OC GRADUATE SCHOOL: June 2013

Practice Problems: Measures of Central Tendency

Mean, Median, and Mode

In many real-life situations, it is helpful to describe data by a single number that is

most representative of the entire collection of numbers. Such a number is called a

measure of central tendency. The most commonly used measures are as follows.

1. The mean, or average, of numbers is the sum of the numbers divided by .

2. The median of numbers is the middle number when the numbers are written

in order. If is even, the median is the average of the two middle numbers.

3. The mode of numbers is the number that occurs most frequently. If two

numbers tie for most frequent occurrence, the collection has two modes and is

called bimodal.

1. On an interview for a job, the interviewer tells you that the average annual income

of the company’s 25 employees is $60,849. The actual annual incomes of the 25

employees are shown below. What are the mean, median, and mode of the

incomes? Was the person telling you the truth?

$17,305, $478,320, $45,678, $18,980, $17,408,

$25,676, $28,906, $12,500, $24,540, $33,450,

$12,500, $33,855, $37,450, $20,432, $28,956,

$34,983, $36,540, $250,921, $36,853, $16,430,

$32,654, $98,213, $48,980, $94,024, $35,671

2. In January of 2006, your family moved to a tropical climate. For the year that followed, you recorded the number of rainy days that occurred each month. Your data contained 14, 14, 10, 12, 11, 13, 11, 11, 14, 10, 13, 8.

a. Find the mean, mode, median and range for your data set of rainy days.
b. If the number of rainy days doubles each month in the year 2007, what will be
the mean, mode, median and range for the 2007 data?
c. If, instead, there are three more rainy days per month in the year 2007, what will
be the mean, mode, median and range for the 2007 data?

3. Test scores for a class of 20 students are as follows:
93, 84, 97, 98, 100, 78, 86, 100, 85, 92, 72, 55, 91, 90, 75, 94, 83, 60, 81, 95

Test Scores	Frequency
91-100
81-90
71-80
61-70
51-60

a) Copy and complete the table shown at the left.b) Find the modal interval.
The "modal interval" is the interval containing the greatest frequency. It is not the mode.

c) Find the interval that contains the median.

4. A group committed to quality television has been concerned about a new talk show. For two weeks, they decide to count the number of words that must be "bleeped" as too obscene for television and the number of physical altercations. They hope that after recording this data that they will be able to argue that the show is inappropriate for television particularly during the day. The data for number of words censored is provided below.

342
267
321
157
33
349
254
166
132
289

Find the Mean, Median, and Mode for the above data.
What does this information tell you about the talk show?
Is this data skewed?

5. Midterm exam scores for a small advanced neuroanatomy class are provided below. Scores represent percent of items marked correct on the exam.

87
99
75
87
94
75
35
88
87
93

Find the Mean, Median, and Mode for the above data.
What does this information tell you about students' performance on the midterm (the traditional grading scale is used)?
Is this data skewed?

6. Bailey has been playing golf on the weekends for the past three years. Recently, she started keeping track of her recorded scores. Her scores for June and July at her favorite 9-hole (par 36) golf course are provided below.

45
49
42
56
41
36
34
38
41
40
42
41
39
38
40
39
36
41

Find the Mean, Median, and Mode for the above data.
What does this information tell you about Bailey's golf game?
Is this data skewed?

7. A high school teacher at a small private school assigns trigonometry practice problems to be worked via the net. Students must use a password to access the problems and the time of log-in and log-off are automatically recorded for the teacher. At the end of the week, the teacher examines the amount of time each student spent working the assigned problems. The data is provided below in minutes.

15
28
25
48
22
43
49
34
22
33
27
25
22
20
39

Find the Mean, Median, and Mode for the above data.
What does this information tell you about students' length of time on the computer solving trigonometry problems?
Is this data skewed?

Exercises

1. Find the mean, median, mode and range of each set of numbers below.

(a) 3, 4, 7, 3, 5, 2, 6, 10

(b) 8, 10, 12, 14, 7, 16, 5, 7, 9, 11

(d) 108, 99, 112, 111, 108

(e) 64, 66, 65, 61, 67, 61, 57

(f) 21, 30, 22, 16, 24, 28, 16, 17

2. Twenty students were asked their shoe sizes. The results are given below.

8, 6, 7, 6, 5, 4 1

2 , 7 1

2 , 6 1

2 , 8 1

2, 10

7, 5, 5 1

2 8, 9, 7, 5, 6, 8 1

2 6

For this data, find

(a) the mean (b) the median (c) the mode (d) the range.

3. Eight people work in an office. They are paid hourly rates of

$12, $15, $15, $14, $13, $14, $13, $13

(a) Find

(i) the mean (ii) the median (iii) the mode.

(b) Which average would you use if you wanted to claim that the staff were:

(i) well paid (ii) badly paid?

4. Two people work in a factory making parts for cars. The table shows how many

complete parts they make in one week.

Worker Mon Tue Wed Thu Fri

Randy 20 21 22 20 21

John 30 15 12 36 28

(a) Find the mean and range for Randy and John.

(b) Who is more consistent?

5. A gardener buys 10 packets of seeds from two different companies. Each pack

contains 20 seeds and he records the number of plants which grow from each pack.

Company A 20 5 20 20 20 6 20 20 20 8

Company B 17 18 15 16 18 18 17 15 17 18

(a) Find the mean, median and mode for each company's seeds.

(b) Which company does the mode suggest is best?

(d) Find the range for each company's seeds.

6. Lionel takes four tests and scores the following marks.

65, 72, 58, 77

(a) What are his median and mean scores?

(b) If he scores 70 in his next test, does his mean score increase or decrease?

Find his new mean score.

7. Deran keeps a record of the number of fish he catches over a number of fishing

trips. His records are:

1, 0, 2, 0, 0, 0, 12, 0, 2, 0, 0, 1, 18, 0, 2, 0, 1.

(a) Why does he object to talking about the mode and median of the number of

fish caught?

MEP Jamaica: STRAND F UNIT 17 Measures of Central Tendency: Student Text

17.1

(b) What are the mean and range of the data?

he has caught is also 0 and his range is 15.

What is the largest number of fish that Evan has caught?

8. A gas station owner records the number of cars which visit his premises on

10 days. The numbers are:

204, 310, 279, 314, 257, 302, 232, 261, 308, 217

(a) Find the mean number of cars per day.

(b) The owner hopes that the mean will increase if he includes the number of

cars on the next day. If 252 cars use the gas station on the next day, will the

mean increase or decrease?

9. The students in a class state how many children there are in their family.

The numbers they state are given below.

1, 2, 1, 3, 2, 1, 2, 4, 2, 2, 1, 3, 1, 2,

2, 2, 1, 1, 7, 3, 1, 2, 1, 2, 2, 1, 2, 3

(a) Find the mean, median and mode for this data.

(b) Which is the most sensible average to use in this case?

10. In a beauty contest, the scores awarded by eight judges were:

5.9 6.7 6.8 6.5 6.7 8.2 6.1 6.3

(i) Using the eight scores, determine:

(a) the mean

(b) the median

(ii) Only six scores are to be used. Which two scores may be omitted to leave

the value of the median the same? (CXC)

11. The table shows the maximum and minimum temperatures recorded in six cities

one day last year.

City Maximum Minimum

Los Angeles 22°C 12°C

Boston 22°C − 3°C

Moscow 18°C − 9°C

Atlanta 27°C 8°C

Archangel 13°C −15°C

Cairo 28°C 13°C

(a) Work out the range of temperature for Atlanta.

(b) Which city in the table had the lowest temperature?

minimum temperature for Moscow.

MEP Jamaica: STRAND F UNIT 17 Measures of Central Tendency: Student Text

17.1

12. The weights, in grams, of seven sweet potatoes are

260, 225, 205, 240, 232, 205, 214

What is the median weight?

13. Here are the number of goals scored by a school football team in their matches this

term.

3, 2, 0, 1, 2, 0, 3, 4, 3, 2

(a) Work out the mean number of goals.

(b) Work out the range of the number of goals scored.

14.

(a) The weights, in kilograms, of the 8 members of Hereward House tug-of-war

team at a school sports event are

75, 73, 77, 76, 84, 76, 77, 78.

Calculate the mean weight of the team.

(b) The 8 members of Nelson House tug of war team have a mean weight of

64 kilograms.

Which team do you think will win a tug-of-war between Hereward House

and Nelson House? Give a reason for your answer.

15. Students in Grade 8 are arranged in eleven classes. The class sizes are

23, 24, 24, 26, 27, 28, 30, 24, 29, 24, 27.

(a) What is the modal class size?

(b) Calculate the mean class size.

The range of the class sizes for Grade 9 is 3.

in Grade 8?

16. A school has to select one student to take part in a General Knowledge Quiz.

Kelly and Rory took part in six trial quizzes. The following lists show their scores.

Kelly 28 24 21 27 24 26

Rory 33 19 16 32 34 18

Kelly had a mean score of 25 with a range of 7.

(a) Calculate Rory's mean score and range.

(b) Which student would you choose to represent the school? Explain the

reason for your choice, referring to the mean scores and ranges.

MEP Jamaica: STRAND F UNIT 17 Measures of Central Tendency: Student Text

17. Eight judges each give a mark out of 6 in a gymnastics competition.

Nicole is given the following marks.

5.3, 5.7, 5.9, 5.4, 4.5, 5.7, 5.8, 5.7

The mean of these marks is 5.5, and the range is 1.4.

The rules say that the highest mark and the lowest mark are to be deleted.

5.3, 5.7, 5.9, 5.4, 4.5, 5.7, 5.8, 5.7

(a) (i) Find the mean of the six remaining marks.

(ii) Find the range of the six remaining marks.

(b) Do you think it is better to count all eight marks, or to count only the six

remaining marks? Use the means and the ranges to explain your answer.

of 5.2 and a range of 0.6. Explain why none of her marks could be as high

as 5.9.

MEASURES OF VARIABILITY

1. The following scores were obtained by a class in a 15-item test in social science.

Boys: 9, 10, 13, 6, 10, 6, 9, 10, 8

Girls: 9, 7, 5, 12, 3, 8, 10, 2, 8, 10, 14, 7

Find the range, mean deviation, standard deviation, and variance.

a) For the boys

b) For the girls

c) For the whole class

Use a calculator and interpret the results.

2. Data on mental ability of a third-year high school class were recorded and are given below.

85 92 100 104 110 119 125 86 95 102 106

112 120 130 90 98 103 107 115 124

Assuming that the mean ±0.50 of a standard deviaton is considered average, how many in the class are considered average? Use a calculator.

3. A mental ability test was administered to ten students. The results are shown below:

110 105 120 115 137 122 108 112 114 124

Find the mean and standard deviation.

4. The numbers of hours worked last week by employees at a San Rafael Plant have been grouped in the following frequency distribution. Determine the standard deviation.

Hours Number of Workers

50 - 55 45

46 – 50 44

41 – 45 50

36 – 40 40

31 – 35 16

26 – 30 5

5. The following is the frequency distribution of the weights of 50 patients admitted in a hospital during the week.

Weight (kilos) Number of Patients

71 – 75 7

66 – 70 9

61 – 65 14

56 – 60 10

51 – 55 6

46 – 50 4

6. A hospital administrator compared the age with the emergency admissions on a given Monday with those Friday of the same week.

Age Monday Friday

61 – 70 8 3

51 – 60 6 6

41 – 50 5 12

31 – 40 9 14

21 – 30 10 10

11 – 20 9 8

1 - 10 7 7

Calculate the standard deviation for each set of data and compare the age dispersion in the Monday and Friday admissions.

TABLE 2.4

COMPUTATION OF THE MANAGEMENT PRACTICES OF SECONDARY SCHOOL ADMINISTRATORS IN TERMS OF PANNING AND ORGANIZING WORK AS ASSESSED BY MASTER TEACHER

FUNCTIONS	5	4	3	2	1	TOTAL	WM	DESCRIPTION
1. Develop project proposal for development	12	17	4	0	0	33/140	4.24	Very Satisfactory
2. Prepare the yearly school program	15	17	10	0	0	33/146	4.42	Very Satisfactory
3. Plans and administer school test	13	15	3	1	0	33/136	4.12	Very Satisfactory
4. Plans the implementation of Secondary School Curriculum	13	16	3	1	0	33/140	4.24	Very satisfactory
5. Plans and organize the government thrust such as Alay-Tanim, Cleanliness, etc.	11	19	3	0	0	33/140	4.24	Very Satisfactory
6. Prepares administrative and supervisory plans	14	12	7	0	0	33/139	4.21	Very Satisfactory
Total						198	4.24	Very Satisfactory

Scale Range Interpretation

5 4.51 – 5.00 Outstanding

4 3.51 – 4.50 Very Satisfactory

3 2.51 – 3.50 Satisfactory

2 1.51 – 2.50 Fair

1 1.00 – 1.50 Needs Improvement

Table 2.4 shows that the secondary school administrators were rated by the master teachers very satisfactory in “Planning and Organizing Work” as indicated by the grand weighted mean of 4.24. This tends to show that the secondary administrators were proficient in organizing the schools’ year long activities.

METHODS AND STRATEGIES OF TEACHING	5	4	3	2	1
1. Starts the lesson in an interesting manner	35	75	85	12	14
2. Clarifies the objective/s of the lesson	124	25	147	85	90
3. Makes the lesson practical and relevant by relating it with everyday living	3	25	85	1	4
4. Relates the lesson with current issues and other subject areas whenever possible	25	14	36	96	45
5. Facilitates students’ understanding by rephrasing questions using synonyms, giving examples and the like	75	31	47	90	4
6. Formulates questions that challenge the students’ imagination or reasoning ability	84	75	102	14	8
7. Gives students’ learning tasks; exercises related to the topics learned/being learned	147	25	14	3	7
8. Uses instructional time productively by properly apportioning it to all class activities	45	54	24	98	4
9. Encourages maximum participation of the students in all activities	23	25	74	32	105
10. Make use of blackboard and other audiovisual materials like pictures, graphs, and outlines to make lesson more interesting	45	74	25	80	205
11. Provides students with more opportunities for learning by giving assignments, research work and the like which could be accomplished within allotted time	32	12	98	25	6
12. Encourages students to ask intelligent and relevant questions	98	25	14	5	6
13. Establishes continuity and unity of present and previous lessons; summarizes the lessons to bring out important points	45	85	145	6	8
14. Evaluates student’s performance by giving quizzes from time to time	21	17	105	6	7
15. Informs students of test results not later than two weeks after the exam	125	64	35	5	4
16. Explains the grading system and gives grades fairly	35	98	105	85	5

MASTERY OF THE SUBJECT MATTER	5	4	3	2	1
1. Presents the lessons systematically	85	12	4	9	4
2. Explains the lessons/objectives clearly	95	25	145	7	5
3. Answers questions of the students intelligently and satisfactorily	35	85	96	14	4

COMMUNICATION SKILLS	5	4	3	2	1
1. Speaks correct English and/or Filipino and communicates his ideas well	35	24	6	1	85
2. Has a loud voice enough to be heard by everyone in class	45	85	3	4	6

CLASSROOM MANAGEMENT	5	4	3	2	1
1. Sees to it that the room is clean and orderly at all times	4	85	45	106	6
2. Maintains classroom discipline	4	90	4	6	14
3. Checks attendance of students	145	85	6	8	4
4. Sees to it that the students wear the prescribed uniform	85	95	105	6	8
5. Sees to it that all teaching materials like chalk and eraser are prepared	90	85	106	4	2
6. Puts off electric fans, lights and makes sure that blackboards are clean before dismissing the class	85	95	104	8	4

PERSONAL AND SOCIAL TRAITS	5	4	3	2	1
1. Projects self-confidence, respects opinion, suggestions, and comments of students	13	85	95	10	4
2. Projects respectable and decent image	45	75	98	12	6
3. Practices what he/she preaches/advises	13	25	74	6	7
4. Manifests and promotes charism of work and pray	35	74	106	4	8
5. Integrates positive values in the lesson	145	6	25	10	8
6. Recognizes students’ participation in institutional activities by proper considerations	63	85	106	4	5
7. Comes to class regularly	14	85	95	110	68
8. Comes to class on time	85	4	5	6	4
9. Approachable and encourages students to consult him/her for assistance	5	85	10	45	1

OC GRADUATE SCHOOL

Sunday, June 30, 2013

MEASURES OF CENTRAL TENDENCY

Practice Problems: Measures of Central Tendency

MEASURES OF VARIABILITY