2.02f Measures of average and spread

At a stall in a fair, contestants have to estimate the mass of a cake. A group of 10 people made estimates, \(m\) kg, and for each person the value of \((m - 5)\) was recorded. The mean and standard deviation of \((m - 5)\) were found to be 0.74 and 0.13 respectively.

1. Write down the mean and standard deviation of \(m\). [2]

The mean and standard deviation of the estimates made by another group of 15 people were found to be 5.6 kg and 0.19 kg respectively.

1. Calculate the mean of all 25 estimates. [2]
2. Fiona claims that if a group's estimates are more consistent, they are likely to be more accurate. Given that the true mass of the cake is 5.65 kg, comment on this claim. [2]

A camera records the speeds in miles per hour of 15 vehicles on a motorway. The speeds are given below. $$73 \quad 67 \quad 75 \quad 64 \quad 52 \quad 63 \quad 75 \quad 81 \quad 77 \quad 72 \quad 68 \quad 74 \quad 79 \quad 72 \quad 71$$

1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of 50, 60, ... . [4]
2. Write down the median and midrange of the data. [2]
3. Which of the median and midrange would you recommend to measure the central tendency of the data? Briefly explain your answer. [2]

A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears. \includegraphics{figure_7}

1. Calculate the number of pears which are between 90 and 100 mm long. [2]
2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate. [4]
3. Calculate an estimate of the standard deviation. [3]
4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers. [4]
5. Name the type of skewness of the distribution. [1]
6. Illustrate the data using a cumulative frequency diagram. [5]

The stem and leaf diagram shows the weights, rounded to the nearest 10 grams, of 25 female iguanas. \begin{align} 8 &| 3 \quad 9
9 &| 3 \quad 5 \quad 6 \quad 6 \quad 6 \quad 8 \quad 9 \quad 9
10 &| 0 \quad 2 \quad 2 \quad 3 \quad 4 \quad 6 \quad 9
11 &| 2 \quad 4 \quad 7 \quad 8
12 &| 3 \quad 4 \quad 5
13 &| 2 \end{align} Key: \(11|2\) represents 1120 grams

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

The incomes of a sample of 918 households on an island are given in the table below.

1. Draw a histogram to illustrate the data. [5]
2. Calculate an estimate of the mean income. [3]
3. Calculate an estimate of the standard deviation of the incomes. [4]
4. Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation. [4]
5. The incomes were converted into another currency using the formula \(y = 1.15x\). Calculate estimates of the mean and variance of the incomes in the new currency. [3]

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

Number of eggs	1	2	3	4
Frequency	10	40	15	5

1. Find the mean and standard deviation of the number of eggs laid per gull. [5]
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? [2]

The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.

Heating quality (\(x\))	9.1 \(\leqslant x <\) 9.3	9.3 \(< x \leqslant\) 9.5	9.5 \(< x \leqslant\) 9.7	9.7 \(< x \leqslant\) 9.9	9.9 \(< x \leqslant\) 10.1
Frequency	5	7	15	16	7

1. Draw a cumulative frequency diagram to illustrate these data. [5]
2. Use the diagram to estimate the median and interquartile range of the data. [3]
3. Show that there are no outliers in the sample. [3]
4. Three of these 50 sacks are selected at random. Find the probability that
   1. in all three, the heating quality \(x\) is more than 9.5, [3]
   2. in at least two, the heating quality \(x\) is more than 9.5. [4]

The ages, \(x\) years, of the senior members of a running club are summarised in the table below.

Age (\(x\))	\(20 \leqslant x < 30\)	\(30 \leqslant x < 40\)	\(40 \leqslant x < 50\)	\(50 \leqslant x < 60\)	\(60 \leqslant x < 70\)	\(70 \leqslant x < 80\)	\(80 \leqslant x < 90\)
Frequency	10	30	42	23	9	5	1

1. Draw a cumulative frequency diagram to illustrate the data. [5]
2. Use your diagram to estimate the median and interquartile range of the data. [3]

The weights, \(w\) grams, of a random sample of 60 carrots of variety A are summarised in the table below.

Weight	\(30 \leqslant w < 50\)	\(50 \leqslant w < 60\)	\(60 \leqslant w < 70\)	\(70 \leqslant w < 80\)	\(80 \leqslant w < 90\)
Frequency	11	10	18	14	7

1. Draw a histogram to illustrate these data. [5]
2. Calculate estimates of the mean and standard deviation of \(w\). [4]
3. Use your answers to part (ii) to investigate whether there are any outliers. [3]

The weights, \(x\) grams, of a random sample of 50 carrots of variety B are summarised as follows. $$n = 50 \quad \sum x = 3624.5 \quad \sum x^2 = 265416$$

1. Calculate the mean and standard deviation of \(x\). [3]
2. Compare the central tendency and variation of the weights of varieties A and B. [2]

A group of 60 children were each asked to choose an integer value between 1 and 9 inclusive. Their choices are summarised in the table below.

Value chosen	1	2	3	4	5	6	7	8	9
Number of children	3	4	5	10	12	13	7	4	2

1. Calculate the mean and standard deviation of the values chosen. [6]

It is suggested that the value chosen could be modelled by a discrete uniform distribution.

1. Write down the mean that this model would predict. [2]

Given also that the standard deviation according to this model would be 2.58,

1. explain why this model is not suitable and suggest why this is the case. [2]

The number of patients attending a hospital trauma clinic each day was recorded over several months, giving the data in the table below.

Number of patients	10 - 19	20 - 29	30 - 34	35 - 39	40 - 44	45 - 49	50 - 69
Frequency	2	18	24	30	27	14	5

These data are represented by a histogram. Given that the bar representing the 20 - 29 group is 2 cm wide and 7.2 cm high,

1. calculate the dimensions of the bars representing the groups
   1. 30 - 34
   2. 50 - 69
   [6]
2. Use linear interpolation to estimate the median and quartiles of these data. [6]

The lowest and highest numbers of patients recorded were 14 and 67 respectively.

1. Represent these data with a boxplot drawn on graph paper and describe the skewness of the distribution. [5]

Jane and Tahira play together in a basketball team. The list below shows the number of points that Jane scored in each of 30 games.

1. Construct a stem and leaf diagram for these data. [3 marks]
2. Find the median and quartiles for these data. [4 marks]
3. Represent these data with a boxplot. [3 marks]

Tahira played in the same 30 games and her lowest and highest points total in a game were 19 and 41 respectively. The quartiles for Tahira were 27, 31 and 35 respectively.

1. Using the same scale draw a boxplot for Tahira's points totals. [2 marks]
2. Compare and contrast the number of points scored per game by Jane and Tahira. [3 marks]

A call-centre dealing with complaints collected data on how long customers had to wait before an operator was free to take their call. The lower quartile of the data was 12.7 minutes and the interquartile range was 5.8 minutes.

1. Find the value of the upper quartile of the data. [1 mark]

It is suggested that a normal distribution could be used to model the waiting time.

1. Calculate correct to 3 significant figures the mean and variance of this normal distribution based on the values of the quartiles. [8 marks]

The actual mean and variance of the data were 15.3 minutes and 20.1 minutes\(^2\) respectively.

1. Comment on the suitability of the model. [2 marks]

A College offers evening classes in GCSE Mathematics and English. In order to assess which age groups were reluctant to use the classes, the College collected data on the age in completed years of those currently attending each course. The results are shown in this back-to-back stem and leaf diagram. \includegraphics{figure_4} Key: \(1 | 3 | 2\) means age 31 doing Mathematics and age 32 doing English

1. Find the median and quartiles of the age in completed years of those attending the Mathematics classes. [4 marks]
2. On graph paper, draw a box plot representing the data for the Mathematics class. [3 marks]

The median and quartiles of the age in completed years of those attending the English classes are 25, 41 and 57 years respectively.

1. Draw a box plot representing the data for the English class using the same scale as for the data from the Mathematics class. [3 marks]
2. Using your box plots, compare and contrast the ages of those taking each class. [4 marks]

In a packet of 40 biscuits, the number of currants in each biscuit is as follows

1. Find the mean and variance of the random variable \(X\) representing the number of currants per biscuit. [4 marks]
2. State an appropriate model for the distribution of \(X\), giving two reasons for your answer. [2 marks]

Another machine produces biscuits with a mean of 1.9 currants per biscuit.

1. Determine which machine is more likely to produce a biscuit with at least two currants. [5 marks]

As part of a research project, the masses, \(m\) grams, of a random sample of 1000 pebbles from a certain beach were recorded. The results are summarised in the table.

Mass (g)	\(50 \leq m < 150\)	\(150 \leq m < 200\)	\(200 \leq m < 250\)	\(250 \leq m < 350\)
Frequency	162	318	355	165

1. Calculate estimates of the mean and standard deviation of these masses. [2]

The masses, \(x\) grams, of a random sample of 1000 pebbles on a different beach were also found. It was proposed that the distribution of these masses should be modelled by the random variable \(X \sim N(200, 3600)\).

1. Use the model to find \(P(150 < X < 210)\). [1]
2. Use the model to determine \(x_1\) such that \(P(160 < X < x_1) = 0.6\), giving your answer correct to five significant figures. [3]

It was found that the smallest and largest masses of the pebbles in this second sample were 112 g and 288 g respectively.

1. Use these results to show that the model may not be appropriate. [1]
2. Suggest a different value of a parameter of the model in the light of these results. [2]

A student wishes to prove that, for all positive integers \(a\) and \(b\), \(a^2 - 4b \neq 2\).

1. Prove that \(a^2 - 4b = 2 \Rightarrow a\) is even. [2]
2. Hence or otherwise prove that, for all positive integers \(a\) and \(b\), \(a^2 - 4b \neq 2\). [3]

A comparison of the masses (in kg) of convertible cars was made using the Large Data Set. A sample of 20 masses was chosen from both the 2002 data and the 2016 data. The masses of the 20 cars in each sample were used to create a box plot for each year. The box plots were labelled Box Plot A and Box Plot B as shown in the diagram below. \includegraphics{figure_19}

1. Estimate the median of the masses from Box Plot A [1 mark]
2. It is claimed that Box Plot B must be incorrectly drawn.
   1. Give a reason why this claim was made. [1 mark]
   2. Comment on the validity of this claim. [1 mark]
3. It is claimed that Box Plot B must be from the 2002 data. Give a reason why this claim is correct. [1 mark]

Which one of the following is not a measure of spread? Circle your answer. [1 mark] median \(\qquad\) range \(\qquad\) standard deviation \(\qquad\) variance

The boxplot below represents the time spent in hours by students revising for a history exam. \includegraphics{figure_16}

1. Use the information in the boxplot to state the value of a measure of central tendency of the revision times, stating clearly which measure you are using. [1 mark]
2. Use the information in the boxplot to explain why the distribution of revision times is negatively skewed. [1 mark]

The box plot below summarises the CO\(_2\) emissions, in g/km, for cars in the Large Data Set from the London and North West regions. \includegraphics{figure_12}

1. Using the box plot, give one comparison of central tendency and one comparison of spread for the two regions. [2 marks]
2. Jaspal, an environmental researcher, used all of the data in the Large Data Set to produce a statistical comparison of the CO\(_2\) and CO emissions in regions of England. Using your knowledge of the Large Data Set, give two reasons why his conclusions may be invalid. [2 marks]

The table below is an extract from the Large Data Set.

1. 1. Calculate the mean and standard deviation of CO₂ emissions in the table. [2 marks]
   2. Any value more than 2 standard deviations from the mean can be identified as an outlier. Determine, using this definition of an outlier, if there are any outliers in this sample of CO₂ emissions. Fully justify your answer. [2 marks]
2. Maria claims that the last line in the table must contain two errors. Use your knowledge of the Large Data Set to comment on Maria's claim. [2 marks]

The box plot below shows summary data for the number of minutes late that buses arrived at a rural bus stop. \includegraphics{figure_12} Identify which term best describes the distribution of this data. Circle your answer. [1 mark] negatively skewed \quad\quad normal \quad\quad positively skewed \quad\quad symmetrical

A random sample of 84 students was asked how many revision websites they had visited in the past month. The data is summarised in the table below. Find the interquartile range of the number of websites visited by these 84 students. Circle your answer. [1 mark] 3 \quad 4 \quad 19 \quad 42