Adult Basic Education

Advanced Level

MATHEMATICS

Data Analysis

Ministry of Advanced Education,

Training and Technology

Adult Basic Education

Advanced Level Mathematics

Data Analysis

Prepared by

Paul Grinder, Okanagan University College

with

Pat Corbett-Labatt, North Island College

Bob Darling, Malaspina University-College

Peter Robbins, Kwantlen University College

Ada Sarsiat, Northwest Community College

for the

Province of British Columbia

Ministry of Advanced Education, Training and Technology

and the

Centre for Curriculum, Transfer and Technology

Advanced Education, Skills & Training

Republished by BCcampus with permission.

Victoria, B.C.

Data Analysis by Paul Grinder is released under a

Creative Commons Attribution 4.0 International Licence,

except where otherwise noted.

The CC licence permits you to retain, reuse, copy,

redistribute, and revise this book—in whole or in part—for

free providing the authors are attributed as follows:

If you redistribute all or part of this book, it is

recommended the following statement be added to the

at no cost:

This textbook can be referenced. In APA citation style, it

would appear as follows:

Visit BCcampus Open Education to learn about open

education in British Columbia.

Data Analysis by Paul Grinder is under a CC BY 4.0 Licence.

Download for free from the B.C. Open Textbook Collection: https://open.bccampus.ca

Grinder, P. (2020). Data analysis. BCcampus.

Contents

Learning outcomes .................................................................... ii

Glossary ................................................................................... iii

Unit 1: The uses and abuses of statistics ...................................1

Unit 2: Introduction: Mean, median, mode, range and

graphs ...............................................................................5

Unit 3: Measures of position: quartiles and percentiles .........16

Unit 4: The standard deviation ................................................26

Unit 5: The normal distribution ..............................................33

Unit 6: The normal curve ........................................................44

Unit 7: Analysing survey data .................................................55

Unit 8: A statistics project .......................................................62

Appendix A ..............................................................................66

Appendix B ..............................................................................67

Answers....................................................................................69

Learning outcomes

The word statistics is derived from the Latin word status which means “state”. Governments

were the first to use statistics. They used statistics to collect and interpret data about their

countries. Today, statistics are used in almost every major field of study.

Upon completion of this Module, you should be able to:

• explain the uses and misuses of statistics

• demonstrate an understanding of mean, median, mode, range, quartiles, percentiles,

standard deviation, the normal curve, z scores, sampling error and confidence intervals

• graphically present data in the form of frequency tables, line graphs, bar graphs and stem

and leaf plots

• design and conduct a statistics project, analyze the data and communicate your

observations about the data

Procedure for independent study

1. Read each of the units in order and complete all of the exercises. If you need

assistance, contact your instructor.

2. Complete the Activity Exercises wherever possible.

3. Study the terminology in the Glossary to become familiar with the definitions.

4. If recommended by your instructor, complete additional problem sets.

5. Complete the Project for this Module.

iii

Glossary

Bar graph

A graph that uses side by side bars of different lengths to represent ranked data.

Confidence interval

The interval in which a statistic will likely fall, a certain percent of the time, after repeated

experimentation.

Data

The information collected for statistical analysis.

Deviation

The difference between one data value and the mean.

Frequency

The number of times that a particular value occurs in a set of data.

Frequency graph

Sometimes called a broken line graph. A graph with a horizontal axis representing data

values and a vertical axis representing frequency values.

Frequency histogram

Also known as a bar graph.

Measures of central tendency

Statistics that describe where the data is centred. The mean, median and mode are measures

of central tendency.

Measures of position

Statistics that describe how one data value compares to another. Percentiles, quartiles and z

scores are measures of position.

Measures of variation

Statistics that describe how the data is spread out or dispersed. The range, deviation and

standard deviation are measures of variation.

Mean

The average. The mean is obtained by finding the sum of the data values and dividing by the

number of data values.

Median

The middle value, or the average of the two middle values, of a set of ranked data.

Mode

The data value that occurs most frequently.

Normal curve

Also called a bell curve. Data that is distributed symmetrically about the mean so that most

of the data is close to the mean.

Normal distribution

A distribution that takes the shape of a normal curve when graphed. Approximately 68% of

the data values will fall within one standard deviation of the mean, 95.5% will fall within two

standard deviations of the mean and 99.7% of the data will fall within three standard

deviations of the mean.

Percentile

One of the 100 values that divide a set of ranked data into 100 equal intervals. The 48

percentile is a value that is higher than 48% of all the data values.

Population

A large group from which samples are taken for statistical analysis.

Quartile

One of four values that divide a set of ranked data into four equal intervals. The first quartile

is equal to the 25

percentile.

Random

A value is random if it has an equal chance of occurring as any other value from the same set.

Random sample

A sample that has the same probability of being chosen as any other sample of the same size.

Range

The difference between the largest data value and the smallest data value.

Ranked data

Data that is listed from highest to lowest or lowest to highest.

Sample

A small set of data chosen from a larger set of data.

Sampling error

The amount of error associated with a calculated value as determined by the size of the

sample.

Standard deviation

The square root of the average squared deviation of a set of data.

Statistic

A value calculated from a set of data. The mean and z scores are statistics.

Statistics

A branch of mathematics that collects, organizes and analyzes data.

Stem and leaf plot

A table of data values where the last digits of data values (leaves) are strung out behind their

first digits (or stem values).

Survey

Information derived from a sampling of a certain population.

Tally

A method of counting data using “tic” marks.

Yes population

A 40% yes population is one that has responded yes to a particular question 40% of the time.

z score

Also known as a standard score. The value obtained by dividing the deviation by the standard

deviation.

Unit 1: The uses and abuses of statistics

The word statistics has two meanings. A statistic is a numerical measurement describing

some characteristic of a set of data. For example, a statistic like 290 pounds could be used to

describe the average or mean weight of a football team. Statistics is also a collection of

methods for planning experiments, collecting data, analyzing the data and drawing

conclusions.

The uses of statistics

It is hard to read a magazine or newspaper without coming across some statistical survey or

analysis. Sportscasts, TV documentaries and newscasts also have their share of statistics. The

uses of statistics include applications in business, sports, medicine, agriculture, psychology,

sociology, education and political science. Governments use statistics to monitor everything

from life style preferences to crime rates. New drugs are statistically analyzed to determine

their effectiveness on patients. The statistical technique of random selection is employed to

guarantee that a small sample of a larger population group is actually an unbiased

representation of the whole population. Statistics, such as plus-minus records, can even be

used to determine whether a certain hockey player should be given more or less ice time.

The abuses of statistics

Just as statistics can be used to provide a solid quantitative analysis of a set of data, statistics

can be misused to distort data. The abuse of statistics is what Benjamin Disraeli (nineteenth

century British prime minister) was referring to when he made the famous comment, “There

are three kinds of lies – lies, damned lies and statistics.”

Statistics can be used to misrepresent a situation. Suppose a small store employs 6 people

who earn an average, or mean, wage of $8.50 per hour as calculated below,

Now suppose the store owner, who earns $40 per hour, includes his wages in the calculation,

If the store owner reports that the average wage earned at the store is $13, he or she is

misrepresenting the situation since the store owner is the only person making $13 per hour or

more.

$8.50

$10$9$8$8$8$8

++++

$13

$40$10$9$8

$8$8$8

++++++

Another source of deceptive statistics results from the faulty collection of data. Companies

that conduct public opinion polls have to be extremely careful that they survey a large

enough sample of the population and also an unbiased segment of the population. For

example, suppose a poll was conducted in BC to determine whether a luxury tax should be

imposed on buyers of new pick-up trucks. The citizens of Prince George might respond quite

differently to the poll than the residents of Victoria. The poll could be quite biased if it was

only conducted in Victoria, or only conducted in Prince George.

Statistical graphs can be presented in a deceptive manner. Consider the two bar graphs

below depicting the same data.

Men Women

Hours

watching

Men Women

Hours

watching

Hours spent per week watching TV

Without a close inspection of the vertical scale, the first bar graph creates the impression that

men watch twice as much TV as women do. In the second graph, the vertical scale starts at 0,

and the length of the bars are proportional to the actual hour of TV watching.

The above examples illustrate only a few of the abuses of statistics. To avoid the “lies and

damned lies”, every step of the statistical process must be scrupulously carried out; from the

collection of the data, to the calculation of a statistic, to the presentation of conclusions.

Now complete Exercise 1 and check your answers.

Exercise 1

1. Why is the following bar graph misleading?

Women

Men

75 80

Life expectancy from birth

2. What factor or factors might cause the following surveys to be biased?

a. TV news watchers are asked to phone in their opinion on whether marijuana

smoking should be legalized.

b. A questionnaire asking family members to list the number of books they read in

the last year is mailed to 1000 homes in the city of Vancouver.

c. To determine how many college students are smokers, Butler asks the first 20

students he sees standing outside the main entrance to the college, “Are you a

smoker?”

Answers are on page 69.

Activity 1: Watching TV

Ask every student in the room to write down, on a small piece of paper, an estimate of the

number of minutes they spent watching TV yesterday. Collect the data (pieces of paper) in

some sort of container.

1. a. Draw one piece of paper and record the number.

b. Do you think that this one piece of data is a good representation of the actual (yet

to be calculated) average?

2. Replace the first piece of paper and draw two pieces of data. Find the mean of these

two.

3. Replace the two pieces of data and now draw four pieces of paper. What is the

average time spent watching TV based on just these four pieces of data?

4. Replace the four pieces of paper and draw one half (or one half plus one) of the data.

Find the average time for one half the data.

5. Now find the mean using all the data.

a. How do the previous calculations of the mean, using smaller samples of the total

data, compare to the actual mean?

b. Some of the students may have recorded 0 minutes for the time they spent

watching TV yesterday. How did these zeros affect the mean time?

c. Now calculate the mean for only those students who actually watched some TV

yesterday.

Unit 2: Introduction: Mean, median, mode, range

and graphs

Statistics is the science of collecting, classifying, presenting and interpreting numerical data.

The data are numbers or measurements collected by a statistician. For example, the data

below are scores obtained by 12 students on a math quiz out of 40 marks.

32, 39, 32, 27, 30, 34, 32, 35, 40, 36, 32, 36

In order to statistically describe the above data, we might ask the following questions.

1. What is the average, or mean, score?

2. What is the middle, or median, score?

3. What score occurs most often, or what is the mode?

4. What is the difference between the highest and lowest score, or what is the range?

6. How can the data be represented graphically, with a line graph, bar graph or stem

and leaf plot?

The mean, median, mode, and range are four statistics which can be used to describe a set of

data. The mean, median, and mode are called measures of central tendency because they tell

us where the data is centered. The range is a measure of variation because it tells us how

much the data is spread out.

The mean is the most important measure of central tendency. It is calculated as follows:

valuesdata ofnumber theisn and value,data a is x mean, x where

or values,data ofnumber by the divided valuesdata theall of sum theis The

mean

The symbol “

” is the Greek letter “sigma” and means “the sum of all”. Here “

x” means the sum of all x (or data) values.

Example 1

Find the mean score of the following 12 math test scores;

32, 39, 32, 27, 30, 34, 32, 35, 40, 36, 32 and 36.

Solution

Using the formula, (note that n = 12),

363236403532343027323932 +++++++++++

33.75

405

The mean score is

.75.33

The median is the middle value when the data is arranged from highest to

lowest. If there are two middle values, then the median is the mean of these two

values.

Example 2

Find the median score of the above 12 math scores.

Solution

Arrange the data from the highest to lowest.

middle

values

Note that because there are an even number (twelve) of data values, we have two middle

values. The mean of these two values is

3234

Hence the median math score is 33.