The following are the house prices in thousands of dollars, arranged in ascending order, for 51 houses from a certain area.
| 253 | 270 | 310 | 354 | 386 | 428 | 433 | 468 | 472 | 477 | 485 | 520 | 520 | 524 | 526 | 531 | 535 |
| 536 | 538 | 541 | 543 | 546 | 548 | 549 | 551 | 554 | 572 | 583 | 590 | 605 | 614 | 638 | 649 | 652 |
| 666 | 670 | 682 | 684 | 690 | 710 | 725 | 726 | 731 | 734 | 745 | 760 | 800 | 854 | 863 | 957 | 986 |
An expensive house is defined as a house which has a price that is more than 1.5 times the interquartile range above the upper quartile.
The following back-to-back stem-and-leaf diagram shows the annual salaries of a group of 39 females and 39 males.
| Females | Males | |||
|---|---|---|---|---|
| (4) | 5 2 0 0 | 20 | 3 | (1) |
| (9) | 9 8 8 7 6 4 0 0 0 | 21 | 0 0 7 | (3) |
| (8) | 8 7 5 3 3 1 0 0 | 22 | 0 0 0 4 5 6 6 | (6) |
| (6) | 6 4 2 1 0 0 | 23 | 0 0 2 3 3 5 6 7 7 | (9) |
| (6) | 7 5 4 0 0 0 | 24 | 0 1 1 2 5 5 6 8 8 9 | (10) |
| (4) | 9 5 0 0 | 25 | 3 4 5 7 7 8 9 | (7) |
| (2) | 5 0 | 26 | 0 4 6 | (3) |
Key: 2 | 20 | 3 means $20 200 for females and $20 300 for males.
(i) Find the median and the quartiles of the femalesโ salaries.
You are given that the median salary of the males is \($24 000\), the lower quartile is \($22 600\) and the upper quartile is \($25 300.\)
(ii) Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.
The back-to-back stem-and-leaf diagram shows the values taken by two variables A and B.
| A | B | |||
|---|---|---|---|---|
| (3) | 3 1 0 | 15 | 1 3 3 5 | (4) |
| (2) | 4 1 | 16 | 2 2 3 4 4 5 7 7 8 | (10) |
| (3) | 8 3 3 | 17 | 0 1 3 3 3 4 6 6 7 9 9 | (11) |
| (12) | 9 8 8 6 5 5 4 3 2 1 1 0 | 18 | 2 4 7 | (3) |
| (8) | 9 9 8 8 6 5 4 2 | 19 | 1 5 | (2) |
| (5) | 9 8 7 1 0 | 20 | 4 | (1) |
Key: \(4 \mid 16 \mid 7\) means \(A = 0.164\) and \(B = 0.167\).
The marks of the pupils in a certain class in a History examination are as follows.
28, 33, 55, 38, 42, 39, 27, 48, 51, 37, 57, 49, 33
The marks of the pupils in a Physics examination are summarised as follows.
Lower quartile: 28, Median: 39, Upper quartile: 67. The lowest mark was 17 and the highest mark was 74.
The lengths of some insects of the same type from two countries, X and Y, were measured. The stem-and-leaf diagram shows the results.
| Country X | Country Y | |||
|---|---|---|---|---|
| (10) | 9 7 6 6 6 4 4 3 2 0 | 80 | ||
| (18) | 8 8 7 7 6 6 5 5 5 4 4 3 3 2 2 0 | 81 | 1 1 2 2 3 3 3 5 5 6 7 8 9 | (13) |
| (16) | 9 9 9 8 8 7 7 6 5 5 3 2 1 1 0 0 | 82 | 0 0 1 2 3 3 q 4 5 6 6 7 8 8 | (15) |
| (16) | 8 7 6 5 5 5 3 3 2 2 1 1 1 0 0 | 83 | 0 1 2 2 4 4 4 5 5 6 7 7 7 8 9 | (17) |
| (11) | 8 7 6 5 5 4 4 3 3 1 1 | 84 | 0 1 2 4 4 5 5 6 7 7 7 8 9 | (15) |
| 85 | 1 2 r 3 3 5 5 6 7 8 8 | (12) | ||
| 86 | 0 1 2 2 3 5 5 8 9 9 | (11) |
Key: 5 | 81 | 3 means an insect from country X has length 0.815 cm and an insect from country Y has length 0.813 cm.
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.
| 3 | 6 9 9 | (4) |
| 4 | 6 7 | (2) |
| 5 | 0 1 2 2 | (4) |
| 6 | 0 1 1 2 3 4 4 4 5 5 6 6 6 7 8 9 | (20) |
| 7 | 1 1 3 3 3 5 6 7 8 9 9 | (12) |
| 8 | 0 2 4 5 5 6 8 | (7) |
| 9 | 0 1 2 4 4 4 5 5 6 7 7 8 8 9 9 | (18) |
Key: \( 3 \mid 6 \) represents 36 books.
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 | 1 2 2 3 3 4 5 6 6 6 7 9 | (13) |
| 3 | 0 1 1 2 3 4 5 6 6 7 7 8 8 | (15) |
| 4 | 2 2 3 5 7 8 9 | (8) |
Key: \( 3 \mid 6 \) represents 36 books.
The pulse rates, in beats per minute, of a random sample of 15 small animals are shown in the following table.
| 115 | 120 | 158 | 132 | 125 |
| 104 | 142 | 160 | 145 | 104 |
| 162 | 117 | 109 | 124 | 134 |
(ii) Find the median and the quartiles.
(iii) On graph paper, using a scale of 2 cm to represent 10 beats per minute, draw a box-and-whisker plot of the data.
The back-to-back stem-and-leaf diagram shows the diameters, in cm, of 19 cylindrical pipes produced by each of two companies, A and B.
| Company A | Company B | |
|---|---|---|
| 9 8 3 2 0 | 33 | 1 2 8 |
| 8 7 5 4 1 | 34 | 1 6 8 9 9 |
| 9 6 5 2 | 35 | 1 2 2 3 |
| 4 3 1 | 36 | 5 6 |
| 37 | 0 3 4 | |
| 38 | 2 8 |
Key: 1 | 35 | 3 means the pipe diameter from company A is 0.351 cm and from company B is 0.353 cm.
In a survey, people were asked how long they took to travel to and from work, on average. The median time was 3 hours 36 minutes, the upper quartile was 4 hours 42 minutes and the interquartile range was 3 hours 48 minutes. The longest time taken was 5 hours 12 minutes and the shortest time was 30 minutes.
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.
| People who exercise | People who do not exercise | |||
|---|---|---|---|---|
| (9) | 9 8 7 6 4 3 2 2 1 | 3 | 1 5 7 7 | (4) |
| (12) | 9 8 8 7 6 6 5 3 3 2 2 | 4 | 2 3 4 4 5 8 | (6) |
| (9) | 8 7 7 7 6 5 3 3 1 | 5 | 1 2 2 2 3 4 4 5 6 7 8 8 9 | (13) |
| (7) | 6 6 6 6 4 3 2 | 6 | 1 2 3 3 4 5 5 5 7 7 8 9 9 | (14) |
| (3) | 8 4 1 | 7 | 2 4 5 5 6 7 8 8 | (9) |
| (4) | 9 5 5 2 | 8 | 1 3 3 4 6 7 9 9 9 | (9) |
| (1) | 4 | 9 | 1 4 5 5 8 | (5) |
| (0) | 10 | 3 3 6 | (3) |
Key: \( 2 \mid 8 \mid 1 \) represents a cholesterol count of \( 8.2 \) in the group who exercise and \( 8.1 \) in the group who do not exercise.
The weights in kilograms of two groups of 17-year-old males from country P and country Q are displayed in the following back-to-back stem-and-leaf diagram. In the third row of the diagram, 4 | 7 | 1 denotes weights of 74 kg for a male in country P and 71 kg for a male in country Q.
| Country P | Country Q | |
|---|---|---|
| 5 | 1 5 | |
| 6 | 2 3 4 8 | |
| 9 8 7 6 4 | 7 | 1 3 4 5 6 7 7 8 8 9 |
| 8 8 6 6 5 3 | 8 | 2 3 6 7 7 8 8 |
| 9 7 7 6 5 5 5 4 2 | 9 | 0 2 2 4 |
| 5 4 4 3 1 | 10 | 4 5 |
The weights, in kg, of 15 rugby players in the Rebels club and 15 soccer players in the Sharks club are shown below.
| Rebels | 75 | 78 | 79 | 80 | 82 | 82 | 83 | 84 | 85 | 86 | 89 | 93 | 95 | 99 | 102 |
| Sharks | 66 | 68 | 71 | 72 | 74 | 75 | 75 | 76 | 78 | 83 | 83 | 84 | 85 | 86 | 92 |
(a) Represent the data by drawing a back-to-back stem-and-leaf diagram with Rebels on the left-hand side of the diagram.
(b) Find the median and the interquartile range for the Rebels.
A box-and-whisker plot for the Sharks is shown below.
(c) On the same diagram, draw a box-and-whisker plot for the Rebels.
(d) Make one comparison between the weights of the players in the Rebels club and the weights of the players in the Sharks club.

Two machines, A and B, produce metal rods of a certain type. The lengths, in metres, of 19 rods produced by machine A and 19 rods produced by machine B are shown in the following back-to-back stem-and-leaf diagram.
Stem-and-leaf diagram:
| A | B | |
|---|---|---|
| 21 | 1 2 4 | |
| 7 6 3 0 | 22 | 2 4 5 5 6 |
| 8 7 4 3 1 1 | 23 | 0 2 6 8 9 9 |
| 5 5 5 3 2 | 24 | 3 3 4 6 |
| 4 3 1 0 | 25 | 6 |
Key: 7 | 22 | 4 means 0.227 m for machine A and 0.224 m for machine B.
Another group of 33 people ran the same marathon. Their times (in minutes) are:
| 190 | 203 | 215 | 246 | 249 | 253 | 255 | 254 | 258 | 260 | 261 |
| 263 | 267 | 269 | 274 | 276 | 280 | 288 | 283 | 287 | 294 | 300 |
| 307 | 318 | 327 | 331 | 336 | 345 | 351 | 353 | 360 | 368 | 375 |
In a survey 55 students were asked to record, to the nearest kilometre, the total number of kilometres they travelled to school in a particular week. The results are shown below.
| 5 | 5 | 9 | 10 | 13 | 13 | 13 | 15 | 15 | 15 | 15 |
| 16 | 18 | 18 | 18 | 19 | 19 | 20 | 20 | 20 | 20 | 21 |
| 21 | 21 | 23 | 25 | 25 | 25 | 27 | 27 | 29 | 30 | 33 |
| 35 | 38 | 39 | 40 | 42 | 45 | 48 | 50 | 50 | 51 | 51 |
| 52 | 55 | 57 | 57 | 60 | 61 | 64 | 65 | 66 | 69 | 70 |
The number of Olympic medals won in the 2012 Olympic Games by the top \(27\) countries is shown below.
| 104 | 88 | 82 | 65 | 44 | 38 | 35 | 34 | 28 |
| 28 | 18 | 18 | 17 | 17 | 14 | 13 | 13 | 12 |
| 12 | 10 | 10 | 10 | 9 | 6 | 5 | 2 | 2 |
Find the median and quartiles of the data and draw a box-and-whisker plot on the grid.
The weights in kilograms of packets of cereal were noted correct to 4 significant figures. The following stem-and-leaf diagram shows the data.
The weights in kilograms of packets of cereal were noted correct to 4 significant figures.
| 747 | 3 | (1) |
| 748 | 1 2 5 7 7 9 | (6) |
| 749 | 0 2 2 2 3 5 5 5 6 7 8 9 | (12) |
| 750 | 1 1 2 2 2 3 4 4 5 6 7 7 8 8 9 | (15) |
| 751 | 0 0 2 3 3 4 4 5 5 7 7 9 | (13) |
| 752 | 0 0 0 1 1 2 3 4 4 4 | (11) |
| 753 | 2 | (1) |
Key: 748 | 5 represents 0.7485 kg.
The weights, x kg, of 120 students in a sports college are recorded. The results are summarised in the following table.
| Weight (x kg) | x โค 40 | x โค 60 | x โค 65 | x โค 70 | x โค 85 | x โค 100 |
|---|---|---|---|---|---|---|
| Cumulative frequency | 0 | 14 | 38 | 60 | 106 | 120 |
Calculate estimates for the mean and standard deviation of the weights of the 120 students.
The Quivers Archery club has 12 Junior members and 20 Senior members. For the Junior members, the mean age is 15.5 years and the standard deviation of the ages is 1.2 years. The ages of the Senior members are summarised by \(\Sigma y = 910\) and \(\Sigma y^2 = 42\,850\), where \(y\) is the age of a Senior member in years.
(i) Find the mean age of all 32 members of the club.
(ii) Find the standard deviation of the ages of all 32 members of the club.
Farfield Travel and Lacket Travel are two travel companies which arrange tours abroad. The numbers of holidays arranged in a certain week are recorded in the table below, together with the means and standard deviations of the prices.
| Number of holidays | Mean price ($) | Standard deviation ($) | |
|---|---|---|---|
| Farfield Travel | 30 | 1500 | 230 |
| Lacket Travel | 21 | 2400 | 160 |
(i) Calculate the mean price of all 51 holidays.
(ii) The prices of individual holidays with Farfield Travel are denoted by $x_F$ and the prices of individual holidays with Lacket Travel are denoted by $x_L$. By first finding $\sum x_F^2$ and $\sum x_L^2$, find the standard deviation of the prices of all 51 holidays.