Analysis of Variance
one-way ANOVA
Comparing more than two samples
ANOVA allows us to compare more than two
samples at a time.
Even though we could always run t tests for
every two samples we want to compare, the
possibility of type I error rises very q
The Standard Error of the Mean
The random sampling
distribution of the mean
is a probability
distribution. That is, a
distribution that shows
us the values that
sample means can take
and the probability that
they will occur.
Probability of occurrence
Ran
Two-way ANOVA
Interactions and Nested Designs
Assumptions for a Two-Way ANOVA
Normality: data come from a population that is
normally distributed.
Random sampling: data were sampled randomly.
Independence: the data were sampled
independent of each othe
Repeated Measures ANOVA
Assumptions
The populations are normally distributed: as
sample size increases, ANOVA can deal with
slight skewing of the data. Large sample sizes
are best.
Assumptions
The populations are normally distributed
The variances are
Hypothesis Testing
Statistical Inference
In statistical inference we are interested in
drawing conclusions about a population based
on a sample.
This is because it is either impractical or
impossible to obtain the populations
parameters.
Statistical Inf
z scores
Fear of dogs
Dogs are our preferred pets
because of their sociality.
But, of course, they are also
carnivores and can defend
themselves.
Fear of dogs
Dogs are our preferred pets
because of their sociality.
But, of course, they are also
carnivores
More on Correlation
Correlation IS NOT causation
The fact that two variables are related does
not mean that one of them necessarily causes
the other.
That is, a relationship only means that the
variables have something in common.
We know this because c
Independent Samples t-Test
Part 2
Look at your data
The first thing you want to do when
conducting an analysis is to look at your data.
Ballistic Movement Data
30
25
20
Inaccuracy Scores (count)
15
slow
fast
10
5
0
1
2
3
4
5
Participant
6
7
8
9
Total par
Chi Square
Chi Square
While the other tests we looked at during the
course involved the use of interval and ratio
scales, 2 (ki square) only requires the frequency
of occurrence of a set of events.
2 can be used to calculate the rarity of categorical
va
Graphing Frequency
Distributions
Graphing Distributions
Y-axis or ordinate
The Graph
3
2
1
Origin
0
0
1
2
X-axis or abscissa
3
Tick marks
The Histogram
Start at zero or make sure your reader knows you have
manipulated the axes.
The Histogram
Rule 1: A ver
z scores 2
Obtaining scores from areas
This is the reverse problem from finding a
percentage limited by a score under the normal
curve.
Here we want to know which score (or scores)
is the limit for a given percentage in the
population.
Single limit
Fin
Central Tendency
Shapes of different distributions
What is the distribution
of the rolls of a single
die?
Shapes of different distributions
What is the distribution
of the rolls of a single
die?
1
6
Shapes of different distributions
What is the distributi
Quantitative Methods
PSYC 120
Luis F. Schettino
Statistics
Why do we need statistics?
The natural world is full of patterns, but many of
those patterns are not easy to detect.
Science uses mathematics to bring those patterns
to the fore.
Statistics al
Central Tendency and Variability
The Normal Curve
Mean
Mode
Median
In a perfect bell-curve, the mean, mode and
median are all at the same central location.
The Mean
The mean is the best descriptor of data in a
symmetrical bell-shaped distribution.
Since
Power
Significance
When we set out to test a hypothesis using hypothesis
testing methods, we essentially carry out a bet.
Because we are comparing two and only two
hypotheses, the results can only favor one of them.
That is, you either retain or reject
Variables, scales, distributions
Scales of Measurement
Scales of measurement describe the
properties of numbers.
They tells us how informative a number is.
Scales of Measurement
Scales of measurement describe the
properties of numbers.
They tells us h
Directional Tests, Effect Size
Directional Tests
We can test hypotheses that claim a direction
to our data.
Instead of using the = sign, we use > or <.
Directional tests allow us to move the region
of rejection to one of the tails of the sampling
distr
Confidence Intervals
A t test
The chancellor oat City
University is worried that
the grades have fallen in
the past year. The grade
point average for
graduating students in the
previous 5 years was 2.8.
The chancellor randomly
samples 10 seniors from
th
Cumulative Distributions and
Percentiles
Cumulative Frequency Distribution
Sometimes we want to know a bit more about
our data by looking at how they are
distributed.
One way to learn more is to see how many
cases occur at consecutively higher intervals
Correlation
Correlation and Prediction
Correlation refers to the level in which two
variables are related.
If two variables are strongly correlated, when
we know a value for one of them we have a
better chance at predicting the value in the
other.
The
Two-way ANOVA
Factorial Designs
Factorial Designs
While a one-way ANOVA can tell us
something about the differences between
different levels of a single variable, factorial
designs can do that for more than one variable
plus provide information regarding
Chi Square 2
Independence between variables
Chi square can be used to
determine whether two
variables are somehow related
(having one affects the other,
similar to a correlation).
The null hypothesis is that they
are not related and so, the
presence of
Prediction
Questions
According to the regression
equation for a particular
problem in prediction, when X
= 81, Y= 317; and when X=92,
Y = 317.
What is the value of r? How
can you tell?
What is the value of MY ?
Can you infer the value of
MX ?
Explain.
The Standard Deviation
The Standard Deviation
The standard deviation is the square root of the
variance. This simplifies both the calculations
and the conceptualization of variability.
Smaller standard deviations mean thinner
distributions and larger me
Probability and the Normal
Distribution
Probability
The probability of an event taking place can be
expressed as percentages (e. g. candidate x has
a 60% probability to win the election) or as
odds (e.g. a person has a 5-in-1 chance of
winning a prize).
One Sample t Test
From z to t
While we can run a hypothesis testing
calculation using z values, it always depends on
knowing the populations standard deviation.
Unfortunately, we rarely (if ever) know that
value. So we cant really know the SEM.
The bes
ANOVA 2
Recap
ANOVA allows us to test for significance
between more than two groups.
The F ratio is the quotient between the betweengroups variance and the error (within-groups)
variance.
An F value larger than the critical F value
indicates a signific