2.02g Calculate mean and standard deviation

1. The lengths, \(x \mathrm {~mm}\), of 50 pebbles are summarised in the table below.

A histogram is drawn to represent these data.
The bar representing the class \(32 \leqslant x < 36\) is 2.5 cm wide and 7.5 cm tall.

1. Calculate the width and the height of the bar representing the class \(30 \leqslant x < 32\)
2. Using linear interpolation, estimate the median of \(x\) The weight, \(w\) grams, of each of the 50 pebbles is coded using \(10 y = w - 20\) These coded data are summarised by $$\sum y = 104 \quad \sum y ^ { 2 } = 233.54$$
3. Show that the mean of \(w\) is 40.8
4. Calculate the standard deviation of \(w\) The weight of a pebble recorded as 40.8 grams is added to the sample.
5. Without carrying out any further calculations, state, giving a reason, what effect this would have on the value of
   1. the mean of \(w\)
   2. the standard deviation of \(w\)

1. The stem and leaf diagram gives the blood pressure, \(x \mathrm { mmHg }\), for a random sample of 19 female patients.

Key: 10 | 1 means blood pressure of 101 mmHg

1. Find the median and the quartiles for these data.
2. Find the interquartile range ( \(Q _ { 3 } - Q _ { 1 }\) ) An outlier is a value that is greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or less than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\)
3. Showing your working clearly, identify any outliers for these data.
4. On the grid on page 21 draw a box and whisker plot to represent these data. Show any outliers clearly. The above data can be summarised by $$\sum x = 2299 \text { and } \sum x ^ { 2 } = 279709$$
5. Calculate the mean and the standard deviation for these data. For a random sample taken from a normal distribution, a rule for determining outliers is: an outlier is more than \(2.7 \times\) standard deviation above or below the mean.
6. Find the limits to determine outliers using this rule.
7. State, giving a reason based on some of the above calculations, whether or not a normal distribution is a suitable model for these data. \includegraphics[max width=\textwidth, alt={}, center]{8ff7539e-fa44-4388-af8c-80656f081528-21_2281_73_308_15}
   Turn over for a spare diagram if you need to redraw your plot.
   \includegraphics[max width=\textwidth, alt={}]{8ff7539e-fa44-4388-af8c-80656f081528-24_2639_1830_121_121}

1. The weights, to the nearest kilogram, of a sample of 33 female spotted hyenas living in the Serengeti are summarised in the stem and leaf diagram below.

\begin{table}[h] \end{table} Totals

1. Find the median and quartiles for the weights of the female spotted hyenas. An outlier is defined as any value greater than \(c\) or any value less than \(d\) where $$\begin{aligned} & c = Q _ { 3 } + 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \\ & d = Q _ { 1 } - 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \end{aligned}$$
2. Showing your working clearly, identify any outliers for these data.
   (3) The weights, to the nearest kilogram, of a sample of male spotted hyenas living in the Serengeti are summarised below. \includegraphics[max width=\textwidth, alt={}, center]{0377c6e9-ab4f-477d-9236-0732fe81f25e-06_755_1568_1537_185}
3. In the space provided in the grid above, draw a box and whisker plot to represent the weights of female spotted hyenas living in the Serengeti. Indicate clearly any outliers. (A copy of this grid is on page 9 if you need to redraw your box and whisker plot.)
4. Compare the weights of male and female spotted hyenas living in the Serengeti. Key: 3|2 means 32
   \includegraphics[max width=\textwidth, alt={}, center]{0377c6e9-ab4f-477d-9236-0732fe81f25e-09_2658_101_107_9}
   

3. The parking times, \(t\) hours, for cars in a car park are summarised below. $$\text { (You may use } \sum \mathrm { fm } = 182 \text { and } \sum \mathrm { fm } ^ { 2 } = 883 \text { ) }$$ A histogram is drawn to represent these data.
The bar representing the time \(1 \leqslant t < 2\) has a width of 1.5 cm and a height of 6 cm .

1. Calculate the width and the height of the bar representing the time \(4 \leqslant t < 6\)
2. Use linear interpolation to estimate the median parking time for the cars in the car park.
3. Estimate the mean and the standard deviation of the parking time for the cars in the car park.
4. Describe, giving a reason, the skewness of the data. One of these cars is selected at random.
5. Estimate the probability that this car is parked for more than 75 minutes.

1. The stem lengths of a sample of 120 tulips are recorded in the grouped frequency table below.

A histogram is drawn to represent these data.
The area of the bar representing the \(40 \leqslant x < 42\) class is \(16.5 \mathrm {~cm} ^ { 2 }\)

1. Calculate the exact area of the bar representing the \(42 \leqslant x < 45\) class. The height of the tallest bar in the histogram is 10 cm .
2. Find the exact height of the second tallest bar. \(Q _ { 1 }\) for these data is 45 cm .
3. Use linear interpolation to find an estimate for
   1. \(Q _ { 2 }\)
   2. the interquartile range. One measure of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
   (d) By calculating this measure, describe the skewness of these data.

1. Morgan is investigating the body length, \(b\) centimetres, of squirrels.

A random sample of 8 squirrels is taken and the data for each squirrel is coded using $$x = \frac { b - 21 } { 2 }$$ The results for the coded data are summarised below $$\sum x = - 1.2 \quad \sum x ^ { 2 } = 5.1$$

1. Find the mean of \(b\)
2. Find the standard deviation of \(b\) A 9th squirrel is added to the sample. Given that for all 9 squirrels \(\sum x = 0\)
3. find
   1. the body length of the 9th squirrel,
   2. the standard deviation of \(x\) for all 9 squirrels.

1. The weights, to the nearest kilogram, of a sample of 33 red kangaroos taken in December are summarised in the stem and leaf diagram below.

Key: 3 | 2 represents 32 kg

1. Find
   1. the value of the median
   2. the value of \(Q _ { 1 }\) and the value of \(Q _ { 3 }\) for the weights of these red kangaroos. For these data an outlier is defined as a value that is
      greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or smaller than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\)
2. Show that there are 2 outliers for these data. Figure 1 on page 7 shows a box plot for the weights of the same 33 red kangaroos taken in February, earlier in the year.
3. In the space on Figure 1, draw a box plot to represent the weights of these red kangaroos in December.
4. Compare the distribution of the weights of red kangaroos taken in February with the distribution of the weights of red kangaroos taken in December of the same year. You should interpret your comparisons in the context of the question.
   \includegraphics[max width=\textwidth, alt={}]{f94b29e0-081f-45e8-99a7-ac835eec91e5-07_2267_51_307_36}
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f94b29e0-081f-45e8-99a7-ac835eec91e5-07_766_1803_1777_132} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Turn over for a spare grid if you need to redraw your box plot. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f94b29e0-081f-45e8-99a7-ac835eec91e5-09_901_1833_1653_114} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} \begin{verbatim} (Total for Question 2 is 13 marks) \end{verbatim}

1. A researcher recorded the time, \(t\) minutes, spent using a mobile phone during a particular afternoon, for each child in a club.

The researcher coded the data using \(v = \frac { t - 5 } { 10 }\) and the results are summarised in the table below. $$\text { (You may use } \sum f m = 825 \text { and } \sum f m ^ { 2 } = 12012.5 \text { ) }$$

1. Write down the value of \(a\) and the value of \(b\).
2. Calculate an estimate of the mean of \(v\).
3. Calculate an estimate of the standard deviation of \(v\).
4. Use linear interpolation to estimate the median of \(v\).
5. Hence describe the skewness of the distribution. Give a reason for your answer.
6. Calculate estimates of the mean and the standard deviation of the time spent using a mobile phone during the afternoon by the children in this club. \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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1. A teacher selects a random sample of 56 students and records, to the nearest hour, the time spent watching television in a particular week.

1. Find the mid-points of the 21-25 hour and 31-40 hour groups. A histogram was drawn to represent these data. The 11-20 group was represented by a bar of width 4 cm and height 6 cm .
2. Find the width and height of the 26-30 group.
3. Estimate the mean and standard deviation of the time spent watching television by these students.
4. Use linear interpolation to estimate the median length of time spent watching television by these students. The teacher estimated the lower quartile and the upper quartile of the time spent watching television to be 15.8 and 29.3 respectively.
5. State, giving a reason, the skewness of these data.
   

5. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 200 motorists were delayed by roadworks on a stretch of motorway.

1. Using graph paper represent these data by a histogram.
2. Give a reason to justify the use of a histogram to represent these data.
3. Use interpolation to estimate the median of this distribution.
4. Calculate an estimate of the mean and an estimate of the standard deviation of these data. One coefficient of skewness is given by $$\frac { 3 ( \text { mean - median } ) } { \text { standard deviation } } .$$
5. Evaluate this coefficient for the above data.
6. Explain why the normal distribution may not be suitable to model the number of minutes that motorists are delayed by these roadworks.

4. A restaurant owner is concerned about the amount of time customers have to wait before being served. He collects data on the waiting times, to the nearest minute, of 20 customers. These data are listed below.

1. Find the median and inter-quartile range of the waiting times. An outlier is an observation that falls either \(1.5 \times\) (inter-quartile range) above the upper quartile or \(1.5 \times\) (inter-quartile range) below the lower quartile.
2. Draw a boxplot to represent these data, clearly indicating any outliers.
3. Find the mean of these data.
4. Comment on the skewness of these data. Justify your answer.

2. The number of caravans on Seaview caravan site on each night in August last year is summarised in the following stem and leaf diagram.

1. Find the three quartiles of these data. During the same month, the least number of caravans on Northcliffe caravan site was 31. The maximum number of caravans on this site on any night that month was 72 . The three quartiles for this site were 38,45 and 52 respectively.
2. On graph paper and using the same scale, draw box plots to represent the data for both caravan sites. You may assume that there are no outliers.
3. Compare and contrast these two box plots.
4. Give an interpretation to the upper quartiles of these two distributions.

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.

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. The labelling on bags of garden compost indicates that the bags weigh 20 kg . The weights of a random sample of 50 bags are summarised in the table below.

1. On graph paper, draw a histogram of these data.
2. Using the coding \(y = 10\) (weight in \(\mathrm { kg } - 14\) ), find an estimate for the mean and standard deviation of the weight of a bag of compost.
   [0pt] [Use \(\Sigma f y ^ { 2 } = 171\) 503.75]
3. Using linear interpolation, estimate the median. The company that produces the bags of compost wants to improve the accuracy of the labelling. The company decides to put the average weight in kg on each bag.
4. Write down which of these averages you would recommend the company to use. Give a reason for your answer.

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. The following table summarises the distances, to the nearest km , that 134 examiners travelled to attend a meeting in London.

1. Give a reason to justify the use of a histogram to represent these data.
2. Calculate the frequency densities needed to draw a histogram for these data.
   (DO NOT DRAW THE HISTOGRAM)
3. Use interpolation to estimate the median \(Q _ { 2 }\), the lower quartile \(Q _ { 1 }\), and the upper quartile \(Q _ { 3 }\) of these data. The mid-point of each class is represented by \(x\) and the corresponding frequency by \(f\). Calculations then give the following values $$\Sigma f _ { x } = 8379.5 \quad \text { and } \quad \Sigma f _ { x ^ { 2 } } = 557489.75$$
4. Calculate an estimate of the mean and an estimate of the standard deviation for these data. One coefficient of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
5. Evaluate this coefficient and comment on the skewness of these data.
6. Give another justification of your comment in part (e).