NOTES - 1
Introduction to Statistics
Outline
•Populations and Samples
•Sampling Error
•Measurement and Scales
•Experimental Method
•Order of Operations
Research Question: How many hours of sleep do CBU
students get?
Characteristics of populations and samples
•Variable
• Characteristic or condition that changes or
has different values for different
individuals
Characteristics of populations and samples
•Variable
• Characteristic or condition that changes or
has different values for different
individuals
•Data
• Measurements or observations of a
variable
◦ Single data point commonly called a
score or raw score
Population and Sample
Population
Sample
Population
• Set of all individuals of interest
• Often large (but not always)
Sample
• Set of individuals selected from
a population
• Intended to represent the
population
Population and Sample
All CBU
students
This class
Question: How many hours of sleep
do CBU students get?
Population
• All CBU students
Sample
• Students in this Quant I class
Parameters and Statistics
• Parameter: A value that describes a population.
◦ Derived from measurements of the
individuals in the population
• Statistic: A value that describes a sample
• Derived from measurements of the individuals
in the sample
Descriptive Statistics
•Descriptive statistics can tell us the
characteristics of sample
• What is the average score?
• What is the range of observed values?
(highest and lowest)
• How are the scores spread out?
•Unit 1 (Chapters 2-4) will focus on descriptive
statistics
Inferential Statistics
•Inferential statistics allow us to use
samples and make generalizations to the
population
• Is the observed outcome due to chance?
Sampling Error
•Sample is an approximation of the population
◦ Never exactly the same
•Sampling error: the discrepancy (amount of
error) between and sample statistic and the
corresponding population parameter
• Unavoidable
• Usually random
Sampling Error: Application
Population:
CBU
Students
7 hrs
Group 1
8 hrs
Group 2
8 hrs
How many hours of
sleep do CBU
students get per
night?
Group 3
4 hrs
Sampling Error or an effect?
1. Is the difference due to sampling error (“due
to chance”)?
2. Is the difference not “due to chance”?
◦ Usually explained by “an effect”
Inferential statistics will help us determine
this
Quick Review So Far
What is a value that describes a
population?
What is a value that describes a sample?
Why are values for a sample not equal to
the values of population?
Outline
•Populations and Samples
•Sampling Error
•Measurement and Scales
•Experimental Method
•Order of Operations
Discrete and Continuous Variables
•Discrete Variables
• Have separate categories
• No values exist between two
neighboring categories
• Examples: Brand, number of people,
letter grade in class (A-F)
Discrete and Continuous Variables
•Continuous Variables
• Infinite number of possible values
between any two observed values
• Every interval is divisible into an infinite
number of equal parts
◦ E.g., 5 → 2.5 → 1.25 → .625
• Examples: Height, Weight, Number-lines
Discrete and Continuous Variables
•Discrete Variables
• No values exist between “apple” and
“orange”
• Continuous Variable
• An infinite number of values exist
between 5 and 6
◦ 5.2, 5.08, 5.0001, . . .
Scales of Measurement
•Nominal scale
•Ordinal scale
•Interval scale
•Ratio scale
Nominal scale
• Categories with names
• Does: Label data
• Does NOT: do anything else
• Examples:
• Room numbers
• Colors
• Brand
Ordinal scale
• Categories in an ordered sequence
• Does: Place data in a ranked order
• Does NOT: Tell us anything about
quantitative differences
• Examples:
• Olympic medals
• French fry size (S/M/L)
Interval scale
• Ordered categories with equal intervals
between categories
• Does: Place data in order, with equivalent
distances between the ordered categories
• Does NOT: Have a meaningful zero
◦ A value of “0” does not indicate a true
absence of the variable
• Examples:
• Fahrenheit/Celsius, IQ
Ratio scale
• Ordered categories with equal intervals between
categories AND a “meaningful” or “true” zero
point
• A value of “0” indicates a true absence of that
variable
• Does: Basically an Interval scale where zero
means something
• Does NOT: This is as good as it gets for us
• Examples: Money, Speed, Gas tank
Scales of Measurement Overview
Why do scales matter?
•Certain statistical procedures are only
appropriate for some types of data
• Most of the tests we’ll be working are
only for interval or ratio data
Learning Check
A study assesses the optimal size (number of
other members) for study groups. The variable
“Size of group” is …
Learning Check – Answer
A study assesses the optimal size (number of
other members) for study groups. The variable
“Size of group” is …
Outline
•Populations and Samples
•Sampling Error
•Measurement and Scales
•Experimental Method
•Order of Operations
Three data structures
1.Correlational Method
Relationships between variables/groups
2.Experimental Method
Demonstrate cause and effect
Experimental Method
•Goal of Experimental Method
• To demonstrate a cause-and-effect
relationship between two variables
•One variable is determined by the
experimenter
•The other variable is measured
Independent/Dependent Variables
•Independent Variable is the variable
manipulated by the researcher
•Dependent Variable is observed to assess
the effect
• Dependent because its value is thought
to depend on the value of the
independent variable
Experimental Method
•Experimental condition
• Individuals do receive the experimental
treatment
•Control condition
• Individuals do not receive the experimental
treatment.
• Purpose: to provide a baseline for
comparison with the experimental condition
Experimental Method: Example
•Does drinking coffee cause you to study
more?
Experimental Method: Example
•Does drinking coffee cause you to study
more?
• Change the level of “coffee”
◦ 0 oz
◦ 8 oz
◦ 16 oz
Experimental Method: Example
•Does drinking coffee cause you to study
more?
• Change the level of “coffee”
◦ 0 oz
◦ 8 oz
◦ 16 oz
• Measure number of hours spent
studying
Statistical Notation
Statistical
Notation
Notation
Definition
X
Individual scores for a
particular variable
Y
Individual scores for a
particular variable
N
Number of scores in a
population
n
Number of scores in a
sample
Σ
Summation of a symbol or
equation
Order of Operations
Please Excuse My Dear Silly Aunt Sally
Summation Notation Example
ΣX = ?
◦ “Sum of X”
◦ 1+1+1+1+1+1 = 6
ΣY = ?
◦ “Sum of Y”
◦ 1+2+3+1+2+3 = 12
X
1
1
1
1
1
1
6
Y
1
2
3
1
2
3
12
Why aren’t these equations equal?
a) ΣY2 vs. b) (ΣY)2
◦ a) Square each Y score, add up the
squared scores
◦ b) Add the Y scores, square that
sum
Why aren’t these equations equal?
a) ΣY2 vs. b) (ΣY)2
◦ a) Square each Y
score, add up the
squared scores
◦ b) Add the Y
scores, square
that sum
a) ΣY2
Y
1
2
3
1
2
3
Y2
12
22
32
12
22
32
Y2
1
4
9
1
4
9
28
Why aren’t these equations equal?
ΣY2
(ΣY)2
a)
vs. b)
◦ a) Square each Y
score, add up the
squared scores
◦ b) Add the Y
scores, square
that sum
b) (ΣY)2
Y
1
2
3
1
2
3
12
2
= 144
Review
Research Question: “Does giving cats more treats lead
to less biting?”
◦ What is the population?
◦ What is the Independent variable?
◦ What is the Dependent variable?
◦ What type of variable is this (discrete or
continuous)?
◦ What scale of measurement? (nominal, ordinal,
interval, or ratio)
Review
What is ΣX + 5 ?
What is Σ(X – 2)
X
7
3
9
Review
What is ΣX + 5 ?
Sum of all X, then add 5
7+3+9 = 19
19+5 = 24
What is Σ(X – 2)?
Subtract 2 from each X, then sum
7 - 2 = 5, 3 - 2 = 1, 9 - 2 = 7
5+1+7 = 13
X
7
3
9
Questions?