2.02f Measures of average and spread

4 The times, to the nearest minute, of 150 athletes taking part in a charity run are recorded. The results are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-08_1493_1397_936_415}
2. Calculate estimates for the mean and standard deviation of the times taken by the athletes.

4 The heights, in cm, of the 11 players in each of two teams, the Aces and the Jets, are shown in the following table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information with the Aces on the left-hand side of the diagram.
2. Find the median and the interquartile range of the heights of the players in the Aces.
3. Give one comment comparing the spread of the heights of the Aces with the spread of the heights of the Jets.

4 The weights, \(x \mathrm {~kg}\), of 120 students in a sports college are recorded. The results are summarised in the following table.

1. Draw a cumulative frequency graph to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{82c36c11-878c-47d1-a07f-fbf8b2a22d97-06_1390_1389_660_418}
2. It is found that \(35 \%\) of the students weigh more than \(W \mathrm {~kg}\). Use your graph to estimate the value of \(W\).
3. Calculate estimates for the mean and standard deviation of the weights of the 120 students. [6]

3 The time taken, in minutes, to walk to school was recorded for 200 pupils at a certain school. These times are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-04_1217_1509_705_278}
2. Use your graph to estimate the median and the interquartile range of the data. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-05_2723_35_101_20}
3. Calculate an estimate for the mean value of the times taken by the 200 pupils to walk to school.

6 Teams of 15 runners took part in a charity run last Saturday. The times taken, in minutes, to complete the course by the runners from the Falcons and the runners from the Kites are shown in the table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with the Falcons on the left-hand side.
2. Find the median and the interquartile range of the times for the Falcons.
   Let \(x\) and \(y\) denote the times, in minutes, of a runner from the Falcons and a runner from the Kites respectively. It is given that $$\sum x = 792 , \quad \sum x ^ { 2 } = 43504 , \quad \sum y = 783 , \quad \sum y ^ { 2 } = 42223 .$$
3. Find the mean and the standard deviation of the times taken by all 30 runners from the two teams.

4 On a certain day, the heights of 150 sunflower plants grown by children at a local school are measured, correct to the nearest cm . These heights are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-06_1600_1301_760_383}
2. Use your graph to estimate the 30th percentile of the heights of the sunflower plants. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-07_2723_35_101_20}
3. Calculate estimates for the mean and the standard deviation of the heights of the 150 sunflower plants.

7 A random sample of 97 people who own mobile phones was used to collect data on the amount of time they spent per day on their phones. The results are displayed in the table below.

1. Calculate estimates of the mean and standard deviation of the time spent per day on these mobile phones.
2. On graph paper, draw a fully labelled histogram to represent the data.

7 The numbers of chocolate bars sold per day in a cinema over a period of 100 days are summarised in the following table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{3ada18de-c4f7-4049-9032-46b796be83c3-12_1203_1399_833_415}
2. What is the greatest possible value of the interquartile range for the data?
3. Calculate estimates of the mean and standard deviation of the number of chocolate bars sold.
   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 Two cricket teams kept records of the number of runs scored by their teams in 8 matches. The scores are shown in the following table.

1. Find the mean and standard deviation of the scores for team \(A\). The mean and standard deviation for team \(B\) are 130.75 and 29.63 respectively.
2. State with a reason which team has the more consistent scores.

2 The following table shows the results of a survey to find the average daily time, in minutes, that a group of schoolchildren spent in internet chat rooms. The mean time was calculated to be 27.5 minutes.

1. Form an equation involving \(f\) and hence show that the total number of children in the survey was 26 .
2. Find the standard deviation of these times.

4 The following back-to-back stem-and-leaf diagram shows the cholesterol count for a group of 45 people who exercise daily and for another group of 63 who do not exercise. The figures in brackets show the number of people corresponding to each set of leaves. Key: 2 | 8 | 1 represents a cholesterol count of 8.2 in the group who exercise and 8.1 in the group who do not exercise.

1. Give one useful feature of a stem-and-leaf diagram.
2. Find the median and the quartiles of the cholesterol count for the group who do not exercise. You are given that the lower quartile, median and upper quartile of the cholesterol count for the group who exercise are 4.25, 5.3 and 6.6 respectively.
3. On a single diagram on graph paper, draw two box-and-whisker plots to illustrate the data.

1 The salaries, in thousands of dollars, of 11 people, chosen at random in a certain office, were found to be: $$40 , \quad 42 , \quad 45 , \quad 41 , \quad 352 , \quad 40 , \quad 50 , \quad 48 , \quad 51 , \quad 49 , \quad 47 .$$ Choose and calculate an appropriate measure of central tendency (mean, mode or median) to summarise these salaries. Explain briefly why the other measures are not suitable.

1 The stem-and-leaf diagram below represents data collected for the number of hits on an internet site on each day in March 2007. There is one missing value, denoted by \(x\). Key: 1 | 5 represents 15 hits

1. Find the median and lower quartile for the number of hits each day.
2. The interquartile range is 19 . Find the value of \(x\).

6 During January the numbers of people entering a store during the first hour after opening were as follows.

1. Find the values of \(a\) and \(b\).
2. Draw a cumulative frequency graph to represent this information. Take a scale of 2 cm for 10 minutes on the horizontal axis and 2 cm for 50 people on the vertical axis.
3. Use your graph to estimate the median time after opening that people entered the store.
4. Calculate estimates of the mean, \(m\) minutes, and standard deviation, \(s\) minutes, of the time after opening that people entered the store.
5. Use your graph to estimate the number of people entering the store between ( \(m - \frac { 1 } { 2 } s\) ) and \(\left( m + \frac { 1 } { 2 } s \right)\) minutes after opening.

2 The numbers of people travelling on a certain bus at different times of the day are as follows.

1. Draw a stem-and-leaf diagram to illustrate the information given above.
2. Find the median, the lower quartile, the upper quartile and the interquartile range.
3. State, in this case, which of the median and mode is preferable as a measure of central tendency, and why.

4 The numbers of rides taken by two students, Fei and Graeme, at a fairground are shown in the following table.

1. The mean cost of Fei's rides is \(\\) 2.50\( and the standard deviation of the costs of Fei's rides is \)\\( 0\). Explain how you can tell that the roller coaster and the water slide each cost \(\\) 2.50\( per ride. [2]
2. The mean cost of Graeme's rides is \)\\( 3.76\). Find the standard deviation of the costs of Graeme's rides.

2 The heights, \(x \mathrm {~cm}\), of a group of 82 children are summarised as follows. $$\Sigma ( x - 130 ) = - 287 , \quad \text { standard deviation of } x = 6.9 .$$

1. Find the mean height.
2. Find \(\Sigma ( x - 130 ) ^ { 2 }\).

6 The lengths of some insects of the same type from two countries, \(X\) and \(Y\), were measured. The stem-and-leaf diagram shows the results. Key: 5 | 81 | 3 means an insect from country \(X\) has length 0.815 cm and an insect from country \(Y\) has length 0.813 cm .

1. Find the median and interquartile range of the lengths of the insects from country \(X\).
2. The interquartile range of the lengths of the insects from country \(Y\) is 0.028 cm . Find the values of \(q\) and \(r\).
3. Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.
4. Compare the lengths of the insects from the two countries.

6 There are 5000 schools in a certain country. The cumulative frequency table shows the number of pupils in a school and the corresponding number of schools.

1. Draw a cumulative frequency graph with a scale of 2 cm to 100 pupils on the horizontal axis and a scale of 2 cm to 1000 schools on the vertical axis. Use your graph to estimate the median number of pupils in a school.
2. \(80 \%\) of the schools have more than \(n\) pupils. Estimate the value of \(n\) correct to the nearest ten.
3. Find how many schools have between 201 and 250 (inclusive) pupils.
4. Calculate an estimate of the mean number of pupils per school.

1 Red Street Garage has 9 used cars for sale. Fairwheel Garage has 15 used cars for sale. The mean age of the cars in Red Street Garage is 3.6 years and the standard deviation is 1.925 years. In Fairwheel Garage, \(\Sigma x = 64\) and \(\Sigma x ^ { 2 } = 352\), where \(x\) is the age of a car in years.

1. Find the mean age of all 24 cars.
2. Find the standard deviation of the ages of all 24 cars.

3 The following cumulative frequency table shows the examination marks for 300 candidates in country \(A\) and 300 candidates in country \(B\).

1. Without drawing a graph, show that the median for country \(B\) is higher than the median for country \(A\).
2. Find the number of candidates in country \(A\) who scored between 20 and 34 marks inclusive.
3. Calculate an estimate of the mean mark for candidates in country \(A\).

5 The lengths of the diagonals in metres of the 9 most popular flat screen TVs and the 9 most popular conventional TVs are shown below.

1. Represent this information on a back-to-back stem-and-leaf diagram.
2. Find the median and the interquartile range of the lengths of the diagonals of the 9 conventional TVs.
3. Find the mean and standard deviation of the lengths of the diagonals of the 9 flat screen TVs.

4 The back-to-back stem-and-leaf diagram shows the values taken by two variables \(A\) and \(B\). Key: \(4 | 16 | 7\) means \(A = 0.164\) and \(B = 0.167\).

1. Find the median and the interquartile range for variable \(A\).
2. You are given that, for variable \(B\), the median is 0.171 , the upper quartile is 0.179 and the lower quartile is 0.164 . Draw box-and-whisker plots for \(A\) and \(B\) in a single diagram on graph paper.

3 The following back-to-back stem-and-leaf diagram shows the annual salaries of a group of 39 females and 39 males. Key: 2 | 20 | 3 means \\(20200 for females and \\)20300 for males.

1. Find the median and the quartiles of the females' salaries. You are given that the median salary of the males is \(\\) 24000\(, the lower quartile is \)\\( 22600\) and the upper quartile is \(\\) 25300$.
2. Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.

5 The following are the annual amounts of money spent on clothes, to the nearest \(\\) 10$, by 27 people.

1. Construct a stem-and-leaf diagram for the data.
2. Find the median and the interquartile range of the data. 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.
3. List the outliers.