Sampling Distributions Suppose I throw a dice

10000 times and count the number of times each face turns up:

Frequency of Single Dice Throws

1200

1600

2000

1 2 3 4 5 6

Dice Face

Fre

qu

ency

Frequency

Each score has a similar frequency (uniform distribution)

Sampling Distributions

If instead you throw the dice 10 times (or throw ten dice) and take the average score each time, you get something like this:

Average of 10 Throws

0

500

1000

1500

2000

2500

3000

3500

1 2 3 4 5 6

Average of 10 Throws

Fre

qu

ency

Frequency

++ + +

+

++ + + }

{

10

Sampling Distributions

Average of 10 Throws

0

500

1000

1500

2000

2500

3000

3500

Average of 10 Throws

Fre

qu

ency

Frequency

Average of 20 Throws

0500

10001500200025003000350040004500

1 2 3 4 5 6

Average of 20

Fre

qu

en

cy

Frequency

Compare averaging 10 vs 20 throws each go:

Frequency of Single Dice Throws

1200

1600

2000

1 2 3 4 5 6

Dice Face

Fre

qu

ency

Frequency

Average of 10 Throws

0

500

1000

1500

2000

2500

3000

3500

1 2 3 4 5 6

Average of 10 Throws

Fre

qu

ency

Frequency

Average of 20 Throws

0500

10001500200025003000350040004500

Average of 20

Fre

qu

ency

Frequency

10 x 20 x

Note what happens to the spread and shape of the distribution of average scores

1 x

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 Average of N ThrowsLight Bulb Life Average Life

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:

Population distribution of Individual Heights

5’5”

4’0” 7’0”

(Population SD)approx = 7”

Population distribution of raw scores

68% of scores lie within 1 standard deviationOf the mean

5’5”

4’0” 7’0”

68% of people have a height between 4’10” and 6’0”

6’0”4’10”

Suppose we take a random sample of 10 people and measure their heights:

5’5”

4’0” 7’0”

The mean of the sample (x ) will tend to be quite close to the average height:

x

Keep taking samples of 10 people and measure average height:

5’5”

4’0” 7’0”

x

Back to17

Keep taking samples of 50 people and measure average height:

5’5”

4’0” 7’0”

x

Back to17

Distribution of Sample Means x cluster around the population mean more closely than the raw scores do:

5’5”

4’0” 7’0”

The degree of spread (standard deviation of the sample means) around the population mean depends on the number (n) in each sample.

5’5”

4’0” 7’0”

n=10n=20n=30

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 is equal to the population standard deviation

divided by √n and is called the standard error of the mean (se)

5’5”

4’0” 7’0”

Raw scores SD= 7”

Samples of size 10SD of the sample means = 7/sqrt(10)= 7/3.16 = 2.2

7”

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:

5’5”

4’0” 7’0”

Parent population of individual scores

Expected variation of samples of size 10

Not:

5’5”

4’0” 7’0”

Expected variation of samples of size 10

Parent population of individual scores

It is more likely that the real situation is that the two samples come from different parent populations:

5’5”

4’0” 7’0”

6’5”

So an experiment selects 8 babies at random and feeds half Marmite and half Bovril. Heights measured at 20 years.

Vs.

6’5”

It is more likely that the real situation is that the two samples come from different parent populations:

5’5”

4’0” 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 and2. How they should vary given their size and the individual

variationIf these two estimates differ widely then H0 is rejected.