Download Descriptive Statistics: Variability

Descriptive Statistics:
Variability
Lesson 5
Theories & Statistical Models
Theories
 Describe, explain, & predict realworld 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?
 Amount of error in model
 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)

~
Xi  

Distributions: 3 useful features

Summarizes important
characteristics of data

1. What is shape of the
distribution?


2. Where is middle of distribution?

3. How wide is distribution?
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
 Variability
 How much do scores vary from the
mean? ~

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 ~

From the Dictionary
Deviation: departure from a standard or
norm.
 Variance: the state, quality, or fact of
being variable, divergent, different, or
anomalous.
 Error: a deviation from accuracy or
correctness
 Variability: something that may or does
vary; a variable feature or factor
 Variation: something that may or does
vary; a variable feature or factor ~

Calculating the Standard Deviation
Why only conceptually mean of
deviation scores?
Xi
 If
Xi




What is mean deviation?
 S(Xi – ) = 0 ~

1
2
3
4
5
Xi -
4 Steps to Standard Deviation




1. Calculate deviation scores
2. Sums of squared deviations
 Or Sums of squares (SS)
3. Variance
 mean of squared deviations
(MS)
4. Standard deviation
 square root of variance ~
Xi  
SS   ( X i   )2

2

(X


i
 )
2
N
(X
i
 )
N
2
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 s
 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 ~

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 ~
