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Research Methods
Part 4
T- Statistics
Partly based on material by Sherry O’Sullivan
Revision
• General terms
– Population
– Sample
– Parameter
– Statistic
• Measures of central
tendency
– Mean
– Median
– Mode
• Measures of spread
– Range
– Inter-quartile range
– Variance
– Standard deviation
• Population
• Sample
Revision of notation
• Numbers describing a
population are called
parameters
• Notation uses Greek
letters
• Population mean = μ
• Population standard
deviation = σ
• Numbers describing a
sample are called
statistics
• Notation uses ordinary
letters
• Sample mean =
• Sample standard
deviation = s
Revision: Z - Scores
• A specific method for describing a specific
location within a distribution
– Used to determine precise location of an in
individual score within the distribution
– Used to compare relative positions of 2 or more
scores
Revision:
Standard Deviation
• Measures the spread of scores within the data
set
– Population standard deviation is used when you
are only interested in your own data
– Sample standard deviation is used when you want
to generalise from your sample to the rest of the
population
Normal Distribution (Bell shaped)
Normal distribution
• Many data sets follow a Normal distribution
– Defined mathematically by its mean and standard
deviation
2
 x    
1
f (x) 
exp
2

 2
2 

Many statistical tests assume that data follows
the Normal distribution
Strictly, you can’t use these tests unless you can
show that your data follows a Normal distribution
Other possible distributions
• Poisson distribution – for very rare events
– e.g. number of BSCs (blue screen crashes) per
hour of computer use
– Mean is small, often less than 1
– Mode and median often zero
• Binomial distribution
– Very similar to the Normal distribution, but a
discrete distribution (as opposed to a continuous
distribution)
• There are lots of others…
Poisson distribution
(μ = mean = variance)

x
f ( x )  e  / x!
Poisson Distribution, μ = 2; n = 1000
300
250
200
150
100
50
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Distribution of the sample Means
(simple example)
• Frequency Distribution of 4 scores (2, 4, 6, 8)
X
0
1
2
X
3
4
X
5
6
X
7
8
9
• Distribution looks flat and not bell shaped
•(actually not enough data to decide what
the distribution might be)
• Mean of population is (2+4+6+8)/4 = 5
Distribution of the sample means
•Take all possible samples of two scores
•Calculate average for each sample
(2+2)/2 = 2
(2+4)/2 = 3
(2+6)/2 = 4
(2+8)/2 = 5
0
(4+2)/2 = 3
(4+4)/2 = 4
(4+6)/2 = 5
(4+8)/2 = 6
1
X
2
X
X
3
(6+2)/2 = 4
(6+4)/2 = 5
(6+6)/2 = 6
(6+8)/2 = 7
X
X
X
4
X
X
X
X
5
(8+2)/2 = 5
(8+4)/2 = 6
(8+6)/2 = 7
(8+8)/2 = 8
X
X
X
6
X
X
7
X
8
9
Central Limit Theorem
• “For any population with a mean μ and standard deviation σ ,
the distribution of sample means for sample size n will have a
mean of μ and standard deviation of σ/√n and will approach
a normal distribution as n gets very large.”
• How big should the sample size be? n=30
X
0
1
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
2
3
4
5
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8
9
Standard Error
σ/√n is used to calculate the Standard Error of the sample
mean
Sample data = x
The mean of each sample =
Then the standard error becomes
It identifies how much the observed sample mean is likely
to differ from the un-measurable population mean μ.
So to be more confident that our sample mean is a good
measure of the population mean, then the standard error
should be small. One way we can ensure this is to take large
samples (large n).
Example
The population of SATs scores is normal with μ= 500, σ =100.
What is the chance that a sample of n=25 students has a mean
score = 540? Since the distribution is normal, we can use
the z-score.
First calculate the Standard Error:
Then the Z-Score:
= 100/5 = 20
= (540-500)/20 =2
The z-value is 2, therefore around 98% of the sample means
are below this and only 2% are above.
So we conclude that the chance of getting a sample mean of
540 or more is about 2%,
so we are about 98% confident that this sample mean (if
recorded in an experiment) is not due to random variation, but
that the 25 students are (on average) brighter than average.
t - Statistics
• So far we’ve looked at mean and sd of
populations and our calculations have had
parameters
• But how do we deduce something about the
population using our sample?
• We can use the t-Statistic
t - Statistics

Remember SD from last week?
 x   
SS


N
N
2

Great for population of N but not for sample
of n
2
x  x 
SS
s

n 1
n 1
Why n -1?
 Because we can only freely choose n-1 (Degree
of freedom = df)

t - Statistics
Standard Error
 t statistic is z-score redone using the
above:


And for the t-statistic, we substitute σ (SD
of population) with s (SD of sample)
But what about μ ?
 An example…

Hypothesis Testing
• Sample of computer game players n =16
• Intervention = inclusion of rich graphical
elements
• Level has 2 rooms
– Room A = lots of visuals
– Room B = very bland
• Put them in level 60 minutes
• Record how long they spend in B
Results
• Average time spent in B = 39 minutes
• Observed “sum of squares” for the
sample is SS = 540.
A
B
Stage1: Formulation of Hypothesis
• H0: “null hypothesis”, that the visuals
have no effect on the behaviour.
• H1: “alternate hypothesis”, that the
visuals do have an effect on the players’
behaviour.
• If visuals have no effect, how long on
average should they be in room B?
• Null hypothesis is crucial; here we can
infer that μ = 30 and get rid of the
population mean
Stage 2: Locate the critical region
• We use the t-table to help us locate this,
enabling us to reject or accept the null
hypothesis. To get we need:
– Number of degree of freedom (df) 16 -1 =15
– We choose a significance or a level of
confidence: α = 0.05 (95% confidence)
– Locate in t-table (2 tails): critical value of
t=2.131,
Stage 3: Calculate statistics
Calculate sample sd
Sample Standard Error
t-Statistic
=6
= 6 / 4 =1.5
=6
The μ = 30 came from the null hypothesis:
if visuals had no effect, then the player
would spend 30 minutes in both rooms A
and B.
Stage 4: Decision
• Can we reject the H0, that the visuals have no
effect on the behaviour?
t = 6 which is well beyond the value of 2.131 which
indicates where chance kicks in.
• So “yes”, we can safely reject it and say it does
affect behaviour
• Which room do they prefer?
– They spent on average 39 minutes in Room B
which is bland
Another use of t (part 1)
• Our example was a comparison of what was
observed with what was expected
• Our analysis gave a confidence with which the
observations were different from the expected
– Note: cannot be used to confirm similarity…
• Another use of t: comparison of two samples
– e.g. male and female performance on a game, or
opinion of a website….
Another use of t (part 2)
• In this case, we have two sample means, and
we are testing for a difference
Recall: before, we had
This time, it gets messy because the two
standard deviations might not be the same,
and we finish up with
x1  x 2
t
2
2
s1 / n1   s2 / n2 
Yet another complication:
• When looking for a difference, we may have no
reason to suppose that one sample gives higher
values than the other
– We don’t know which mean might be higher
• This is a two-tailed test: we test both tails of the
distribution
• If we have a good reason to suppose that one set of
results has to be higher than the other
– e.g. game scores before and after a practice session
• Then we have a one-tailed test
Fortunately, it can all be done on
a computer…
End