2.02f Measures of average and spread

3 In a survey, people were asked how long they took to travel to and from work, on average. The median time was 3 hours 36 minutes, the upper quartile was 4 hours 42 minutes and the interquartile range was 3 hours 48 minutes. The longest time taken was 5 hours 12 minutes and the shortest time was 30 minutes.

1. Find the lower quartile.
2. Represent the information by a box-and-whisker plot, using a scale of 2 cm to represent 60 minutes.

1 A summary of 24 observations of \(x\) gave the following information: $$\Sigma ( x - a ) = - 73.2 \quad \text { and } \quad \Sigma ( x - a ) ^ { 2 } = 2115 .$$ The mean of these values of \(x\) is 8.95 .

1. Find the value of the constant \(a\).
2. Find the standard deviation of these values of \(x\).

5 The arrival times of 204 trains were noted and the number of minutes, \(t\), that each train was late was recorded. The results are summarised in the table.

1. Explain what \(- 2 \leqslant t < 0\) means about the arrival times of trains.
2. Draw a cumulative frequency graph, and from it estimate the median and the interquartile range of the number of minutes late of these trains.

1 Rachel measured the lengths in millimetres of some of the leaves on a tree. Her results are recorded below. $$\begin{array} { l l l l l l l l l l } 32 & 35 & 45 & 37 & 38 & 44 & 33 & 39 & 36 & 45 \end{array}$$ Find the mean and standard deviation of the lengths of these leaves.

5 The pulse rates, in beats per minute, of a random sample of 15 small animals are shown in the following table.

1. Draw a stem-and-leaf diagram to represent the data.
2. Find the median and the quartiles.
3. On graph paper, using a scale of 2 cm to represent 10 beats per minute, draw a box-and-whisker plot of the data.

4 A library has many identical shelves. All the shelves are full and the numbers of books on each shelf in a certain section are summarised by the following stem-and-leaf diagram. Key: 3 | 6 represents 36 books

1. Find the number of shelves in this section of the library.
2. Draw a box-and-whisker plot to represent the data. In another section all the shelves are full and the numbers of books on each shelf are summarised by the following stem-and-leaf diagram.

   2	12222334566679	\(( 13 )\)
   3	01112334456677788	\(( 15 )\)
   4	223357789

   Key: 3 | 6 represents 36 books
3. There are fewer books in this section than in the previous section. State one other difference between the books in this section and the books in the previous section.

6 The following table gives the marks, out of 75, in a pure mathematics examination taken by 234 students.

1. Draw a histogram on graph paper to represent these results.
2. Calculate estimates of the mean mark and the standard deviation.

1 Anita made observations of the maximum temperature, \(t ^ { \circ } \mathrm { C }\), on 50 days. Her results are summarised by \(\Sigma t = 910\) and \(\Sigma ( t - \bar { t } ) ^ { 2 } = 876\), where \(\bar { t }\) denotes the mean of the 50 observations. Calculate \(\bar { t }\) and the standard deviation of the observations.

4 The weights in grams of a number of stones, measured correct to the nearest gram, are represented in the following table. A histogram is drawn with a scale of 1 cm to 1 unit on the vertical axis, which represents frequency density. The \(1 - 10\) rectangle has height 3 cm .

1. Calculate the value of \(x\) and the height of the 51-70 rectangle.
2. Calculate an estimate of the mean weight of the stones.

3 The table summarises the times that 112 people took to travel to work on a particular day.

1. State which time interval in the table contains the median and which time interval contains the upper quartile.
2. On graph paper, draw a histogram to represent the data.
3. Calculate an estimate of the mean time to travel to work.

4 In a survey, the percentage of meat in a certain type of take-away meal was found. The results, to the nearest integer, for 193 take-away meals are summarised in the table.

1. Calculate estimates of the mean and standard deviation of the percentage of meat in these take-away meals.
2. Draw, on graph paper, a histogram to illustrate the information in the table.

3 Swati measured the lengths, \(x \mathrm {~cm}\), of 18 stick insects and found that \(\Sigma x ^ { 2 } = 967\). Given that the mean length is \(\frac { 58 } { 9 } \mathrm {~cm}\), find the values of \(\Sigma ( x - 5 )\) and \(\Sigma ( x - 5 ) ^ { 2 }\).

4 The following histogram summarises the times, in minutes, taken by 190 people to complete a race. \includegraphics[max width=\textwidth, alt={}, center]{df246a50-157b-49f7-bba0-f9b86960b8b9-2_1210_1125_1251_513}

1. Show that 75 people took between 200 and 250 minutes to complete the race.
2. Calculate estimates of the mean and standard deviation of the times of the 190 people.
3. Explain why your answers to part (ii) are estimates.

4 Barry weighs 20 oranges and 25 lemons. For the oranges, the mean weight is 220 g and the standard deviation is 32 g . For the lemons, the mean weight is 118 g and the standard deviation is 12 g .

1. Find the mean weight of the 45 fruits.
2. The individual weights of the oranges in grams are denoted by \(x _ { o }\), and the individual weights of the lemons in grams are denoted by \(x _ { l }\). By first finding \(\Sigma x _ { o } ^ { 2 }\) and \(\Sigma x _ { l } ^ { 2 }\), find the variance of the weights of the 45 fruits.

1 Find the mean and variance of the following data. $$\begin{array} { l l l l l l l l l l } 5 & - 2 & 12 & 7 & - 3 & 2 & - 6 & 4 & 0 & 8 \end{array}$$

4 The following back-to-back stem-and-leaf diagram shows the times to load an application on 61 smartphones of type \(A\) and 43 smartphones of type \(B\).
(7) Key: 3 | 2 | 1 means 0.23 seconds for type \(A\) and 0.21 seconds for type \(B\).

1. Find the median and quartiles for smartphones of type \(A\). You are given that the median, lower quartile and upper quartile for smartphones of type \(B\) are 0.46 seconds, 0.36 seconds and 0.63 seconds respectively.
2. Represent the data by drawing a pair of box-and-whisker plots in a single diagram on graph paper.
3. Compare the loading times for these two types of smartphone.

2 A traffic camera measured the speeds, \(x\) kilometres per hour, of 8 cars travelling along a certain street, with the following results. $$\begin{array} { l l l l l l l l } 62.7 & 59.6 & 64.2 & 61.5 & 68.3 & 66.9 & 62.0 & 62.3 \end{array}$$

1. Find \(\Sigma ( x - 62 )\).
2. Find \(\Sigma ( x - 62 ) ^ { 2 }\).
3. Find the mean and variance of the speeds of the 8 cars.

4

1. Amy measured her pulse rate while resting, \(x\) beats per minute, at the same time each day on 30 days. The results are summarised below. $$\Sigma ( x - 80 ) = - 147 \quad \Sigma ( x - 80 ) ^ { 2 } = 952$$ Find the mean and standard deviation of Amy's pulse rate.
2. Amy's friend Marok measured her pulse rate every day after running for half an hour. Marok's pulse rate, in beats per minute, was found to have a mean of 148.6 and a standard deviation of 18.5. Assuming that pulse rates have a normal distribution, find what proportion of Marok's pulse rates, after running for half an hour, were above 160 beats per minute.

1 For \(n\) values of the variable \(x\), it is given that \(\Sigma ( x - 100 ) = 216\) and \(\Sigma x = 2416\). Find the value of \(n\).

5 The weights, in kilograms, of the 15 rugby players in each of two teams, \(A\) and \(B\), are shown below.

1. Represent the data by drawing a back-to-back stem-and-leaf diagram with team \(A\) on the lefthand side of the diagram and team \(B\) on the right-hand side.
2. Find the interquartile range of the weights of the players in team \(A\).
3. A new player joins team \(B\) as a substitute. The mean weight of the 16 players in team \(B\) is now 93.9 kg . Find the weight of the new player.

6 The heights to the nearest metre of 134 office buildings in a certain city are summarised in the table below.

1. Draw a histogram on graph paper to illustrate the data.
2. Calculate estimates of the mean and standard deviation of these heights.

7 The masses, in grams, of components made in factory \(A\) and components made in factory \(B\) are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent the masses of components made in the two factories.
2. Find the median and the interquartile range for the masses of components made in factory \(B\).
3. Make two comparisons between the masses of components made in factory \(A\) and the masses of those made in factory \(B\).

5 The number of people a football stadium can hold is called the 'capacity'. The capacities of 130 football stadiums in the UK, to the nearest thousand, are summarised in the table.

1. On graph paper, draw a histogram to represent this information. Use a scale of 2 cm for a capacity of 10000 on the horizontal axis.
2. Calculate an estimate of the mean capacity of these 130 stadiums.
3. Find which class in the table contains the median and which contains the lower quartile.

5 The tables summarise the heights, \(h \mathrm {~cm}\), of 60 girls and 60 boys.

1. On graph paper, using the same set of axes, draw two cumulative frequency graphs to illustrate the data.
2. On a school trip the students have to enter a cave which is 165 cm high. Use your graph to estimate the percentage of the girls who will be unable to stand upright.
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3. The students are asked to compare the heights of the girls and the boys. State one advantage of using a pair of box-and-whisker plots instead of the cumulative frequency graphs to do this. [1]

5 The number of Olympic medals won in the 2012 Olympic Games by the top 27 countries is shown below.

1. Draw a stem-and-leaf diagram to illustrate the data.
2. Find the median and quartiles and draw a box-and-whisker plot on the grid. \includegraphics[max width=\textwidth, alt={}, center]{4c2afa86-960c-473e-970c-ed16c8434fec-07_1006_1406_1007_411}