + 24.89 } { 3 } \approx 16.83$$
   2. Without doing any calculations, explain whether the birth rate per 1000 for Australasia as a whole is higher or lower than 16.83 . The scatter diagram in Fig. 9.2 shows birth rate per 1000 and physicians/ 1000 population for all the countries in the pre-release data set. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-09_898_1698_386_274} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
      \end{figure}
3. Describe the correlation in the scatter diagram.
4. Discuss briefly whether the scatter diagram shows that high birth rates would be reduced by increasing the number of physicians in a country.

4 A survey of the number of cars per household in a certain village generated the data in Fig. 4. \begin{table}[h] \end{table}

1. Calculate the mean number of cars per household.
2. Calculate the standard deviation of the number of cars per household.

9 At the end of each school term at North End College all the science classes in year 10 are given a test. The marks out of 100 achieved by members of set 1 are shown in Fig. 9. \begin{table}[h] \end{table} Key \(5 \quad\) 2 represents a mark of 52

1. Describe the shape of the distribution.
2. The teacher for set 1 claimed that a typical student in his class achieved a mark of 95. How did he justify this statement?
3. Another teacher said that the average mark in set 1 is 76 . How did she justify this statement? Benson's mark in the test is 35 . If the mark achieved by any student is an outlier in the lower tail of the distribution, the student is moved down to set 2 .
4. Determine whether Benson is moved down to set 2 .

3 Fig. 3 shows the time Lorraine spent in hours, \(t\), answering e-mails during the working day. The data were collected over a number of months. \begin{table}[h] \end{table}

1. Calculate an estimate of the mean time per day that Lorraine spent answering e-mails over this period.
2. Explain why your answer to part (a) is an estimate. When Lorraine accepted her job, she was told that the mean time per day spent answering e-mails would not be more than 3 hours.
3. Determine whether, according to the data in Fig. 3, it is possible that the mean time per day Lorraine spends answering e-mails is in fact more than 3 hours.

5 Fig. 5 shows the number of times that students at a sixth form college visited a recreational mathematics website during the first week of the summer term. \begin{table}[h] \end{table}

1. State the value of the mid-range of the data.
2. Describe the shape of the distribution.
3. State the value of the mode.

14 The pre-release material includes data concerning crude death rates in different countries of the world. Fig. 14.1 shows some information concerning crude death rates in countries in Europe and in Africa. \begin{table}[h] \end{table}

1. Use your knowledge of the large data set to suggest a reason why the statistics in Fig. 14.1 refer to only 48 of the 51 European countries.
2. Use the information in Fig. 14.1 to show that there are no outliers in either data set. The crude death rate in Libya is recorded as 3.58 and the population of Libya is recorded as 6411776.
3. Calculate an estimate of the number of deaths in Libya in a year. The median age in Germany is 46.5 and the crude death rate is 11.42. The median age in Cyprus is 36.1 and the crude death rate is 6.62 .
4. Explain why a country like Germany, with a higher median age than Cyprus, might also be expected to have a higher crude death rate than Cyprus. Fig. 14.2 shows a scatter diagram of median age against crude death rate for countries in Africa and Fig. 14.3 shows a scatter diagram of median age against crude death rate for countries in Europe. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-10_678_1221_1975_248} \captionsetup{labelformat=empty} \caption{Fig. 14.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-11_588_1248_223_228} \captionsetup{labelformat=empty} \caption{Fig. 14.3}
   \end{figure} The rank correlation coefficient for the data shown in Fig. 14.2 is - 0.281206 .
   The rank correlation coefficient for the data shown in Fig. 14.3 is 0.335215 .
5. Compare and contrast what may be inferred about the relationship between median age and crude death rate in countries in Africa and in countries in Europe.