Post on 08-Mar-2018
Basic Statistical Concepts, Research
Design, & Notation
01:830:200:10-13 Spring 2013
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
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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.
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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)
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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.
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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.
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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.
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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.
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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.
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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.
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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.
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
Example Data from a Correlational Study
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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
Basic Statistical Concepts
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
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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)
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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
01:830:200:10-13 Spring 2013
Basic Statistical Concepts
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