Basic Statistical Concepts, Research Design, & Notationmmm431/quant_methods_S13/QM_Lecture1.pdf ·...

Basic Statistical Concepts, Research

Design, & Notation

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Basic Statistical Concepts

Variables, Scores, & Data

• A variable is a characteristic or condition that can change or

take on different values.

– Most research begins with a general question about the relationship

between two variables for a specific group of individuals.

– Example from book: time spent playing video games & time spent

exercising.

• A score is the value of a variable measured for a particular

individual

• Data are collections of scores measured for multiple individuals

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Populations

• A population is the set of all individuals or events of interest

in a particular study.

• Populations:

– Are generally very large

– Can consist of arbitrary categories of people, objects, and events

– Can include hypothetical or counterfactual events

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Samples

• It is usually impractical for a researcher to examine every

individual in the population

• Instead researchers typically select a small

representative group—a sample—from the population

and limit their studies to individuals in the sample

• The goal is to use the results obtained from the sample

to help answer questions about the population.

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Descriptive Statistics

• Descriptive statistics are methods for organizing and summarizing data. – Tables and graphs organize data

– Descriptive values (averages, frequencies, proportions) summarize data.

• A descriptive value for a population is called a parameter and a descriptive value for a sample is called a statistic. – Parameters are generally represented as Greek letters

(e.g., µ,σ), while statistics are represented as Roman letters (e.g.,M,s)

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Inferential Statistics

• Inferential statistics are methods for using sample data

to make general conclusions (inferences) about

populations.

• A sample typically contains only a small part of the whole

population.

• As a result, sample statistics are generally imperfect

representatives of the corresponding population parameters.

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Sampling Error

• The discrepancy between a sample statistic and its population

parameter is called sampling error.

• Sampling error depends critically on

1. The amount of variability in the population (e.g., number of legs on a

cow versus volume of milk produced)

2. The number of individuals in the sample

• Defining and measuring sampling error is a large part of

inferential statistics. We’ll look more closely at sampling error

in later lectures.

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Measuring Variables

• To establish relationships between variables, researchers

must observe the variables and record their observations.

This requires that the variables be measured.

• The process of measuring a variable requires a set of

categories called a scale of measurement and a process

that classifies each individual into one category.

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Four Types of Measurement Scales

1. A nominal scale is an unordered set of categories identified

only by name. Nominal measurements only permit you to

determine whether two individuals are the same or different.

2. An ordinal scale is an ordered set of categories. Ordinal

measurements tell you the direction of difference between

two individuals, but contain no information about the

magnitude of the difference between neighboring categories.

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Four Types of Measurement Scales

3. An interval scale is an ordered series of equal-sized

categories. Interval measurements identify the direction and

magnitude of a difference. However, the zero point is

located arbitrarily on an interval scale.

4. A ratio scale is an interval scale where a value of zero

indicates none of the variable. Ratio measurements identify

the direction and magnitude of differences and allow ratio

comparisons of measurements.

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Types of Variables

• Variables can be classified as discrete or continuous.

• Discrete variables (such as class size) consist of indivisible

categories

• Continuous variables (such as time or weight) are infinitely

divisible into whatever units a researcher may choose. For

example, time can be measured to the nearest minute,

second, half-second, etc.

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Types of Data

• Another useful distinction is that of qualitative versus

quantitative data

• Qualitative/Categorical data occur when we assign

objects/events into labeled (i.e., nominal or ordinal) groups,

representing only frequencies of occurrence

– E.g., race, gender, yes/no response

• Quantitative/Measurement data occur when we obtain some

number that describes the quantitative trait of interest.

– These numbers can be either discrete or continuous

– E.g., height, weight, income

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Examples of Variables and Their Classifications

Variables

Continuous vs.

Discrete

Qualitative vs.

Quantitative

Scale of

Measurement

Gender (male, female) Discrete Qualitative Nominal

Seasons (spring, summer, fall, winter) Discrete Qualitative Nominal

Number of dreams recalled Discrete Quantitative Ratio

Number of errors Discrete Quantitative Ratio

Duration of drug abuse (in years) Continuous Quantitative Ratio

Ranking of favorite foods Discrete Quantitative Ordinal

Ratings of satisfaction (1 to 7) Discrete Quantitative Interval or Ordinal

Body type (slim, average, heavy) Discrete Qualitative Nominal

Score on a multiple-choice exam Discrete Quantitative Ratio

Number of students in your class Discrete Quantitative Ratio

Temperature (degrees Fahrenheit) Continuous Quantitative Interval

Time (in seconds) to memorize a list Continuous Quantitative Ratio

The size of a reward (in grams) Continuous Quantitative Ratio

Position standing in line Discrete Quantitative Ordinal

Political Affiliation (Republican, Democrat) Discrete Qualitative Nominal

Type of distraction (auditory, visual) Discrete Qualitative Nominal

A letter grade (A, B, C, D, F) Discrete Qualitative Ordinal

Weight (in pounds) of a newborn infant Continuous Quantitative Ratio

A college students' SAT score Discrete Quantitative Interval

Number of lever presses per minute Discrete Quantitative Ratio

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Basic Research Designs

• Correlational Studies

• Experimental Studies

• Quasi-Experimental Studies

• Different research designs

– produce different forms of data

– answer different types of questions

– require different statistical techniques

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Correlational Studies

• The goal of a correlational study is to determine whether

there is a relationship between two variables and to describe

the relationship.

• A correlational study simply observes the two variables as

they exist naturally.

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Example Data from a Correlational Study

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Experiments

• The goal of an experiment is to demonstrate a cause-and-effect

relationship between two (or more) variables

– I.e., to show that changing the value of one variable causes changes to

occur in a second variable.

• In a simple experiment:

– One variable (the independent variable) is manipulated to create treatment

conditions.

– A second variable (the dependent variable) is observed and measured to

obtain scores for a group of individuals in each of the treatment conditions.

• The critical elements of an experiment are:

– Manipulation of an independent variable

– Control of all extraneous variables (e.g., using random assignment)

– Measurement and comparison of dependent variable across conditions

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Example Data from an Experiment

Variable 1 (independent):

Distraction Condition

Variable 2 (dependent):

Exam Score

Low

Distraction

High

Distraction

92 78

77 80

75 82

82 64

84 67

93 85

96 75

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Quasi-Experimental Studies

• Quasi-experimental studies are correlational studies that

look similar to experiments because they also compare

groups of scores. However:

– These studies do not use a manipulated variable to differentiate the

groups.

– The variable that differentiates the groups is usually a pre-existing

participant variable (such as male/female) or a time variable (such as

before/after).

– Because these studies do not use the manipulation and control of true

experiments, they cannot demonstrate cause and effect relationships.

01:830:200:10-13 Spring 2013


Example Data from a Quasi-Experiment

Variable 1 (quasi-independent):

Gender

Variable 2 (dependent):

Number of tasks completed

Male Female

9 10

8 7

9 8

7 9

5 11

6 9

6 11

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Random Sampling & Assignment

• Random sampling occurs when individuals are selected

such that each member of the population has an equal

chance of inclusion

– Failure to sample randomly may result in statistics that don’t reflect the

whole population

– E.g., average height computed for a sample consisting only of women is

unlikely to reflect the average height of all adults

• Random assignment occurs when individuals are assigned

to different groups using a random process

– Failure to assign randomly confounds the independent variable; any

measured difference in a dependent variable could be due solely to the

assignment

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Statistical Notation

• The individual measurements or scores obtained for a research participant will be identified by the letter x (or x and y if there are multiple scores for each individual).

• The number of scores in a data set will be identified by N.

• Summing a set of values is a common operation in statistics and has its own notation. The Greek letter sigma, Σ, is used to mean "the sum of."

– For example, (or simply ΣX) identifies the sum of the N scores.

1

i

N

i

x

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Order of Operations

PEMDAS (Please excuse my dear Aunt Sally) 1. All calculations within parentheses are done first.

2. Squaring or raising to other exponents is done second.

3. Multiplying, and dividing are done third, and should be completed

in order from left to right.

4. Summation with the Σ notation is done next.

5. Any additional adding and subtracting is done last and should be completed in order from left to right.

Note: in the interest of consistency, always report results to two decimal places beyond the precision of the original data (including in intermediate calculations)

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Useful Summation Identities

•

•

•

i i i

N N N

i

j

i j

x y x y

i i

N N

i i

Cx C x

N N

i i

i i

x C x NC

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Example Problems

• Given the following values for x and y:

– x ={11,14,10,13,12}

– y ={3,2,2,5,1}

• Compute the following:

– Σ2x

– Σ(x-1)

– Σy

– Σy2

– (Σy)2

– (Σ(x –y))2

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