Measures of Location and Spread

3 The test marks of 14 students are displayed in a stem-and-leaf diagram, as shown below.
\includegraphics[max width=\textwidth, alt={}, center]{e23cb28b-49e5-436a-942d-e6320029c634-2_234_261_1425_482} Key: 1 | 6 means 16 marks

1. Find the lower quartile.
2. Given that the median is 32 , find the values of \(w\) and \(x\).
3. Find the possible values of the upper quartile.
4. State one advantage of a stem-and-leaf diagram over a box-and-whisker plot.
5. State one advantage of a box-and-whisker plot over a stem-and-leaf diagram.

1 A computer can generate random numbers which are either 0 or 2 . On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers.

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

1 The values of 5 independent observations from a population can be summarised by $$\Sigma x = 75.8 , \quad \Sigma x ^ { 2 } = 1154.58 .$$ Find unbiased estimates of the population mean and variance.

3 The mean and standard deviation of 20 values of \(x\) are 60 and 4 respectively.

1. Find the values of \(\Sigma x\) and \(\Sigma x ^ { 2 }\).
   Another 10 values of \(x\) are such that their sum is 550 and the sum of their squares is 40500 .
2. Find the mean and standard deviation of all these 30 values of \(x\).

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

14 Given that \(\sum x = 364 , \sum x ^ { 2 } = 19412 , n = 10\), find \(\sigma\), the standard deviation of \(X\). Circle your answer.
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24.844 .1616 .21941 .2

4 The weights in kilograms of 11 bags of sugar and 7 bags of flour are as follows.
Sugar: 1.9611 .98312 .00812 .0141 .9681 .9941 .2 .0112 .0171 .9771 .9841 .989
Flour: \(\begin{array} { l l l l l l l } 1.945 & 1.962 & 1.949 & 1.977 & 1.964 & 1.941 & 1.953 \end{array}\)

1. Represent this information on a back-to-back stem-and-leaf diagram with sugar on the left-hand side.
2. Find the median and interquartile range of the weights of the bags of sugar.

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.

4 The marks, \(x\), of a random sample of 50 students in a test were summarised as follows. $$n = 50 \quad \Sigma x = 1508 \quad \Sigma x ^ { 2 } = 51825$$

1. Calculate unbiased estimates of the population mean and variance.
2. Each student's mark is scaled using the formula \(y = 1.5 x + 10\). Find estimates of the population mean and variance of the scaled marks, \(y\).

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.

2 The values, \(x\), in a particular set of data are summarised by $$\Sigma ( x - 25 ) = 133 , \quad \Sigma ( x - 25 ) ^ { 2 } = 3762 .$$ The mean, \(\bar { x }\), is 28.325 .

1. Find the standard deviation of \(x\).
2. Find \(\Sigma x ^ { 2 }\).

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.

1 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.

16

1. (ii) &

   16

   where \(n\) is the total number of cars which had a measured hydrocarbon emission in the Large Data Set.

   16

   Find the mean of \(X\)

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   16

   
   \hline &
   \hline \end{tabular} \end{center} 16

(ii) State one type of emission where more than 80\% of the data is known for cars in the entire UK Department for Transport Stock Vehicle Database.
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1 Twelve values of \(x\) are shown below. Find the mean and standard deviation of \(( x - 1760 )\). Hence find the mean and standard deviation of \(x\). [4]

3 A sample of 36 data values, \(x\), gave \(\Sigma ( x - 45 ) = - 148\) and \(\Sigma ( x - 45 ) ^ { 2 } = 3089\).

1. Find the mean and standard deviation of the 36 values.
2. One extra data value of 29 was added to the sample. Find the standard deviation of all 37 values.

1 The stem and leaf diagram illustrates the weights in grams of 20 house sparrows. Key: \(\quad 27 \quad \mid \quad 7 \quad\) represents 27.7 grams

1. Find the median and interquartile range of the data.
2. Determine whether there are any outliers.

1
\includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-03_1372_1194_260_479} The times taken by 120 children to complete a particular puzzle are represented in the cumulative frequency graph.

1. Use the graph to estimate the interquartile range of the data.
   35\% of the children took longer than \(T\) seconds to complete the puzzle.
2. Use the graph to estimate the value of \(T\).

3 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.

5 In a game it is known that:

• 25\% of players score 0
• 30\% of players score 5
• 35\% of players score 10
• 10\% of players score 20

Players receive prize money, in pounds, equal to 100 times their score.
5

1. State the modal score.
   [0pt] [1 mark] 5
2. Find the median score.
   5
3. Find the mean prize money received by a player.

3 Jagdeesh measured the lengths, \(x\) minutes, of 60 randomly chosen lectures. His results are summarised below.

1. Calculate unbiased estimates of the population mean and variance.
2. Calculate a \(98 \%\) confidence interval for the population mean.

1 Each of a group of 10 boys estimates the length of a piece of string. The estimates, in centimetres, are as follows. $$\begin{array} { l l l l l l l l l l } 37 & 40 & 45 & 38 & 36 & 38 & 42 & 38 & 40 & 39 \end{array}$$

1. Find the mode.
2. Find the median and the interquartile range.

2 The amounts of money, \(x\) dollars, that 24 people had in their pockets are summarised by \(\Sigma ( x - 36 ) = - 60\) and \(\Sigma ( x - 36 ) ^ { 2 } = 227.76\). Find \(\Sigma x\) and \(\Sigma x ^ { 2 }\).

9 Jo is investigating the popularity of a certain band amongst students at her school. She decides to survey a sample of 100 students.

1. State an advantage of using a stratified sample rather than a simple random sample.
2. Explain whether it would be reasonable for Jo to use her results to draw conclusions about all students in the UK.

4 The times taken, in minutes, to complete a word processing task by 250 employees at a particular company are summarised in the table.

1. Draw a histogram to represent this information.
   \includegraphics[max width=\textwidth, alt={}, center]{3e74785d-5981-480c-a0fd-f43d5d227f2d-06_1201_1198_1050_516} From the data, the estimate of the mean time taken by these 250 employees is 43.2 minutes.
2. Calculate an estimate for the standard deviation of these times.

1 The masses in kilograms of 50 children having a medical check-up were recorded correct to the nearest kilogram. The results are shown in the table.

1. Find which class interval contains the lower quartile.
2. On the grid, draw a histogram to illustrate the data in the table.
   \includegraphics[max width=\textwidth, alt={}, center]{dd75fa20-fead-48d6-aff4-c5e733769f9f-02_1397_1397_1187_415}

1 At a tourist information office the numbers of people seeking information each hour over the course of a 12-hour day are shown below. $$\begin{array} { l l l l l l l l l l l l } 6 & 25 & 38 & 39 & 31 & 18 & 35 & 31 & 33 & 15 & 21 & 28 \end{array}$$

1. Construct a sorted stem and leaf diagram to represent these data.
2. State the type of skewness suggested by your stem and leaf diagram.
3. For these data find the median, the mean and the mode. Comment on the usefulness of the mode in this case.

3 Jack has to choose a random sample of 8 people from the 750 members of a sports club.

1. Explain fully how he can use random numbers to choose the sample. Jack asks each person in the sample how much they spent last week in the club café. The results, in dollars, were as follows. $$\begin{array} { l l l l l l l l } 15 & 25 & 30 & 8 & 12 & 18 & 27 & 25 \end{array}$$
2. Find unbiased estimates of the population mean and variance.
3. Explain briefly what is meant by 'population' in this question.

1 Twelve tourists were asked to estimate the height, in metres, of a new building. Their estimates were as follows. $$\begin{array} { l l l l l l l l l l l l } 50 & 45 & 62 & 30 & 40 & 55 & 110 & 38 & 52 & 60 & 55 & 40 \end{array}$$

1. Find the median and the interquartile range for the data.
2. Give a disadvantage of using the mean as a measure of the central tendency in this case.

2. The mark, \(x\), scored by each student who sat a statistics examination is coded using $$y = 1.4 x - 20$$ The coded marks have mean 60.8 and standard deviation 6.60 Find the mean and the standard deviation of \(x\).
\includegraphics[max width=\textwidth, alt={}, center]{8270bcae-494c-4248-8229-a72e9e84eab0-04_99_97_2613_1784}

6 A sample of 5 randomly selected values of a variable \(X\) is as follows: $$\begin{array} { l l l l l } 1 & 2 & 6 & 1 & a \end{array}$$ where \(a > 0\).
Given that an unbiased estimate of the variance of \(X\) calculated from this sample is \(\frac { 11 } { 2 }\), find the value of \(a\).

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.

4 The Mathematics and English A-level marks of 1400 pupils all taking the same examinations are shown in the cumulative frequency graphs below. Both examinations are marked out of 100 .
\includegraphics[max width=\textwidth, alt={}, center]{be6c6525-a20c-42d0-8fef-1cd254baaa76-06_1682_1246_404_445} Use suitable data from these graphs to compare the central tendency and spread of the marks in Mathematics and English.

13 Denzel wants to buy a car with a propulsion type other than petrol or diesel.
He takes a sample, from the Large Data Set, of the CO2 emissions, in \(\mathrm { g } / \mathrm { km }\), of cars with one particular propulsion type. The sample is as follows $$\begin{array} { l l l l l l l l } 82 & 13 & 96 & 49 & 96 & 92 & 70 & 81 \end{array}$$ 13

1. Using your knowledge of the Large Data Set, state which propulsion type this sample is for, giving a reason for your answer.
   13
2. Calculate the mean of the sample.
   13
3. Calculate the standard deviation of the sample.
   13
4. Denzel claims that the value 13 is an outlier. 13
5. 1. Any value more than 2 standard deviations from the mean can be regarded as an outlier. Verify that Denzel's claim is correct.
      13
6. (ii) State what effect, if any, removing the value 13 from the sample would have on the standard deviation.

4 The heights, \(x \mathrm {~cm}\), of a group of 28 people were measured. The mean height was found to be 172.6 cm and the standard deviation was found to be 4.58 cm . A person whose height was 161.8 cm left the group.

1. Find the mean height of the remaining group of 27 people.
2. Find \(\Sigma x ^ { 2 }\) for the original group of 28 people. Hence find the standard deviation of the heights of the remaining group of 27 people.

2. (a) Explain what you understand by (i) a population and (ii) a sampling frame. The population and the sampling frame may not be the same.
(b) Explain why this might be the case.
(c) Give an example, justifying your choices, to illustrate when you might use

1. a census,
2. a sample.

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.

2 The times taken, in minutes, by 80 people to complete a crossword puzzle are summarised by the box and whisker plot below.
\includegraphics[max width=\textwidth, alt={}, center]{088972e9-bfcd-429c-9145-af274a4c0a58-2_163_857_436_642}

1. Write down the range and the interquartile range of the times.
2. Determine whether any of the times can be regarded as outliers.
3. Describe the shape of the distribution of the times.

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.

6 The independent variables \(X\) and \(Y\) have distributions with the same variance \(\sigma ^ { 2 }\). Random samples of \(N\) observations of \(X\) and \(2 N\) observations of \(Y\) are taken, and the results are summarised by $$\Sigma x = 4 , \quad \Sigma x ^ { 2 } = 10 , \quad \Sigma y = 8 , \quad \Sigma y ^ { 2 } = 102 .$$ These data give a pooled estimate of 10 for \(\sigma ^ { 2 }\). Find \(N\).

2. An estate agent recorded the price per square metre, \(p \pounds / \mathrm { m } ^ { 2 }\), for 7 two-bedroom houses. He then coded the data using the coding \(q = \frac { p - a } { b }\), where \(a\) and \(b\) are positive constants. His results are shown in the table below.

1. Find the value of \(a\) and the value of \(b\) The estate agent also recorded the distance, \(d \mathrm {~km}\), of each house from the nearest train station. The results are summarised below. $$\mathrm { S } _ { d d } = 1.02 \quad \mathrm {~S} _ { q q } = 8.22 \quad \mathrm {~S} _ { d q } = - 2.17$$
2. Calculate the product moment correlation coefficient between \(d\) and \(q\)
3. Write down the value of the product moment correlation coefficient between \(d\) and \(p\) The estate agent records the price and size of 2 additional two-bedroom houses, \(H\) and \(J\).

   House	Price \(( \pounds )\)	Size \(\left( \mathrm { m } ^ { 2 } \right)\)
   \(H\)	156400	85
   \(J\)	172900	95

4. Suggest which house is most likely to be closer to a train station. Justify your answer.

1 A summary of 20 values of \(x\) gives $$\Sigma ( x - 30 ) = 439 , \quad \Sigma ( x - 30 ) ^ { 2 } = 12405 .$$ A summary of another 25 values of \(x\) gives $$\sum ( x - 30 ) = 470 , \quad \sum ( x - 30 ) ^ { 2 } = 11346 .$$

1. Find the mean of all 45 values of \(x\).
2. Find the standard deviation of all 45 values of \(x\).

8 Alison selects 10 of her male friends. For each one she measures the distance between his eyes. The distances, measured in mm , are as follows: The mean of these data is 61.4 . The sample standard deviation is 5.232 , correct to 3 decimal places. One of the friends decides he does not want his measurement to be used. Alison replaces his measurement with the measurement from another male friend. This increases the mean to 62.0 and reduces the standard deviation. Give a possible value for the measurement which has been removed and find the measurement which has replaced it.