1.1 Sampling and Data1
1.1.1 Student Learning Outcomes
By the end of this chapter, the student should be able to:
• Recognize and differentiate between key terms.
• Apply various types of sampling methods to data collection.
• Create and interpret frequency tables.
1.1.2 Introduction
You are probably asking yourself the question, "When and where will I use statistics?". If you read any
newspaper or watch television, or use the Internet, you will see statistical information. There are statistics
about crime, sports, education, politics, and real estate. Typically, when you read a newspaper article or
watch a news program on television, you are given sample information. With this information, you may
make a decision about the correctness of a statement, claim, or "fact." Statistical methods can help you make
the "best educated guess."
Since you will undoubtedly be given statistical information at some point in your life, you need to know
some techniques to analyze the information thoughtfully. Think about buying a house or managing a
budget. Think about your chosen profession. The fields of economics, business, psychology, education,
biology, law, computer science, police science, and early childhood development require at least one course
in statistics.
Included in this chapter are the basic ideas and words of probability and statistics. You will soon under-
stand that statistics and probability work together. You will also learn how data are gathered and what
"good" data are.
1.2 Statistics2
The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see
and use data in our everyday lives.
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CHAPTER 1. SAMPLING AND DATA
1.2.1 Optional Collaborative Classroom Exercise
In your classroom, try this exercise. Have class members write down the average time (in hours, to the
nearest half-hour) they sleep per night. Your instructor will record the data. Then create a simple graph
(called a dot plot) of the data. A dot plot consists of a number line and dots (or points) positioned above
the number line. For example, consider the following data:
5; 5.5; 6; 6; 6; 6.5; 6.5; 6.5; 6.5; 7; 7; 8; 8; 9
The dot plot for this data would be as follows:
Frequency of Average Time (in Hours) Spent Sleeping per Night
Figure 1.1
Does your dot plot look the same as or different from the example? Why? If you did the same example in
an English class with the same number of students, do you think the results would be the same? Why or
why not?
Where do your data appear to cluster? How could you interpret the clustering?
The questions above ask you to analyze and interpret your data. With this example, you have begun your
study of statistics.
In this course, you will learn how to organize and summarize data. Organizing and summarizing data is
called descriptive statistics. Two ways to summarize data are by graphing and by numbers (for example,
finding an average). After you have studied probability and probability distributions, you will use formal
methods for drawing conclusions from "good" data. The formal methods are called inferential statistics.
Statistical inference uses probability to determine how confident we can be that the conclusions are correct.
Effective interpretation of data (inference) is based on good procedures for producing data and thoughtful
examination of the data. You will encounter what will seem to be too many mathematical formulas for
interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to
gain an understanding of your data. The calculations can be done using a calculator or a computer. The
understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more
confident in the decisions you make in life.
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1.3 Probability3
Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of
an event occurring. For example, if you toss a fair coin 4 times, the outcomes may not be 2 heads and 2
tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails.
The expected theoretical probability of heads in any one toss is 1 or 0.5. Even though the outcomes of a
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few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After
reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012
heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction 996 is equal
2000
to 0.498 which is very close to 0.5, the expected probability.
The theory of probability began with the study of games of chance such as poker. Predictions take the form
of probabilities. To predict the likelihood of an earthquake, of rain, or whether you will get an A in this
course, we use probabilities. Doctors use probability to determine the chance of a vaccination causing the
disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of
return on a client’s investments. You might use probability to decide to buy a lottery ticket or not. In your
study of statistics, you will use the power of mathematics through probability calculations to analyze and
interpret your data.
1.4 Key Terms4
In statistics, we generally want to study a population. You can think of a population as an entire collection
of persons, things, or objects under study. To study the larger population, we select a sample. The idea of
sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to
gain information about the population. Data are the result of sampling from a population.
Because it takes a lot of time and money to examine an entire population, sampling is a very practical
technique. If you wished to compute the overall grade point average at your school, it would make sense
to select a sample of students who attend the school. The data collected from the sample would be the
students’ grade point averages. In presidential elections, opinion poll samples of 1,000 to 2,000 people are
taken. The opinion poll is supposed to represent the views of the people in the entire country. Manu-
facturers of canned carbonated drinks take samples to determine if a 16 ounce can contains 16 ounces of
carbonated drink.
From the sample data, we can calculate a statistic. A statistic is a number that is a property of the sample.
For example, if we consider one math class to be a sample of the population of all math classes, then the
average number of points earned by students in that one math class at the end of the term is an example of
a statistic. The statistic is an estimate of a population parameter. A parameter is a number that is a property
of the population. Since we considered all math classes to be the population, then the average number of
points earned per student over all the math classes is an example of a parameter.
One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. The
accuracy really depends on how well the sample represents the population. The sample must contain the
characteristics of the population in order to be a representative sample. We are interested in both the
sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the
sample statistic to test the validity of the established population parameter.
A variable, notated by capital letters like X and Y, is a characteristic of interest for each person or thing in
a population. Variables may be numerical or categorical. Numerical variables take on values with equal
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CHAPTER 1. SAMPLING AND DATA
units such as weight in pounds and time in hours. Categorical variables place the person or thing into a
category. If we let X equal the number of points earned by one math student at the end of a term, then X
is a numerical variable. If we let Y be a person’s party affiliation, then examples of Y include Republican,
Democrat, and Independent. Y is a categorical variable. We could do some math with values of X (calculate
the average number of points earned, for example), but it makes no sense to do math with values of Y
(calculating an average party affiliation makes no sense).
Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single
value.
Two words that come up often in statistics are mean and proportion. If you were to take three exams in
your math classes and obtained scores of 86, 75, and 92, you calculate your mean score by adding the three
exam scores and dividing by three (your mean score would be 84.3 to one decimal place). If, in your math
class, there are 40 students and 22 are men and 18 are women, then the proportion of men students is 22
40
and the proportion of women students is 18 . Mean and proportion are discussed in more detail in later
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chapters.
NOTE: The words "mean" and "average" are often used interchangeably. The substitution of one
word for the other is common practice. The technical term is "arithmetic mean" and "average" is
technically a center location. However, in practice among non-statisticians, "average" is commonly
accepted for "arithmetic mean."
Example 1.1
Define the key terms from the following study: We want to know the average amount of money
first year college students spend at ABC College on school supplies that do not include books. We
randomly survey 100 first year students at the college. Three of those students spent $150, $200,
and $225, respectively.
Solution
The population is all first year students attending ABC College this term.
The sample could be all students enrolled in one section of a beginning statistics course at ABC
College (although this sample may not represent the entire population).
The parameter is the average amount of money spent (excluding books) by first year college stu-
dents at ABC College this term.
The statistic is the average amount of money spent (excluding books) by first year college students
in the sample.
The variable could be the amount of money spent (excluding books) by one first year student.
Let X = the amount of money spent (excluding books) by one first year student attending ABC
College.
The data are the dollar amounts spent by the first year students. Examples of the data are $150,
$200, and $225.
1.4.1 Optional Collaborative Classroom Exercise
Do the following exercise collaboratively with up to four people per group. Find a population, a sample,
the parameter, the statistic, a variable, and data for the following study: You want to determine the average
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number of glasses of milk college students drink per day. Suppose yesterday, in your English class, you
asked five students how many glasses of milk they drank the day before. The answers were 1, 0, 1, 3, and 4
glasses of milk.
1.5 Data5
Data may come from a population or from a sample. Small letters like x or y generally are used to represent
data values. Most data can be put into the following categories:
• Qualitative
• Quantitative
Qualitative data are the result of categorizing or describing attributes of a population. Hair color, blood
type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data.
Qualitative data are generally described by words or letters. For instance, hair color might be black, dark
brown, light brown, blonde, gray, or red. Blood type might be AB+, O-, or B+. Researchers often prefer to
use quantitative data over qualitative data because it lends itself more easily to mathematical analysis. For
example, it does not make sense to find an average hair color or blood type.
Quantitative data are always numbers. Quantitative data are the result of counting or measuring attributes
of a population. Amount of money, pulse rate, weight, number of people living in your town, and the
number of students who take statistics are examples of quantitative data. Quantitative data may be either
discrete or continuous.
All data that are the result of counting are called quantitative discrete data. These data take on only certain
numerical values. If you count the number of phone calls you receive for each day of the week, you might
get 0, 1, 2, 3, etc.
All data that are the result of measuring are quantitative continuous data assuming that we can measure
accurately. Measuring angles in radians might result in the numbers π , π , π ,
, etc. If you and your
6
3
2
π , 3 π
4
friends carry backpacks with books in them to school, the numbers of books in the backpacks are discrete
data and the weights of the backpacks are continuous data.
Example 1.2: Data Sample of Quantitative Discrete Data
The data are the number of books students carry in their backpacks. You sample five students.
Two students carry 3 books, one student carries 4 books, one student carries 2 books, and one
student carries 1 book. The numbers of books (3, 4, 2, and 1) are the quantitative discrete data.
Example 1.3: Data Sample of Quantitative Continuous Data
The data are the weights of the backpacks with the books in it. You sample the same five students.
The weights (in pounds) of their backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that backpacks carrying
three books can have different weights. Weights are quantitative continuous data because weights
are measured.
Example 1.4: Data Sample of Qualitative Data
The data are the colors of backpacks. Again, you sample the same five students. One student has
a red backpack, two students have black backpacks, one student has a green backpack, and one
student has a gray backpack. The colors red, black, black, green, and gray are qualitative data.
NOTE: You may collect data as numbers and report it categorically. For example, the quiz scores
for each student are recorded throughout the term. At the end of the term, the quiz scores are
reported as A, B, C, D, or F.
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CHAPTER 1. SAMPLING AND DATA
Example 1.5
Work collaboratively to determine the correct data type (quantitative or qualitative). Indicate
whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with
the words "the number of."
1. The number of pairs of shoes you own.
2. The type of car you drive.
3. Where you go on vacation.
4. The distance it is from your home to the nearest grocery store.
5. The number of classes you take per school year.
6. The tuition for your classes
7. The type of calculator you use.
8. Movie ratings.
9. Political party preferences.
10. Weight of sumo wrestlers.
11. Amount of money (in dollars) won playing poker.
12. Number of correct answers on a quiz.
13. Peoples’ attitudes toward the government.
14. IQ scores. (This may cause some discussion.)
1.6 Variation6
1.6.1 Variation in Data
Variation is present in any set of data. For example, 16-ounce cans of beverage may contain more or less
than 16 ounces of liquid. In one study, eight 16 ounce cans were measured and produced the following
amount (in ounces) of beverage:
15.8; 16.1; 15.2; 14.8; 15.8; 15.9; 16.0; 15.5
Measurements of the amount of beverage in a 16-ounce can may vary because different people make the
measurements or because the exact amount, 16 ounces of liquid, was not put into the cans. Manufacturers
regularly run tests to determine if the amount of beverage in a 16-ounce can falls within the desired range.
Be aware that as you take data, your data may vary somewhat from the data someone else is taking for the
same purpose. This is completely natural. However, if two or more of you are taking the same data and
get very different results, it is time for you and the others to reevaluate your data-taking methods and your
accuracy.
1.6.2 Variation in Samples
It was mentioned previously that two or more samples from the same population, taken randomly, and
having close to the same characteristics of the population are different from each other. Suppose Doreen and
Jung both decide to study the average amount of time students at their college sleep each night. Doreen and
Jung each take samples of 500 students. Doreen uses systematic sampling and Jung uses cluster sampling.
Doreen’s sample will be different from Jung’s sample. Even if Doreen and Jung used the same sampling
method, in all likelihood their samples would be different. Neither would be wrong, however.
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Think about what contributes to making Doreen’s and Jung’s samples different.
If Doreen and Jung took larger samples (i.e. the number of data values is increased), their sample results
(the average amount of time a student sleeps) might be closer to the actual population average. But still,
their samples would be, in all likelihood, different from each other. This variability in samples cannot be
stressed enough.
1.6.2.1 Size of a Sample
The size of a sample (often called the number of observations) is important. The examples you have seen
in this book so far have been small. Samples of only a few hundred observations, or even smaller, are
sufficient for many purposes. In polling, samples that are from 1200 to 1500 observations are considered
large enough and good enough if the survey is random and is well done. You will learn why when you
study confidence intervals.
Be aware that many large samples are biased.
For example, call-in surveys are invariable biased
because people choose to respond or not.
1.6.2.2 Optional Collaborative Classroom Exercise
Exercise 1.6.1
Divide into groups of two, three, or four. Your instructor will give each group one 6-sided die.
Try this experiment twice. Roll one fair die (6-sided) 20 times. Record the number of ones, twos,
threes, fours, fives, and sixes you get below ("frequency" is the number of times a particular face
of the die occurs):
First Experiment (20 rolls)
Face on Die
Frequency
1
2
3
4
5
6
Table 1.1
Second Experiment (20 rolls)
Face on Die
Frequency
1
2
3
4
5
6
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CHAPTER 1. SAMPLING AND DATA
Table 1.2
Did the two experiments have the same results? Probably not. If you did the experiment a third
time, do you expect the results to be identical to the first or second experiment? (Answer yes or
no.) Why or why not?
Which experiment had the correct results? They both did. The job of the statistician is to see
through the variability and draw appropriate conclusions.
1.6.3 Critical Evaluation
We need to critically evaluate the statistical studies we read about and analyze before accepting the results
of the study. Common problems to be aware of include
• Problems with Samples: A sample should be representative of the population. A sample that is not
representative of the population is biased. Biased samples that are not representative of the popula-
tion give results that are inaccurate and not valid.
• Self-Selected Samples: Responses only by people who choose to respond, such as call-in surveys are
often unreliable.
• Sample Size Issues: Samples that are too small may be unreliable. Larger samples are better if possible.
In some situations, small samples are unavoidable and can still be used to draw conclusions, even
though larger samples are better. Examples: Crash testing cars, medical testing for rare conditions.
• Undue influence: Collecting data or asking questions in a way that influences the response.
• Non-response or refusal of subject to participate: The collected responses may no longer be represen-
tative of the population. Often, people with strong positive or negative opinions may answer surveys,
which can affect the results.
• Causality: A relationship between two variables does not mean that one causes the other to occur.
They may both be related (correlated) because of their relationship through a different variable.
• Self-Funded or Self-Interest Studies: A study performed by a person or organization in order to sup-
port their claim. Is the study impartial? Read the study carefully to evaluate the work. Do not
automatically assume that the study is good but do not automatically assume the study is bad either.
Evaluate it on its merits and the work done.
• Misleading Use of Data: Improperly displayed graphs, incomplete data, lack of context.
• Confounding: When the effects of multiple factors on a response cannot be separated. Confounding
makes it difficult or impossible to draw valid conclusions about the effect of each factor.
1.7 Answers and Rounding Off7
A simple way to round off answers is to carry your final answer one more decimal place than was present
in the original data. Round only the final answer. Do not round any intermediate results, if possible. If it
becomes necessary to round intermediate results, carry them to at least twice as many decimal places as the
final answer. For example, the average of the three quiz scores 4, 6, 9 is 6.3, rounded to the nearest tenth,
because the data are whole numbers. Most answers will be rounded in this manner.
It is not necessary to reduce most fractions in this course. Especially in Probability Topics (Section 4.1), the
chapter on probability, it is more helpful to leave an answer as an unreduced fraction.
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1.8 Frequency8
Twenty students were asked how many hours they worked per day. Their responses, in hours, are listed
below:
5; 6; 3; 3; 2; 4; 7; 5; 2; 3; 5; 6; 5; 4; 4; 3; 5; 2; 5; 3
Below is a frequency table listing the different data values in ascending order and their frequencies.
Frequency Table of Student Work Hours
DATA VALUE
FREQUENCY
2
3
3
5
4
3
5
6
6
2
7
1
Table 1.3
A frequency is the number of times a given datum occurs in a data set. According to the table above,
there are three students who work 2 hours, five