Download Measures of Variation

Building
Statistical
Models
Lesson 4
Theories & Models
Theories
 Describe, explain, & predict real-world
events/objects
 Models
 Replicas of real-world events/objects
 Can test predictions ~

Models & Fit
Model not exact replica
 Smaller, simulated
 Sample
 Model of population
 Introduces error
 Fit
 How well does model
represent population?
 Good fit  more useful ~

Models in Psychology
My research model
 Domestic chicks
 Effects of pre-/postnatal drug use
 Addiction & its consequences
 Who/What do most psychologists study?
 Rats, pigeons, intro. psych. students
 External validity
 Good fit with real-world populations? ~

The General Linear Model

Relationship b/n predictor & outcome
variables form straight line
 Correlation, regression, analysis of
variance
 Other more complex models ~
The Mean as a Statistical Model

Very simple model
 1 number represents all the
observations
 Often hypothetical value
e.g., mean # friends = 2.6
Error introduced
 Actual # friends = mean + error
 Deviation (deviance)

~
X 

i
Assessing the Fit of the Mean

How well does it represent all
observations?
 On average near or far from mean?
Distance from mean

Or width of distribution
Mean Daily Temperature

For which group is the mean
a better fit for the data?
10 20 30 40 50 60 70 80 90

10 20 30 40 50 60 70 80 90
Measures of Variability
Deviation: for a single score
 Range
 Highest value – lowest value + 1
 Standard deviation
 Conceptually: mean of all deviation
scores
 average distance of scores from mean
 Variance
 Used to calculate standard deviation
 Also used in analysis of variance ~

Calculating the Standard Deviation
Why only conceptually
mean of deviation
scores?
 If
Xi




What is mean
deviation?
 S(Xi – ) = 0 ~

Xi
1
2
3
4
5
Xi -
Variability: Notation & Formulas
3 steps to standard deviation
 Sums of squares (squared deviations)
2
 SS = S(Xi – )
 Variance = mean of squared deviations (MS)


 
2
2
(
X


)
 i
N
 square root of variance = standard deviation ~
Standard Deviation (SD)

(X
i
 )
2
N
Conceptually mean deviation score for
all data
 Gives width (dispersion) of distribution
 Describing a distribution
 Report mean & standard deviation

, 
~

Samples & Variability

Usually study samples



to learn about populations
Sampling introduces error
Change symbols & formula
SS   ( X  X ) 2
s 
2
2
(
X

X
)

N 1
s
2
(
X

X
)

N 1
Samples: Degrees of Freedom (df)
df = N – 1
 For a single sample (or group)
 s tends to underestimate 
 Fewer Xi used to calculate
 Dividing by N-1 boosts value of s
 Also used for
 Confidence intervals for sample means
 Critical values in hypothesis testing ~

Degrees of Freedom: Extra
Don’t lose any sleep over this
 df theory
 If n= 4 & sample mean = 10
th can be only
 3 of Xi can be any value, 4
one value
 See Jane Superbrain 2.2 (pg 37) ~

Level Of Measurement & Variability
Which can be used?
 nominal
 none
 ordinal
 range only
 interval/ratio
 all 3 OK
 range, standard deviation, &
variance ~

Statistical Models
Representation of the population
 We will focus on linear models
 Mean is a simple model
 One number represents all data
 Both  and
X
 Standard deviation
 measures fit of model
 Better fit  more useful
 Smaller  and s ~
