9. Centre and Spread

Worksheet

1

Find the median of the following sets of scores:

a

11, \, 11, \, 13, \, 14, \, 18, \, 22, \, 23, \, 25

b

6, \, 8, \, 9, \, 11, \, 16, \, 17, \, 18

2

For each of the following sets of scores:

i

Sort the scores in ascending order.

ii

Calculate the median.

a

23, \, 25, \, 13, \, 9, \, 11, \, 21, \, 24, \, 17, \, 20

b

24, \, 11, \, 1, \, 3, \, 24, \, 28, \, 16, \, 16

c

65.2, \, 64.3, \, 71.6, \, 63.2, \, 45.2, \, 62.2, \, 46.8, \, 58.7

d

44.9, \, 45.6, \, -54.8, \, 74.7, \, -77.6 , \, -42.6, \, 67.9, \, 40.6

3

Find the median for the data in each of the following frequency distribution tables:

a

Score | Frequency |
---|---|

11 | 24 |

12 | 9 |

13 | 21 |

14 | 9 |

15 | 6 |

b

Score | Frequency |
---|---|

3 | 4 |

4 | 8 |

5 | 6 |

6 | 11 |

7 | 10 |

c

Score | Frequency |
---|---|

23 | 2 |

24 | 26 |

25 | 37 |

26 | 24 |

27 | 25 |

d

Score | Frequency |
---|---|

10 | 6 |

14 | 9 |

18 | 6 |

21 | 7 |

24 | 8 |

4

Find the median for the data in the given stem plot:

Leaf | |
---|---|

1 | 1\ 3 |

2 | 0\ 3\ 4 |

3 | 3\ 4\ 6\ 7 |

4 | 0\ 0\ 1\ 5 |

5 | 0\ 1 |

6 | 5 |

7 | 4\ 7 |

8 | 1\ 4 |

Key: 1|2 = 12

5

Which of the following sets of data has the highest median?

Set A: \, 3, \, 9, \, 13, \, 18

Set B: \, 9, \, 16, \, 16, \, 65, \, 86

6

Consider the following sets of data:

Set A: \, 14, \, 15, \, 2, \, 15, \, 26

Set B: \, 7, \, 5, \, 3, \, 2, \, 1, \, 5

a

Which data set has the lowest mean?

b

Which data set has the lowest median?

7

Which of the following dot plots has the highest median?

A

B

8

Which of the following column graphs has the smallest median?

A

B

9

For each of the following column graphs, find:

i

The total number of scores.

ii

The median.

a

b

10

State the position of the median in an ordered set of:

a

69 scores

b

152 scores

11

How many scores are there in a distribution if the number of scores is even and the median lies between the 24th and 25th scores?

12

Find four consecutive odd numbers whose median is 40.

13

Find the mode(s) of the following sets of scores:

a

2, \, 2, \, 6, \, 7, \, 7, \, 7, \, 7, \, 11, \, 11, \, 11, \, 13, \, 13, \, 16, \, 16

b

2, \, 2, \, 6, \, 8, \, 8, \, 8, \, 8, \, 12, \, 14, \, 14, \, 14, \, 14, \, 18, \, 18

c

8, \, 18, \, 5, \, 2, \, 2, \, 10, \, 8, \, 5, \, 14, \, 14, \, 8, \, 8, \, 10, \, 18, \, 14, \, 5

14

Find the mode(s) of the following data sets:

a

b

Score | Frequency |
---|---|

25 | 17 |

26 | 42 |

27 | 35 |

28 | 32 |

29 | 12 |

30 | 20 |

c

Leaf | |
---|---|

2 | 2\ 5\ 6\ 7\ 9 |

3 | 0\ 0\ 5\ 6\ 8 |

4 | 0\ 1\ 3\ 8\ 9 |

**Key:** 2\vert 4=2.4

d

15

Which of the following sets of data has the lowest mode?

Set A: \, 3, \, 9, \, 18, \, 9, \, 65, \, 13

Set B: \, 5, \, 12, \, 16, \, 16, \, 86, \, 3

16

Which of the following dot plots has the lowest mode?

A

B

17

Which of the following column graphs has the largest mode?

A

B

18

State whether the following statements are true or false:

a

If two sets of data have the same median then the data sets must be the same.

b

If two sets of data have different modes then the highest values cannot be the same.

c

Two sets of data that have the same highest and lowest values must have the same mean.

d

Two sets of data have the same highest and lowest values. This means they must have the same mode.

19

Consider the stem and leaf plot below:

a

Find the mode.

b

Find the median.

Leaf | |
---|---|

2 | 4\ 9 |

3 | 1\ 1\ 1\ 9 |

4 | 3\ 7 |

5 | 1\ 6\ 6 |

6 | 2\ 2\ 6 |

7 | 4\ 8 |

8 | 3\ 7\ 7 |

9 | 5 |

20

Consider the dot plot given:

From which score can a dot be removed so that the mean, median and mode remain unchanged?

21

The percentage of faulty computer chips in 42 batches were recorded in the given histogram:

a

Describe the distribution of the data in terms of the mode.

b

State the modal class of the data.

22

Consider the given dot plot:

a

Describe the modality of the data.

b

Find the modal score(s).

23

A consumer group surveys the price of petrol per litre at six different petrol stations and the results are as follows:

\$1.68, \, \$1.64, \, \$1.64, \, \$1.71, \, \$1.64, \, \$1.57

State the mode petrol price.

24

The following stem plot shows the number of hours students spent studying for a science exam.

Find the following, correct to two decimal places if necessary:

a

Mean

b

Median

c

Mode

Leaf | |
---|---|

6 | 0\ 2\ 3\ 6\ 7\ 7\ 8 |

7 | 3\ 6\ 9 |

8 | 0\ 5\ 6\ 7 |

9 | 2 |

25

The following stem plot shows the ages of 20 employees in a company:

a

How many of the employees are in their 30's?

b

What is the age of the oldest employee?

c

What is the age of the youngest employee?

d

Find the median age of the employees.

e

What is the modal age group?

Leaf | |
---|---|

2 | 0\ 2\ 3\ 4\ 6\ 6\ 7 |

3 | 1\ 5\ 6\ 7\ 8 |

4 | 0\ 3\ 4\ 6\ 7 |

5 | 0\ 0\ 8 |

Key: 1 \vert 2 \ = \ 12 years old

26

On Sunday, 20 planes were delayed at the airport. The dot plot shows the number of hours each departure was delayed:

a

What was the median number of minutes a plane was delayed?

b

What percentage of planes were delayed for longer than the median time?

c

If a plane is delayed for more than 30 minutes, the airline must pay \$5000. In total, how much were airlines fined that day?

27

A diver measures how long she can hold her breath underwater over several dives. If the median time is 3.9 minutes, determine whether the following statements are true:

a

The longest she held her breath is 7.8 minutes.

b

Most of the time she held her breath for less than 3.9 minutes.

c

Half the dives she was able to hold her breath longer than 3.9 minutes.

d

The shortest time she held her breath is 1.95 minutes.

e

Most of the time she held her breath for longer than 3.9 minutes.

28

A real estate agent wanted to determine a typical house price in a certain area. He gathered the selling price of some houses (in dollars):

317\,000,\, 320\,000,\, 347\,000,\, 360\,000,\, 378\,000,\, 395\,000,\, 438\,000,\, 461\,000,\, 479\,000,\, 499\,000

a

Find the mean house price.

b

What percentage of the house prices exceeded the mean?

c

Find the median house price.

d

What percentage of house prices exceeded the median?

29

The luggage check-in weight (in kilograms) of passengers is given in the following frequency table:

a

Complete the table.

b

Find the median check-in weight.

c

Find the mean check-in weight, correct to two decimal places.

d

When one of the passengers sees that the weight of their luggage is 21 \text{ kg}, they decide to add some items to the bag. Would this change the mean or the median?

\text{Weight }(x) | \text{Frequency }(f) | fx |
---|---|---|

16 | 12 | |

17 | 17 | |

18 | 21 | |

19 | 10 | |

20 | 18 | |

21 | 22 | |

\text{Total} |

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identify the mode

calculate measures of central tendency, the arithmetic mean and the median