Transcript sample means - the Peninsula MRCPsych Course

Sampling Distributions
Suppose I throw a dice
10000 times and count
the number of times
each face turns up:
Frequency of Single Dice Throws
2000
Frequency

1600
Frequency
1200
1
2
3
4
Dice Face
Each score has a similar frequency (uniform distribution)
5
6
Sampling Distributions
Average of 10 Throws
If instead you throw the
dice 10 times (or throw
ten dice) and take the
average score each
time, you get something
like this:
3500
3000
Frequency

2500
2000
Frequency
1500
1000
500
6
5
4
3
{
2
1
0
Average of 10 Throws
+
+
+
+
+
+
+
+
+
}
10
Sampling Distributions
Compare averaging 10 vs 20 throws each go:
Average of 20 Throws
Average of 10 Throws
500
6
5
4
3
2
1
0
Average of 10 Throws
Average of 20
6
1000
5
1500
Frequency
4
Frequency
3
2000
Frequency
2500
1
Frequency
3000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
2
3500
Frequency of Single Dice Throws
1x
Frequency
2000
1600
Frequency
20 x
10 x
1200
1
2
3
4
5
6
Dice Face
Average of 10 Throws
Average of 20 Throws
3500
2500
2000
Frequency
1500
1000
500
6
5
4
3
2
1
0
Average of 10 Throws
Frequency
Frequency
3000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
Frequency
1
2
3
4
5
6
Average of 20
Note what happens to the spread and shape of the distribution of average scores
1.Central Limit Theorem

This is a theorem of statistics and probability
that implies that the distribution of a sum (or
average) of any set of scores approaches a
Normal Distribution as the number of scores
involved in the sum (or average) gets larger
and larger.
Single
Throws
Light Bulb
Life
AverageLife
of N Throws
Average
2. Relation between the variation between
individual scores and the variation between
the averages of several scores.

If the individual scores (values) in a
population have a Variance of X then the
variance of the averages of samples of size n
has a variance of X/10.

This is intuitive – think of individual heights:
s (Population SD)
approx = 7”
s
4’0”
m
7’0”
5’5”
Population distribution of Individual Heights
68% of scores lie within
1 standard deviation
Of the mean
4’0”
4’10”
s
5’5”
68% of people have a
height between 4’10”
and 6’0”
6’0”
7’0”
m
Population distribution of raw scores
The
Suppose
mean of
wethe
take
sample
a random
(x ) will
sample
tend to
of be
10 quite
people
close
and
tomeasure
the average
theirheight:
heights:
x
4’0”
m
5’5”
7’0”
Keep taking samples of 10 people and measure
average height:
x
4’0”
m
5’5”
7’0”
Back to17
Keep taking samples of 50 people and measure
average height:
x
4’0”
m
5’5”
7’0”
Back to17
Distribution of Sample Means x cluster around the
population mean m more closely than the raw scores
do:
4’0”
m
5’5”
7’0”
The degree of spread (standard deviation of the
sample means) around the population mean
depends on the number (n) in each sample.
n=10
n=20
n=30
4’0”
m
5’5”
7’0”
Variance and SD


As we observed before the Variance of
sample means is the variance of the
population of individual scores divided by the
sample size.
Because the Standard Deviation is the
square root of the Variance, the Standard
Deviation of the sample means is equal to the
Standard Deviation of the individual scores
divided by the square root of the sample size.
The amount of variation (standard deviation of
the sample means) around the population
mean depends on the number (n) in each
sample.
The standard deviation of sample
means of size n around the
population mean m is equal to the
population standard deviation
divided by √n and is called the
standard error of the mean (se)
Raw scores SD= 7”
7”
4’0”
m
7’0”
5’5”
2.2”
Samples of size 10
SD of the sample means
= 7/sqrt(10)
= 7/3.16 = 2.2
Quick Summary

We get an idea of the amount of variation in
the population of individual scores from the
variation within our sample (i.e. the data).

Given that our sample average is from x
number of scores we know how the sample
averages would be expected to vary from one
sample to the next.
T-Test


The T-Test works by assuming the data
collected in two conditions is equivalent to
collecting two samples from the same
‘parent’ population (this is the null hypothesis).
The variation within the data is a good
estimate of the variation in the parent
population. This, together with the size of the
samples, allows one to predict how much
variation to expect in the means of one
sample to the next.
E.g.
T Test


If the two sample means obtained in the
experiment conditions vary by more than
we’d expect from this simple relation between
the variation of individual scores and sample
averages then it is unlikely that the data in
the two conditions is equivalent to two
samples from the same parent population.
It is more likely they reflect two samples from
different parent populations (i.e. one’s with
different means)
I.e. if the data does reflect samples from the same
population we expect our samples, say of size 10, to
cluster around the population mean quite closely:
Parent population of
individual scores
Expected variation of
samples of size 10
4’0”
m
5’5”
7’0”
Not:
Parent population of
individual scores
Expected variation of
samples of size 10
4’0”
m
5’5”
7’0”
It is more likely that the real situation is that the two
samples come from different parent populations:
4’0”
m1
5’5”
m2
6’5”
7’0”
So an experiment selects 8 babies at random and feeds half
Marmite and half Bovril. Heights measured at 20 years.
Vs.
It is more likely that the real situation is that the two
samples come from different parent populations:
4’0”
m1
m2
5’5”
6’5”
7’0”
T-Test & ANOVA



The T-Test works by computing the likelihood of
getting a certain difference between two sample
means.
If you have experiments with more than 2 conditions
there is no single distance between two means.
Instead you can examine the ‘average’ distance or
variation between them. The Variance of those
condition means is just such a measure.
ANOVA works out how likely it is to get the
observed amount of variation (Variance) between
several sample means if they really had been drawn
from the same parent population.
In a nutshell



The data from the conditions of an experiment can be
conceptualised as samples from a parent population.
The null hypothesis assumes that these samples have been
drawn from a single population.
If the variation (or just difference in the case of a T-Test) between
the means of these ‘samples’ is greater than we would expect
given the samples size used, then we conclude that it is unlikely
that they can be thought of as having been drawn from a single
population but instead come from separate ones (i.e. ones that
have different means).


Some minor details:
The T-test actually works out the sampling distribution of
the difference between two means. When the probability of
getting the observed difference is less than 5% H0 is
rejected – i.e. the two populations from which the means
were drawn are assumed not to be equal.
ANOVA works out:
1.
How the sample means vary and
2.
How they should vary given their size and the individual
variation
If these two estimates differ widely then H0 is rejected.
