2.02f Measures of average and spread

1. Over a period of time, the number of people \(x\) leaving a hotel each morning was recorded. These data are summarised in the stem and leaf diagram below.

For these data,

1. write down the mode,
2. find the values of the three quartiles. Given that \(\Sigma x = 1335\) and \(\Sigma x ^ { 2 } = 71801\), find
3. the mean and the standard deviation of these data. One measure of skewness is found using $$\frac { \text { mean - mode } } { \text { standard deviation } } \text {. }$$
4. Evaluate this measure to show that these data are negatively skewed.
5. Give two other reasons why these data are negatively skewed.
   

1. Summarised below are the distances, to the nearest mile, travelled to work by a random sample of 120 commuters.

For this distribution,

1. describe its shape,
2. use linear interpolation to estimate its median. The mid-point of each class was represented by \(x\) and its corresponding frequency by \(f\) giving $$\Sigma f x = 3550 \text { and } \Sigma f x ^ { 2 } = 138020$$
3. Estimate the mean and the standard deviation of this distribution. One coefficient of skewness is given by $$\frac { 3 ( \text { mean - median } ) } { \text { standard deviation } } .$$
4. Evaluate this coefficient for this distribution.
5. State whether or not the value of your coefficient is consistent with your description in part (a). Justify your answer.
6. State, with a reason, whether you should use the mean or the median to represent the data in this distribution.
7. State the circumstance under which it would not matter whether you used the mean or the median to represent a set of data.
   

2. Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke. A doctor tested the blood of 12 patients, who claimed to smoke a packet of cigarettes a day, for cotinine. The results, in appropriate units, are shown below. $$\text { [You may use } \sum x ^ { 2 } = 724 \text { 961] }$$

1. Find the mean and standard deviation of the level of cotinine in a patient's blood.
2. Find the median, upper and lower quartiles of these data. A doctor suspects that some of his patients have been smoking more than a packet of cigarettes per day. He decides to use \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) to determine if any of the cotinine results are far enough away from the upper quartile to be outliers.
3. Identify which patient(s) may have been smoking more than a packet of cigarettes a day. Show your working clearly. Research suggests that cotinine levels in the blood form a skewed distribution.
   One measure of skewness is found using \(\frac { \left( Q _ { 1 } - 2 Q _ { 2 } + Q _ { 3 } \right) } { \left( Q _ { 3 } - Q _ { 1 } \right) }\).
4. Evaluate this measure and describe the skewness of these data.

4. In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week. The total length of calls, \(y\) minutes, for the 11 students were $$17,23,35,36,51,53,54,55,60,77,110$$

1. Find the median and quartiles for these data. A value that is greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or smaller than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) is defined as an outlier.
2. Show that 110 is the only outlier.
3. Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier. The value of 110 is omitted.
4. Show that \(S _ { y y }\) for the remaining 10 students is 2966.9 These 10 students were each asked how many text messages, \(x\), they sent in the same week. The values of \(S _ { x x }\) and \(S _ { x y }\) for these 10 students are \(S _ { x x } = 3463.6\) and \(S _ { x y } = - 18.3\).
5. Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students. A parent believes that a student who sends a large number of text messages will spend fewer minutes on calls.
6. Comment on this belief in the light of your calculation in part (e). \includegraphics[max width=\textwidth, alt={}, center]{d5d000c7-de42-461a-ba05-6c8b2c333780-09_611_1593_297_178}

5. In a shopping survey a random sample of 104 teenagers were asked how many hours, to the nearest hour, they spent shopping in the last month. The results are summarised in the table below. A histogram was drawn and the group ( \(8 - 10\) ) hours was represented by a rectangle that was 1.5 cm wide and 3 cm high.

1. Calculate the width and height of the rectangle representing the group (16-25) hours.
2. Use linear interpolation to estimate the median and interquartile range.
3. Estimate the mean and standard deviation of the number of hours spent shopping.
4. State, giving a reason, the skewness of these data.
5. State, giving a reason, which average and measure of dispersion you would recommend to use to summarise these data.

1. Keith records the amount of rainfall, in mm , at his school, each day for a week. The results are given below.
   0.0
   0.5
   1.8
   2.8
   2.3
   5.6
   9.4

Jenny then records the amount of rainfall, \(x \mathrm {~mm}\), at the school each day for the following 21 days. The results for the 21 days are summarised below. $$\sum x = 84.6$$

1. Calculate the mean amount of rainfall during the whole 28 days. Keith realises that he has transposed two of his figures. The number 9.4 should have been 4.9 and the number 0.5 should have been 5.0 Keith corrects these figures.
2. State, giving your reason, the effect this will have on the mean.
   

3. Over a long period of time a small company recorded the amount it received in sales per month. The results are summarised below. An outlier is an observation that falls
either \(1.5 \times\) interquartile range above the upper quartile or \(1.5 \times\) interquartile range below the lower quartile.

1. On the graph paper below, draw a box plot to represent these data, indicating clearly any outliers.
   (5) \includegraphics[max width=\textwidth, alt={}, center]{c78ec7b6-dd06-4de1-94c2-052a5577dd10-05_933_1226_1283_367}
2. State the skewness of the distribution of the amount of sales received. Justify your answer.
3. The company claims that for \(75 \%\) of the months, the amount received per month is greater than \(\pounds 10000\). Comment on this claim, giving a reason for your answer.
   (2)

5. On a randomly chosen day, each of the 32 students in a class recorded the time, \(t\) minutes to the nearest minute, they spent on their homework. The data for the class is summarised in the following table.

1. Use interpolation to estimate the value of the median. Given that $$\sum t = 1414 \quad \text { and } \quad \sum t ^ { 2 } = 69378$$
2. find the mean and the standard deviation of the times spent by the students on their homework.
3. Comment on the skewness of the distribution of the times spent by the students on their homework. Give a reason for your answer.

1. The marks, \(x\), of 45 students randomly selected from those students who sat a mathematics examination are shown in the stem and leaf diagram below.

1. Write down the modal mark of these students.
2. Find the values of the lower quartile, the median and the upper quartile. For these students \(\sum x = 2497\) and \(\sum x ^ { 2 } = 143369\)
3. Find the mean and the standard deviation of the marks of these students.
4. Describe the skewness of the marks of these students, giving a reason for your answer. The mean and standard deviation of the marks of all the students who sat the examination were 55 and 10 respectively. The examiners decided that the total mark of each student should be scaled by subtracting 5 marks and then reducing the mark by a further \(10 \%\).
5. Find the mean and standard deviation of the scaled marks of all the students.

1. A survey of 100 households gave the following results for weekly income \(\pounds y\).

(You may use \(\sum f y ^ { 2 } = 12452\) 800)
A histogram was drawn and the class \(200 \leqslant y < 240\) was represented by a rectangle of width 2 cm and height 7 cm .

1. Calculate the width and the height of the rectangle representing the class $$320 \leqslant y < 400$$
2. Use linear interpolation to estimate the median weekly income to the nearest pound.
3. Estimate the mean and the standard deviation of the weekly income for these data. One measure of skewness is \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\).
4. Use this measure to calculate the skewness for these data and describe its value. Katie suggests using the random variable \(X\) which has a normal distribution with mean 320 and standard deviation 150 to model the weekly income for these data.
5. Find \(\mathrm { P } ( 240 < X < 400 )\).
6. With reference to your calculations in parts (d) and (e) and the data in the table, comment on Katie's suggestion.

1. Each of the 25 students on a computer course recorded the number of minutes \(x\), to the nearest minute, spent surfing the internet during a given day. The results are summarised below.

$$\Sigma x = 1075 , \Sigma x ^ { 2 } = 44625 .$$

1. Find \(\mu\) and \(\sigma\) for these data. Two other students surfed the internet on the same day for 35 and 51 minutes respectively.
2. Without further calculation, explain the effect on the mean of including these two students.
   (2)
   

6. Three swimmers Alan, Diane and Gopal record the number of lengths of the swimming pool they swim during each practice session over several weeks. The stem and leaf diagram below shows the results for Alan.

1. Find the three quartiles for Alan's results. The table below summarises the results for Diane and Gopal.

   	Diane	Gopal
   Smallest value	35	25
   Lower quartile	37	34
   Median	42	42
   Upper quartile	53	50
   Largest value	65	57

2. Using the same scale and on the same sheet of graph paper draw box plots to represent the data for Alan, Diane and Gopal.
3. Compare and contrast the three box plots.

4. The attendance at college of a group of 18 students was recorded for a 4-week period. The number of students actually attending each of 16 classes are shown below.

1. 1. Calculate the mean and the standard deviation of the number of students attending these classes.
   2. Express the mean as a percentage of the 18 students in the group. In the same 4-week period, the attendance of a different group of 20, students is shown below.

      20	16	18	19
      15	14	14	15
      18	15	16	17
      16	18	15	14

2. Construct a back-to-back stem and leaf diagram to represent the attendance in both groups.
3. Find the mode, median and inter-quartile range for each group of students. The mean percentage attendance and standard deviation for the second group of students are 81.25 and 1.82 respectively.
4. Compare and contrast the attendance of these 2 groups of students.

2. Sunita and Shelley talk to one another once a week on the telephone. Over many weeks they recorded, to the nearest minute, the number of minutes spent in conversation on each occasion. The following table summarises their results. Two of the conversations were chosen at random.

1. Find the probability that both of them were longer than 24.5 minutes. The mid-point of each class was represented by \(x\) and its corresponding frequency by \(f\), giving \(\Sigma f x = 1060\).
2. Calculate an estimate of the mean time spent on their conversations. During the following 25 weeks they monitored their weekly conversations and found that at the end of the 80 weeks their overall mean length of conversation was 21 minutes.
3. Find the mean time spent in conversation during these 25 weeks.
4. Comment on these two mean values.

2. The box plot in Figure 1 shows a summary of the weights of the luggage, in kg, for each musician in an orchestra on an overseas tour. \begin{figure}[h] \end{figure} The airline's recommended weight limit for each musician's luggage was 45 kg . Given that none of the musicians' luggage weighed exactly 45 kg ,

1. state the proportion of the musicians whose luggage was below the recommended weight limit. A quarter of the musicians had to pay a charge for taking heavy luggage.
2. State the smallest weight for which the charge was made.
3. Explain what you understand by the + on the box plot in Figure 1, and suggest an instrument that the owner of this luggage might play.
4. Describe the skewness of this distribution. Give a reason for your answer. One musician of the orchestra suggests that the weights of luggage, in kg, can be modelled by a normal distribution with quartiles as given in Figure 1.
5. Find the standard deviation of this normal distribution.

5. \begin{figure}[h] \end{figure} Figure 2 shows a histogram for the variable \(t\) which represents the time taken, in minutes, by a group of people to swim 500 m .

1. Complete the frequency table for \(t\).

   \(t\)	\(5 - 10\)	\(10 - 14\)	\(14 - 18\)	\(18 - 25\)	\(25 - 40\)
   Frequency	10	16	24

2. Estimate the number of people who took longer than 20 minutes to swim 500 m .
3. Find an estimate of the mean time taken.
4. Find an estimate for the standard deviation of \(t\).
5. Find the median and quartiles for \(t\). One measure of skewness is found using \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\).
6. Evaluate this measure and describe the skewness of these data.

2. The age in years of the residents of two hotels are shown in the back to back stem and leaf diagram below. Abbey Hotel \(8 | 5 | 0\) means 58 years in Abbey hotel and 50 years in Balmoral hotel Balmoral Hotel For the Balmoral Hotel,

1. write down the mode of the age of the residents,
2. find the values of the lower quartile, the median and the upper quartile.
3. 1. Find the mean, \(\bar { x }\), of the age of the residents.
   2. Given that \(\sum x ^ { 2 } = 81213\) find the standard deviation of the age of the residents. One measure of skewness is found using $$\frac { \text { mean - mode } } { \text { standard deviation } }$$
4. Evaluate this measure for the Balmoral Hotel. For the Abbey Hotel, the mode is 39 , the mean is 33.2 , the standard deviation is 12.7 and the measure of skewness is - 0.454
5. Compare the two age distributions of the residents of each hotel.

4. A researcher measured the foot lengths of a random sample of 120 ten-year-old children. The lengths are summarised in the table below.

1. Use interpolation to estimate the median of this distribution.
2. Calculate estimates for the mean and the standard deviation of these data. One measure of skewness is given by $$\text { Coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
3. Evaluate this coefficient and comment on the skewness of these data. Greg suggests that a normal distribution is a suitable model for the foot lengths of ten-year-old children.
4. Using the value found in part (c), comment on Greg's suggestion, giving a reason for your answer.

1. The marks of a group of female students in a statistics test are summarised in Figure 1

\begin{figure}[h] \end{figure}

1. Write down the mark which is exceeded by \(75 \%\) of the female students. The marks of a group of male students in the same statistics test are summarised by the stem and leaf diagram below.

   Mark	(2|6 means 26)	Totals
   1	4	(1)
   2	6	(1)
   3	447	(3)
   4	066778	(6)
   5	001113677	(9)
   6	223338	(6)
   7	008	(3)
   8	5	(1)
   9	0	(1)

2. Find the median and interquartile range of the marks of the male students. An outlier is a mark that is
   either more than \(1.5 \times\) interquartile range above the upper quartile or more than \(1.5 \times\) interquartile range below the lower quartile.
3. In the space provided on Figure 1 draw a box plot to represent the marks of the male students, indicating clearly any outliers.
4. Compare and contrast the marks of the male and the female students.
   

4. The following table summarises the times, \(t\) minutes to the nearest minute, recorded for a group of students to complete an exam. $$\text { [You may use } \sum \mathrm { f } t ^ { 2 } = 134281.25 \text { ] }$$

1. Estimate the mean and standard deviation of these data.
2. Use linear interpolation to estimate the value of the median.
3. Show that the estimated value of the lower quartile is 18.6 to 3 significant figures.
4. Estimate the interquartile range of this distribution.
5. Give a reason why the mean and standard deviation are not the most appropriate summary statistics to use with these data. The person timing the exam made an error and each student actually took 5 minutes less than the times recorded above. The table below summarises the actual times.

   Time (minutes) \(t\)	\(6 - 15\)	\(16 - 20\)	\(21 - 25\)	\(26 - 30\)	\(31 - 40\)	\(41 - 55\)
   Number of students f	62	88	16	13	11	10

6. Without further calculations, explain the effect this would have on each of the estimates found in parts (a), (b), (c) and (d).

1. The table shows the time, to the nearest minute, spent waiting for a taxi by each of 80 people one Sunday afternoon.

1. Write down the upper class boundary for the \(2 - 4\) minute interval. A histogram is drawn to represent these data. The height of the tallest bar is 6 cm .
2. Calculate the height of the second tallest bar.
3. Estimate the number of people with a waiting time between 3.5 minutes and 7 minutes.
4. Use linear interpolation to estimate the median, the lower quartile and the upper quartile of the waiting times.
5. Describe the skewness of these data, giving a reason for your answer.
   

1. A random sample of 35 homeowners was taken from each of the villages Greenslax and Penville and their ages were recorded. The results are summarised in the back-to-back stem and leaf diagram below.

Key: 7 | 3 | 1 means 37 years for Greenslax and 31 years for Penville
Some of the quartiles for these two distributions are given in the table below.

1. Find the value of \(a\) and the value of \(b\). An outlier is a value that falls either $$\begin{aligned} & \text { more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { above } Q _ { 3 } \\ & \text { or more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { below } Q _ { 1 } \end{aligned}$$
2. On the graph paper opposite draw a box plot to represent the data from Penville. Show clearly any outliers.
3. State the skewness of each distribution. Justify your answers. \includegraphics[max width=\textwidth, alt={}, center]{8270bcae-494c-4248-8229-a72e9e84eab0-03_930_1237_1800_367}
   

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. The times, in seconds, spent in a queue at a supermarket by 85 randomly selected customers, are summarised in the table below. A histogram was drawn to represent these data. The \(30 - 60\) group was represented by a bar of width 1.5 cm and height 1 cm .

1. Find the width and the height of the \(70 - 80\) group.
2. Use linear interpolation to estimate the median of this distribution. Given that \(x\) denotes the midpoint of each group in the table and $$\sum f x = 6460 \quad \sum f x ^ { 2 } = 529400$$
3. calculate an estimate for
   1. the mean,
   2. the standard deviation,
      for the above data. One measure of skewness is given by $$\text { coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
4. Evaluate this coefficient and comment on the skewness of these data.

Each of 60 students was asked to draw a \(20 ^ { \circ }\) angle without using a protractor. The size of each angle drawn was measured. The results are summarised in the box plot below. \includegraphics[max width=\textwidth, alt={}, center]{9626e3ce-35d6-41b5-a0bd-1185f38b9e36-02_371_1040_340_461}

1. Find the range for these data.
2. Find the interquartile range for these data. The students were then asked to draw a \(70 ^ { \circ }\) angle.
   The results are summarised in the table below.

   Angle, \(\boldsymbol { a }\), (degrees)	Number of students
   \(55 \leqslant a < 60\)	6
   \(60 \leqslant a < 65\)	15
   \(65 \leqslant a < 70\)	13
   \(70 \leqslant a < 75\)	11
   \(75 \leqslant a < 80\)	8
   \(80 \leqslant a < 85\)	7

3. Use linear interpolation to estimate the size of the median angle drawn. Give your answer to 1 decimal place.
4. Show that the lower quartile is \(63 ^ { \circ }\) For these data, the upper quartile is \(75 ^ { \circ }\), the minimum is \(55 ^ { \circ }\) and the maximum is \(84 ^ { \circ }\) An outlier is an observation that falls either more than \(1.5 \times\) (interquartile range) above the upper quartile or more than \(1.5 \times\) (interquartile range) below the lower quartile.
5. 1. Show that there are no outliers for these data.
   2. Draw a box plot for these data on the grid on page 3.
6. State which angle the students were more accurate at drawing. Give reasons for your answer.
   (3) \includegraphics[max width=\textwidth, alt={}, center]{9626e3ce-35d6-41b5-a0bd-1185f38b9e36-03_378_1059_2067_447}