week 7

Question 1

Why do we use statistics?

Answer 1

 We use statistics as checks on our own
biases and help us better answer our RQ;
rather than just as math. We will never
have to memorize a formula or do
calculations by hand.
 Moving from t-tests which compares 2x
groups into IVs with multiple levels i.e., 2x
control groups, another experimental
condition etc.

Question 2

Do music lessons make kids smarter?
Causal claim (Mozart Effect)

Answer 2

Possible Studies
1. Correlation between IQ and music
lessons? A correlation can be established
but not temporal precedence or control for
confounds (third variable problem).
2. Longitudinal study showing IQ gains with
music lessons? Temporal precedence and
association can be identified with this
design (third variable problem present).

Recap:
Criteria for causal evidence:
1. An association between variables (i.e.,
correlation)
2. Temporal precedence (cause-effect; music
lessons improve intelligence)
3. Control of alternative explanations (i.e.,
rule out confounds)

*Need Random Assignment to obtain control
and make causal claims.

Question 3

Music lessons enhance IQ

Schellenberg (2004)

Answer 3

 Method:
o Over 36 weeks 4 groups of 6-year-old had
music lessons added to their course work.
Children were either taught; keyboard,
voice, drama or no lessons by qualified
instructors.
 Why use three different treatment groups?
o No lessons is a control group but having
three other experimental conditions helps
us gain a better understanding of what it is
about music lessons which cause this
effect on IQ.
o Comparing drama to music lessons helps
understand if it’s something specific about
music or just being creative.
o Comparing two music groups to see if its
music in general or a specific type of
music.
 Matching:
o He matched the four groups on
extraneous variables on age, family
income (SES), IQ before lessons. This is
essentially a pre-test post-test design
which allows us to compare IQ before and
after the experimental manipulation
(lessons).
 Results:
o Indicate that IQ gains were greater for
music lessons (keyboard and voice)
relative to control and drama lessons. This
illustrates that it is something about music
which increases IQ and not just creative
classes.
o How do we know if these effects are
meaningful? We need to use statistics to
find if these between group differences
are statistically significant and not due to
sampling error.

Question 4

Analysis of Variance (ANOVA)

Answer 4

Analysis of Variance (ANOVA)
 F statistic: between-groups variance (how
groups differ from each other) / Within
group variance (how people differ from
others in their group.
 Comparing effect due to IV to the variance
which naturally occurs in your population.
 If the null hypothesis is not true the sample
distribution for each group should not
overlap, the means should be much
different and give us a big F statistics.
 We want between group variance to be
high and within groups differences (noise)
smaller.

Question 5

How do we calculate variance (s2)

Answer 5

 Null Hypothesis: All kids drawn from the
same population (i.e., 36 participants all
from the same population and no effect of
the IV on groups).
 Variance: calculating each participant’s
distance from the mean. Since some with
be + and some -, they can not just be
added together because it will equal zero.
Instead, we add all the distances from the
mean all together and square root it/ n-1,
so we remove the – symbol. Bigger
number = more spread from the mean and
small variance indicates that they fall close
to the mean.
 Total Sums of Squares (SStotal)

Question 6

SSbetween: Total Sums of Squares Between Groups Variance

Answer 6

SSbetween: Total Sums of Squares Between Groups Variance
 Mean for each group comparing it to the
overall grand mean
 Add the three means together, square it to
see how much each group differs from the
overall mean.
 If they are all different from one another
than the SSbetween will be larger. If
they’re very similar it will be small and no
effect of IV present

Question 7

SSwithin groups: Sums of Squares within

Answer 7

SSwithin groups: Sums of Squares within
 We want within group variance to be small
(noise)
 How do each participant differ from their
group mean?

Question 8

Sums of Squares and Mean Squares

Answer 8

Sums of Squares and Mean Squares
§ In ANOVA we calculate variance using a
technique called sums of squares (SS).
o SStotal = how much each participant varies
from the overall mean (squared)
o SSbetween = how much each group varies
from the overall mean (squared).
o SSwithin = how much each person varies
from their own group mean (squared).
Spread of data within one group.
o SStotal = SSb + SSw
§ Mean Squares (MS) are adjusted for n by
dividing SS by df
o dfbetween = #Groups -1
o dfwithin = #Participants - #Groups
§ MSbetween = SSb/dfb (Mean Square
Between groups = sums of squares
between divided by degrees of freedom
between)
§ MSwithin = SSw/dfw (mean square within
groups = sums of squares within divided by
degrees of freedom within).
§ F = MSb/MSw (F statistic = Mean Squared
Between divided by Mean Square Within
§ WARNING: MSw also called MSresidual or
MSerror

Question 9

Mean Squares

Answer 9

Mean Squares
 The more people we have in each 
   group, the bigger the sums of square will 
   be.
 What we want to know is the average how 
   far are people from the mean (mean 
   square).
 Mean Squares (MS) are adjusted for n by 
   dividing SS by df
o dfbetween = #Groups -1
o dfwithin =     #Participants - #Groups
 MSbetween = SSb/dfb
 MSwithin = SSw/dfw

*Two Degrees of freedom in ANOVA
(between; top and within; bottom).

Question 10

ANOVA F Statistic Calculation:

Answer 10

ANOVA F Statistic Calculation:
§ F = Msbetween/Mswithin
§ We compare the F statistic to the F
sampling distribution which tells us how
often the null hypothesis can produce a F
that big.
§ All the F values are +, the Mean = 0 and the
distribution is all above 0 (one tailed; peaks
just under one).
§ The bigger the F statistic the less likely it is
being produced by the null hypothesis.
§ We want the between group variance to be
bigger than within group variance for the F
statistic to be big.
§ F of 1 tells us that the group variance is not
that much bigger than within (no effect of IV
– null is true).
§ Critical region: if the null hypothesis is true
less than 5% of the time it will produce a F
statistic 3.5 or bigger

Question 11

F Distribution

Answer 11

F Distribution
 F is a sampling distribution of possible F
values if the null hypothesis is true.
 The exact size/shape of F depends on the
degrees of freedom
 If the groups differ from each other a lot
compared to how much people (or
animals) differ from others in their
condition, you get a large F.
 Reject the null if p < .05
 F is always positive
 No difference between one-tailed and
two-tailed F

Question 12

Different df for different F distributions

Answer 12

Different df for different F distributions
§ df(x,y)
o X = number of groups – 1
o Y = total N – number of groups
§ In our example, we have 3 groups of 12 =
36 participants so the DF would be (2,33).