Find median and quartiles from stem-and-leaf diagram

1 The stem-and-leaf diagram below represents data collected for the number of hits on an internet site on each day in March 2007. There is one missing value, denoted by \(x\). Key: 1 | 5 represents 15 hits

1. Find the median and lower quartile for the number of hits each day.
2. The interquartile range is 19 . Find the value of \(x\).

4 The weights in kilograms of packets of cereal were noted correct to 4 significant figures. The following stem-and-leaf diagram shows the data. Key: 748 | 5 represents 0.7485 kg .

1. On the grid, draw a box-and-whisker plot to represent the data.
   \includegraphics[max width=\textwidth, alt={}, center]{556a1cc2-47ef-4ef7-a8f6-42850c303531-05_814_1604_1336_299}
2. Name a distribution that might be a suitable model for the weights of this type of cereal packet. Justify your answer.

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.

2 The back-to-back stem-and-leaf diagram below shows the number of hours of television watched per week by each of 15 boys and 15 girls. $$\begin{aligned} & \text { Boys Girls }
& \left. \begin{array} { r r r r r r r r | r r r r r r r r r r r r r } & 677664 & 4 & 3 & 0 & 0 & 5 & 5 & 6 & 677888 \end{array} \right\} \end{aligned}$$ Key: 4 | 2 | 2 means a boy who watched 24 hours and a girl who watched 22 hours of television per week.

1. Find the median and the quartiles of the results for the boys.
2. Give a reason why the median might be preferred to the mean in using an average to compare the two data sets.
3. State one advantage, and one disadvantage, of using stem-and-leaf diagrams rather than box-andwhisker plots to represent the data.

8 The stem-and-leaf diagram shows the age in completed years of the members of a sports club. \section*{Male} \begin{table}[h] \end{table} Key: 1 | 4 | 0 represents a male aged 41 and a female aged 40.

1. Find the median and interquartile range for the males.
2. The median and interquartile range for the females are 27 and 15 respectively. Make two comparisons between the ages of the males and the ages of the females.
3. The mean age of the males is 30.7 and the mean age of the females is 27.5 , each correct to 1 decimal place. Give one advantage of using the median rather than the mean to compare the ages of the males with the ages of the females. A record was kept of the number of hours, \(X\), spent by each member at the club in a year. The results were summarised by $$n = 49 , \quad \Sigma ( x - 200 ) = 245 , \quad \Sigma ( x - 200 ) ^ { 2 } = 9849 .$$
4. Calculate the mean and standard deviation of \(X\).

1 Alice carries out a survey of the 28 students in her class to find how many text messages each sent on the previous day. Her results are shown in the stem and leaf diagram. Key: 2 | 3 represents 23

1. Find the mode and median of the number of text messages.
2. Identify the type of skewness of the distribution.
3. Alice is considering whether to use the mean or the median as a measure of central tendency for these data.
   (A) In view of the skewness of the distribution, state whether Alice should choose the mean or the median.
   (B) What other feature of the distribution confirms Alice's choice?
4. The mean number of text messages is 14.75 . If each message costs 10 pence, find the total cost of all of these messages.

2. The stem and leaf diagram below shows the ages (in years) of the residents in a care home.

1. Find the median age of the residents.
2. Find the interquartile range (IQR) of the ages of the residents. An outlier is defined as a value that is either
   more than \(1.5 \times ( \mathrm { IQR } )\) below the lower quartile or more than \(1.5 \times ( \mathrm { IQR } )\) above the upper quartile.
3. Determine any outliers in these data. Show clearly any calculations that you use.
4. On the grid on page 5, draw a box plot to summarise these data.
   Ages

5 The stem-and-leaf diagram shows the masses, in grams, of 23 plums, measured correct to the nearest gram. \(\quad\) Key \(: 6 \mid 2\) means 62

1. Find the median and interquartile range of these masses.
2. State one advantage of using the interquartile range rather than the standard deviation as a measure of the variation in these masses.
3. State one advantage and one disadvantage of using a stem-and-leaf diagram rather than a box-and-whisker plot to represent data.
4. James wished to calculate the mean and standard deviation of the given data. He first subtracted 5 from each of the digits to the left of the line in the stem-and-leaf diagram, giving the following.

   0	5	6	7	8	8	9
   1	1	2	3	5	6	8	9
   2	0	0	2	4	5	6	7	8
   3	0
   4	7

   The mean and standard deviation of the data in this diagram are 18.1 and 9.7 respectively, correct to 1 decimal place. Write down the mean and standard deviation of the data in the original diagram.

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.

8 The stem-and-leaf diagram shows the heights, in centimetres, of 17 plants, measured correct to the nearest centimetre. Key: 5 | 6 means 56

1. Find the median and inter-quartile range of these heights.
2. Calculate the mean and standard deviation of these heights.
3. State one advantage of using the median rather than the mean as a measure of average for these heights.

6 An app on my new smartphone records the number of times in a day I use the phone. The data for each day since I bought the phone are shown in the stem and leaf diagram. Key: 3|1 means 31

1. Explain whether these data are a sample or a population.
2. Describe the shape of the distribution.
3. Determine the interquartile range.
4. Use your answer to part (c) to determine whether there are any outliers in the lower tail.

9 At the end of each school term at North End College all the science classes in year 10 are given a test. The marks out of 100 achieved by members of set 1 are shown in Fig. 9. \begin{table}[h] \end{table} Key \(5 \quad\) 2 represents a mark of 52

1. Describe the shape of the distribution.
2. The teacher for set 1 claimed that a typical student in his class achieved a mark of 95. How did he justify this statement?
3. Another teacher said that the average mark in set 1 is 76 . How did she justify this statement? Benson's mark in the test is 35 . If the mark achieved by any student is an outlier in the lower tail of the distribution, the student is moved down to set 2 .
4. Determine whether Benson is moved down to set 2 .

2. Chi wanted to summarise the scores of the 39 competitors in a village quiz. He started to produce the following stem and leaf diagram Key: 2|5 is a score of 25 \begin{table}[h] \end{table} He did not complete the stem and leaf diagram but instead produced the following box plot.
\includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-04_357_1237_772_356} Chi defined an outlier as a value that is $$\text { greater than } Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$ or
less than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\)

1. Find
   1. the interquartile range
   2. the range.
2. Describe, giving a reason, the skewness of the distribution of scores. Albert and Beth asked for their scores to be checked.
   Albert's score was changed from 25 to 37
   Beth's score was changed from 54 to 60
3. On the grid on page 5, draw an updated box plot. Show clearly any calculations that you used. Some of the competitors complained that the questions were biased towards the younger generation. The product moment correlation coefficient between the age of the competitors and their score in the quiz is - 0.187
4. State, giving a reason, whether or not the complaint is supported by this statistic. \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-05_360_1242_2238_351} Turn over for a spare grid if you need to redraw your box plot. \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-07_367_1246_2261_351}

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}

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. 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. 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.

7. The following stem and leaf diagram shows the aptitude scores \(x\) obtained by all the applicants for a particular job.

1. Write down the modal aptitude score.
2. Find the three quartiles for these data. Outliers can be defined to be outside the limits \(\mathrm { Q } _ { 1 } - 1.0 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) and \(\mathrm { Q } _ { 3 } + 1.0 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\).
3. On a graph paper, draw a box plot to represent these data. For these data, \(\Sigma x = 3363\) and \(\Sigma x ^ { 2 } = 238305\).
4. Calculate, to 2 decimal places, the mean and the standard deviation for these data.
5. Use two different methods to show that these data are negatively skewed.

5. For a project, a student asked 40 people to draw two straight lines with what they thought was an angle of \(75 ^ { \circ }\) between them, using just a ruler and a pencil. She then measured the size of the angles that had been drawn and her data are summarised in this stem and leaf diagram.

1. Find the median and quartiles of these data. Given that any values outside of the limits \(\mathrm { Q } _ { 1 } - 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) and \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) are to be regarded as outliers,
2. determine if there are any outliers in these data,
3. draw a box plot representing these data on graph paper,
4. describe the skewness of the distribution and suggest a reason for it.

5. Each child in class 3A was given a packet of seeds to plant. The stem and leaf diagram below shows how many seedlings were visible in each child's tray one week after planting.

1. Find the median and interquartile range for these data.
2. Use the quartiles to describe the skewness of the data. Show your method clearly. The mean and standard deviation for these data were 27.2 and 10.3 respectively.
3. Explaining your answer, state whether you would recommend using these values or your answers to part (a) to summarise these data. Outliers are defined to be values outside of the limits \(\mathrm { Q } _ { 1 } - 2 s\) and \(\mathrm { Q } _ { 3 } + 2 s\) where \(s\) is the standard deviation given above.
4. Represent these data with a boxplot identifying clearly any outliers.

5. In a survey unemployed people were asked how many months it had been, to the nearest month, since they were last employed on a full-time basis. The data collected is summarised in this stem and leaf diagram.

1. Write down the values needed to complete the totals column on the stem and leaf diagram.
2. State the mode of these data.
3. Find the median and quartiles of these data. Given that any values outside of the limits \(\mathrm { Q } _ { 1 } - 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) and \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) are to be regarded as outliers,
4. determine if there are any outliers in these data,
5. draw a box plot representing these data on graph paper,
6. describe the skewness of these data and suggest a reason for it.

2. In a study of reaction times, 25 participants completed a test where their reaction times (in milliseconds) were recorded. The results are shown in the stem-and-leaf diagram below: \(20 \mid 3579\)
\(21 \mid 02568\)
\(22 \mid 134579\)
\(23 \mid 0258\)
\(24 \mid 1467\)
\(25 \mid 25\) Key: 21 | 0 represents a reaction time of 210 milliseconds

1. State the median reaction time.
2. Calculate the interquartile range of these reaction times.
3. Find the mean and standard deviation of these reaction times.
4. State one advantage of using a stem-and-leaf diagram to display this data rather than a frequency table.
5. One participant completed the test again and recorded a reaction time of 195 milliseconds. Add this result to the stem-and-leaf diagram and state the effect this would have on:
   i) the median
   ii) the mean
   ii) the standard deviation
   [0pt] [4]
6. Explain why the interquartile range might be preferred to the standard deviation as a measure of spread in this context
   [0pt] [2]

5. In a study of reaction times, 25 participants completed a test where their reaction times (in milliseconds) were recorded. The results are shown in the stem-and-leaf diagram below: \(20 \mid 3579\)
\(21 \mid 02568\)
\(22 \mid 134579\)
\(23 \mid 0258\)
\(24 \mid 1467\)
\(25 \mid 25\) Key: 21 | 0 represents a reaction time of 210 milliseconds

1. State the median reaction time.
2. Calculate the interquartile range of these reaction times.
3. Find the mean and standard deviation of these reaction times.
4. State one advantage of using a stem-and-leaf diagram to display this data rather than a frequency table.
5. One participant completed the test again and recorded a reaction time of 195 milliseconds. Add this result to the stem-and-leaf diagram and state the effect this would have on:
   a. the median
   b. the mean
   c. the standard deviation
   [0pt] [4]
6. Explain why the interquartile range might be preferred to the standard deviation as a measure of spread in this context
   [0pt] [BLANK PAGE]

1. The stem and leaf diagram shows the number of deliveries made by Pat each day for 24 days

\begin{table}[h] \end{table} where \(a\), \(b\) and \(c\) are positive integers with \(a < b < c\)
An outlier is defined as any value greater than \(1.5 \times\) interquartile range above the upper quartile. Given that there is only one outlier for these data,

1. show that \(c = 9\) The number of deliveries made by Pat each day is represented by \(d\)
   The data in the stem and leaf diagram are coded using $$x = d - 125$$ and the following summary statistics are obtained $$\sum x = - 96 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1306$$
2. Find the mean number of deliveries.
3. Find the standard deviation of the number of deliveries. One of these 24 days is selected at random. The random variable \(D\) represents the number of deliveries made by Pat on this day. The random variable \(X = D - 125\)
4. Find \(\mathrm { P } ( D > 118 \mid X < 0 )\)