Compare distributions using stem-and-leaf

4 The times taken, in minutes, to complete a cycle race by 19 cyclists from each of two clubs, the Cheetahs and the Panthers, are represented in the following back-to-back stem-and-leaf diagram. Key: 7 |9| 1 means 97 minutes for Cheetahs and 91 minutes for Panthers

1. Find the median and the interquartile range of the times of the Cheetahs.
   The median and interquartile range for the Panthers are 103 minutes and 14 minutes respectively.
2. Make two comparisons between the times taken by the Cheetahs and the times taken by the Panthers.
   Another cyclist, Kenny, from the Cheetahs also took part in the race. The mean time taken by the 20 cyclists from the Cheetahs was 99 minutes.
3. Find the time taken by Kenny to complete the race.

4 The following back-to-back stem-and-leaf diagram shows the cholesterol count for a group of 45 people who exercise daily and for another group of 63 who do not exercise. The figures in brackets show the number of people corresponding to each set of leaves. Key: 2 | 8 | 1 represents a cholesterol count of 8.2 in the group who exercise and 8.1 in the group who do not exercise.

1. Give one useful feature of a stem-and-leaf diagram.
2. Find the median and the quartiles of the cholesterol count for the group who do not exercise. You are given that the lower quartile, median and upper quartile of the cholesterol count for the group who exercise are 4.25, 5.3 and 6.6 respectively.
3. On a single diagram on graph paper, draw two box-and-whisker plots to illustrate the data.

4 The following back-to-back stem-and-leaf diagram shows the times to load an application on 61 smartphones of type \(A\) and 43 smartphones of type \(B\).
(7) Key: 3 | 2 | 1 means 0.23 seconds for type \(A\) and 0.21 seconds for type \(B\).

1. Find the median and quartiles for smartphones of type \(A\). You are given that the median, lower quartile and upper quartile for smartphones of type \(B\) are 0.46 seconds, 0.36 seconds and 0.63 seconds respectively.
2. Represent the data by drawing a pair of box-and-whisker plots in a single diagram on graph paper.
3. Compare the loading times for these two types of smartphone.

1 Th fb low ing b ck te b ck stem-ad leaf il ag am sw stb a lsalaries \(\mathbf { 6 }\) agp \(\mathbf { 6 } \mathbf { 9 }\) females adgn ales. Key 4 Q 3 m eas ( st \(\mathbf { o }\) females an of \(\mathbf { o }\) males.

1. Fid b med ara d b ɛ rtiles \(\mathbf { 6 }\) th females' salaries. Yo are gie \(n\)th \(t\) th med an salary \(\mathbf { 6 }\) th males is \(\boldsymbol { \otimes } \rho\) th lw er \(\mathbf { q }\) rtile is \(\boldsymbol {
   ) } \boldsymbol { \theta }\( ad th \)\mathbf { p }\( r e rtile is
   )50
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   \includegraphics[max width=\textwidth, alt={}, center]{1fef5f2c-b375-4be2-b8a1-c30136bd0063-02_997_1589_1736_310}

2. Two youth clubs, Eastyou and Westyou, decided to raise money for charity by running a 5 km race. All the members of the youth clubs took part and the time, in minutes, taken for each member to run the 5 km was recorded. The times for the Westyou members are summarised in Figure 1. \begin{figure}[h] \end{figure}

1. Write down the time that is exceeded by \(75 \%\) of Westyou members. The times for the Eastyou members are summarised by the stem and leaf diagram below.

   Stem	Leaf
   2	0	2	3	4											\(( 4 )\)
   2	5	6	8	8	8	9	9
   3	0	0	0	0	0	1	1	1	2	2	2	2	3	4	\(( 14 )\)
   3	5	5	5	7	9										\(( 5 )\)

   Key: 2|0 means 20 minutes
2. Find the value of the median and interquartile range for the Eastyou members. An outlier is a value that falls either
3. On the grid on page 7, draw a box plot to represent the times of the Eastyou members.
4. State the skewness of each distribution. Give reasons for your answers. $$\begin{aligned} & \text { more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { above } Q _ { 3 }
   & \text { or more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { below } Q _ { 1 } \end{aligned}$$
   \includegraphics[max width=\textwidth, alt={}]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-06_2255_50_314_1976}
   \includegraphics[max width=\textwidth, alt={}, center]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-07_406_1390_2224_262} Turn over for a spare grid if you need to redraw your box plot. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Only use this grid if you need to redraw your box plot.} \includegraphics[alt={},max width=\textwidth]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-09_401_1399_2261_258}
   \end{figure}

1. A researcher is investigating the growth of two types of tree, Birch and Maple. The height, to the nearest cm, a seedling grows in one year is recorded for 35 Birch trees and 32 Maple trees. The results are summarised in the back-to-back stem and leaf diagram below.

Key: 5 | 6 | 3 means 65 cm for a Birch tree and 63 cm for a Maple tree
The median height that these Maple trees grow in one year is 45 cm .

1. Find the value of \(\boldsymbol { k }\), used in the stem and leaf diagram.
2. Find the lower quartile and the upper quartile of the height grown in one year for these Birch trees. The researcher defines an outlier as an observation that is $$\text { greater than } Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { or less than } Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
3. Show that there is only one outlier amongst the Birch trees. The grid on page 3 shows a box plot for the heights that the Maple trees grow in one year.
4. On the same grid draw a box plot for the heights that the Birch trees grow in one year.
5. Comment on any difference in the distributions of the growth of these Birch trees and the growth of these Maple trees.
   State the values of any statistics you have used to support your comment. The researcher realises he has missed out 4 pieces of data for the Maple trees. The heights each seedling grows in one year, to the nearest cm, in ascending order, for these 4 Maple trees are \(27 \mathrm {~cm} , a \mathrm {~cm} , 48 \mathrm {~cm} , 2 a \mathrm {~cm}\). Given that there is no change to the box plot for the Maple trees given on page 3
6. find the range of possible values for \(a\) Show your working clearly.
   \includegraphics[max width=\textwidth, alt={}]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-03_1243_1659_1464_210}
   Only use this grid if you need to redraw your answer for part (d)
   \includegraphics[max width=\textwidth, alt={}, center]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-05_1154_1643_1503_217}
   (Total for Question 1 is 13 marks)

7. The back-to-back stem and leaf diagram shows the journey times, to the nearest minute, of the commuter services into a big city provided by the trains of two operating companies. Key : 4| 3 | 6 means 34 minutes for Company \(A\) and 36 minutes for Company \(B\).

1. Write down the numbers needed to complete the diagram.
2. Find the median and the quartiles for each company.
3. On graph paper, draw box plots for the two companies. Show your scale.
4. Use your plots to compare the two sets of data briefly.
5. Describe the skewness of each company's distribution of times.

19 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[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-28_1109_1660_751_191} 19

1. Estimate the median of the masses from Box Plot A
   19
2. It is claimed that Box Plot B must be incorrectly drawn.
   19
3. 1. Give a reason why this claim was made.
      19
4. (ii) Comment on the validity of this claim.
   19
5. It is claimed that Box Plot B must be from the 2002 data. Give a reason why this claim is correct.
   \includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-30_2492_1721_217_150}

12
12
The box plot below summarises the \(\mathrm { CO } _ { 2 }\) emissions, in \(\mathrm { g } / \mathrm { km }\), for cars in the Large Data Set from the London and North West regions.
London
39
119142168
346
North West