MATH 1401 · Dr. Barry Monk

Sampling & Types of Data

Section 1.1 & 1.2

Nine objectives, two sections

Section 1.1 · Sampling
1

Describe the investigative process of statistics

2

Construct a simple random sample

3

Determine when samples of convenience are acceptable

4

Describe stratified sampling, cluster sampling, systematic sampling, and voluntary response sampling

5

Distinguish between statistics and parameters

Section 1.2 · Types of data
6

Understand the structure of a typical data set

7

Distinguish between qualitative and quantitative variables

8

Distinguish between ordinal and nominal variables

9

Distinguish between discrete and continuous variables

Section 1.1 · Objective 1

Describe the investigative process of statistics

Three masked voters marking ballots at separate cardboard voting booths.

The investigative process of statistics

Before an election, polls often tell the percentages of voters that prefer each candidate. Polling follows a process that is typical in statistics.

Statistics is an investigative process that involves these steps:

1Formulate questions
2Collect data needed to answer the questions
3Describe the data
4Draw conclusions, using appropriate methods
Definition

Terminology

Often, we want information about a large group of individuals but are only able to collect information on a small part of that group.

Statistics is the study of procedures for collecting, describing, and drawing conclusions from information.

A population is the entire collection of individuals about which information is sought.

A sample is a subset of a population containing the individuals that are observed.

A pipette releasing a single droplet onto a plate already scattered with dozens of identical droplets.

Simple random sampling

Ideally, we would like our sample to represent the population as closely as possible. Unfortunately, there are no methods that can guarantee this.

The best sampling methods all involve random selection. The most basic, and in many cases the best, sampling method is the method of simple random sampling.

Section 1.1 · Objective 2

Construct a simple random sample

Definition

Simple random sample

A simple random sample of size n is a sample chosen by a method in which each collection of n population items is equally likely to make up the sample.

A simple random sample is analogous to a lottery. Suppose that 10,000 lottery tickets are sold and 5 are drawn as the winning tickets. Each collection of 5 tickets that can be formed is equally likely to comprise the group of 5 that is drawn.

Five numbered lottery balls in a row: 1, 14, 21, 35 and 42.
A large group of students in athletic clothing walking together along an outdoor path.
Example 1

Simple random sample

A physical education professor wants to study the fitness levels of 20,000 students at her university. She numbers the students from 1 to 20,000, then uses a random number generator to select 100 students to participate in the study. Is this a simple random sample?

Yes, this is a simple random sample since any group of 100 students would have been equally likely to have been chosen.
Rows of empty wooden seats in a university lecture hall.
Example 2

Simple random sample

The professor now wants to survey 50 students about which sports they play. She uses her 10:00 am class of 50 students to fill out the questionnaire. Is this a simple random sample?

No, this isn't a simple random sample. Here, only the 10:00 am class was considered.
Tall stacks of precast concrete blocks in a yard, banded and layered so that only the outermost blocks can be reached.
Section 1.1 · Objective 3

Determine when samples of convenience are acceptable

Colored pencils resting on white paper, each one at the end of the line it has drawn.
Definition

Samples of convenience

In some cases, drawing a truly random sample is difficult or impossible. In these cases, a convenient method is sometimes used instead.

A sample of convenience is one that is selected without following a well-defined random process.

Example

Sample of convenience

A construction engineer received 1,000 concrete blocks in a large pile and wants to test the strength of 10 blocks. Why might it be difficult to draw a simple random sample?

To draw a simple random sample, the engineer would need to remove blocks from the center and bottom of the pile. A more convenient method would be to take 10 blocks from the top of the pile.
Section 1.1 · Objective 4

Describe stratified sampling, cluster sampling, systematic sampling, and voluntary response sampling

A sandstone canyon wall, its rock in bands of different colors stacked one above another.
Definition

Stratified random sampling

In stratified random sampling, the population is divided into groups, called strata, and a simple random sample is drawn from each group (stratum).

This method is useful when the strata differ from one another, but the individuals within a stratum tend to be similar.

Example

Stratified random sampling

A company with 800 full-time and 200 part-time employees wants to draw a sample of 100. They select a simple random sample of 80 full-time employees and 20 part-time employees.

GROUP 1 Full-time Employees 800 GROUP 2 Part-time Employees 200 Choose simple random sample of 80 full-time employees Choose simple random sample of 20 part-time employees Stratified Random Sample of 100
Dozens of small white model houses spread across a white surface, with one red-roofed house among them.
Definition

Cluster sampling

In cluster sampling, items are randomly selected in groups, or clusters, from the population. This method is useful when the population is too large and spread out for simple random sampling to be feasible.

Example:

To estimate the unemployment rate, a government agency randomly samples households in a county and asks how many adults live there and how many are unemployed. What are the clusters? Why is this considered a cluster sample?

Example

Cluster sampling

The clusters are the groups of adults in each of the households. This is a cluster sample because a simple random sample of clusters is selected and every individual in each selected cluster is part of the sample.

Household Household Household Household Household Household Household Household Household Household Household Household Household Household Household Household Household Household Household Household Household Random sample of households is taken Interview every individual in each household Cluster Sample of Individuals
Definition

Systematic sampling

In systematic sampling, items are ordered, and every kth item is selected for the sample. This method is often used to sample products on an assembly line to ensure they meet quality standards.

Example:

Automobiles are coming off an assembly line, and a systematic sample will be taken for inspection. Starting with the third car, every fifth car will be sampled. Which cars will be selected?

Cars moving in a line along a factory assembly line.
Example

Systematic sampling

The sample begins with the third car, followed by every fifth car. The cars sampled will be the 3rd, 8th, 13th, 18th, 23rd, and so forth.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 …
Cars sampled 3, 8, 13, 18, 23
A radio presenter at a studio microphone, headphones on, mid-broadcast.

Voluntary response sampling

Voluntary response samples are often used by the media to try to engage the audience. For example, a radio announcer will invite people to call the station to say what they think.

Voluntary response samples are never reliable for the following reasons:

People who volunteer an opinion tend to have stronger opinions than is typical of the population.

People with negative opinions are often more likely to volunteer their response.

What we covered

Simple randomEvery possible sample is equally likely.
StratifiedDivide into strata, then draw from each.
ClusterSelect whole groups, then take everyone in them.
SystematicOrder the items, then take every kth.
ConvenienceSelected without a well-defined random process.
Voluntary responseRespondents volunteer themselves — never reliable.

Stratified, cluster and systematic all use randomness — but none of them is a simple random sample.

Interactive

Draw a sample from 1,000 employees

Selected 0
Method
Full-time 0 of 800 Part-time 0 of 200
Check your understanding 3

What type of sample is it?

a

A podcaster uses social media to ask listeners their opinions on an upcoming election, receiving over 10,000 responses. What type of sample is this?

Voluntary Response Sample
b

Every 10 years, the U.S. Census Bureau counts everyone living in the U.S. To verify accuracy in a specific city, they sample census districts and recount the people in those districts. What type of sample do these recounted individuals form?

Cluster Sample
Check your understanding 4

What type of sample is it?

a

A researcher studying the effect of diet on heart disease selects a simple random sample of 100 men and another of 100 women, knowing their diets differ and that men are more susceptible to heart disease. What type of sample do these 200 people represent?

Stratified Sample
b

A college basketball team held a promotion at one of its games in which every twentieth person who entered the arena won a free basketball. What type of sample do the winners represent?

Systematic Sample
Check your understanding 5

What type of sample is it?

a

A restaurant manager stops by the tables of people in the restaurant and asks them for their opinions on upcoming menu changes. What type of sample is this?

Sample of Convenience
b

To select people to call for jury duty, a numbered list of all residents with driver’s licenses or ID cards is made. Then random numbers are generated, and the people corresponding to those numbers are selected. What type of sample is this?

Simple Random Sample
Section 1.1 · Objective 5

Distinguish between statistics and parameters

Definition

Statistic vs. parameter

We often use numbers to describe, or summarize, a sample or a population.

Statistic

A statistic is a number that describes a sample.

Parameter

A parameter is a number that describes a population.

Note how statistic and sample both start with “s” and parameter and population both start with “p”.

Example

Statistic and parameter

Decide whether the number describes a statistic or a parameter:

a

57% of the teachers at Central High School are female.

The number 57% is a parameter, because it describes the entire population of teachers in the school.
b

In a sample of 100 surgery patients who were given a new pain reliever, 78% of them reported significant pain relief.

The number 78% is a statistic, because it describes a sample.
Section 1.2 · Objective 6

Understand the structure of a typical data set

Definition

Major, final exam score, and grade for several students

Student Major Exam score Grade
1Psychology92A
2Business75B
3Communications82B
4Psychology72C
5Art85B

Information is collected on individuals.

The characteristics of the individuals about which we collect information are called variables.

The values of the variables that we obtain are the data.

The information collected is called a data set.

Section 1.2 · Objective 7

Distinguish between qualitative and quantitative variables

Definition

Qualitative and quantitative variables

Variables can be divided into two types.

Variables
Qualitative variables (or categorical variables)Classify individuals into categories
Quantitative variablesTell how much or how many of something there is
Example

Qualitative and quantitative

Decide whether each of the following describes qualitative variables or quantitative variables.

a

A person’s age

This variable is quantitative because it tells how much time has elapsed since the person was born.
b

A person’s place of birth

Place is a qualitative variable. It includes categories like "New York," "Atlanta," etc.
c

The mileage of a car

This variable is quantitative because it tells how many miles a car will go on a certain amount of gasoline.
d

The color of a car

This variable is qualitative because it consists of the categories of different colors.
Section 1.2 · Objective 8

Distinguish between ordinal and nominal variables

Definition

Ordinal and nominal

Qualitative variables can be further divided into ordinal and nominal variables.

Variables
Qualitative
Ordinal variablesHave a natural ordering
Nominal variablesDo not have a natural ordering
Quantitative
Example

Ordinal and nominal

Decide whether each of the following describes ordinal variables or nominal variables.

a

State of residence

State of residence is nominal, because there is no natural order to the states.
b

Hair color

Hair color is nominal, because there is no natural order to the colors.
c

Letter grade in a class (A, B, C, D, or F)

This variable is ordinal because there is a natural ordering. For example, grades from best to worst are A, B, C, D, F.
d

Size of drink ordered

This variable is ordinal because there is a natural ordering. For example, sizes may be small, medium, large, or extra-large.
Check your understanding 2

Nominal or ordinal?

Categorize each of the following as nominal or ordinal.

a

The names of streets in a town

Nominal
b

Movie ratings G, PG, PG-13, R, and NC-17

Ordinal
Section 1.2 · Objective 9

Distinguish between discrete and continuous variables

Definition

Discrete and continuous

Quantitative variables can be further divided into discrete and continuous variables.

Variables
Qualitative
Ordinal
Nominal
Quantitative
Discrete variablesPossible values can be listed or counted
Continuous variablesCan take on any value within a given interval
Example

Discrete and continuous

Decide whether each of the following describes discrete variables or continuous variables.

a

Age of a person at their last birthday

This variable is discrete because a person’s age at their birthday can be listed.
b

Height of a person

This variable is continuous because height can take on any value in an interval.
c

Number of siblings a person has

This variable is discrete because the number of siblings can be listed.
d

Distance a person commutes to work

This variable is continuous because distance can take on any value in an interval.

Types of variables

Variables
Qualitative
OrdinalHave a natural ordering
NominalDo not have a natural ordering
Quantitative
DiscretePossible values can be listed or counted
ContinuousCan take on any value within a given interval
Interactive

Classify a variable

Pick a variable, or draw one at random.
Call it:
A numeral that names rather than counts is still qualitative.
52 variables 1 / 9
You are ready for:

Sections 1.1 & 1.2 HW: Sampling and Types of Data

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