1. Sampling distributions:

2. A population of scores has: mean (μ), standard deviation (σ) and
In Psychology we generally make inferences about populations on the basis of samples.
We therefore need to know what relationship exists between samples and populations.
A population of scores has:
mean (μ),
standard deviation (σ) and
shape (e.g., normal distribution).
Does violence on television actually influence behavior of children?
Are women being paid less than men for doing the same work?
Who is going to win the next US presidential election?
Can’t study the whole population!

3. CONCRETE BUT MUNDANE EXAMPLE: HEIGHT OF ALL ADULT WOMEN IN ENGLAND
high
µ = 63 in.
σ = 2 in.
frequency
of raw scores
Can’t measure the height of all adult women in England. σ
low 59 61 63 65 67
Height (inches)

4. frequency of raw scores
Sample of 100 adult women from England
high
s = 2.5 in
N = 100
frequency of raw scores s
low
64.2 in

5. If we take repeated samples, each sample has a mean height ( ), a standard deviation (s), and a shape.
Due to random fluctuations, each sample is different - from other samples and from the parent population.
Fortunately, these differences are predictable - and hence we can still use samples in order to make inferences about their parent populations.

6. Sampling Distribution
Taking multiple samples (of a given size) gives rise to:
Sampling Distribution
Frequency
(how many
means of a
given value)
This is the most complex concept concerning sampling.
63.0
Mean sample heights (inches)

7. What are the properties of the distribution of Sample Means (Sampling Distribution)?
(a) The mean of the sample means is the same as the mean of the population of raw scores:
(b) The sampling distribution of the mean also has a standard deviation. The standard deviation of the sample means is called the "standard error". It is NOT the same as the standard deviation of the population (of raw scores).

8. The standard error is smaller than the population SD:
Standard Error = SD when sample size is 1.
The bigger the sample size, the smaller the standard error.
i.e., variation between samples decreases as sample size increases.
(This is because sample mean based on a large sample reduces the influence of any extreme raw scores in a sample.)

9. Suppose we take samples of N = 100
Sampling distribution for N = 100
µ = 63 in.
N = 100
frequency
Staying with the height example. 63
Mean sample heights (inches)

10. Suppose the N = 16 instead of 100
How do we calculate the standard deviation of a sampling distribution of the mean (known as the standard error)? For N =100
Suppose the N = 16 instead of 100

11. (c) -The distribution of sample means is normally distributed if the population of scores is normally distributed.
- The distribution of sample means (for N ≥ 30) is normally distributed no matter what the shape of the original distribution of raw scores is!

12. Example: Annual income of Americans.
Many people in the lower and medium income bracket; very few are ultra rich. (So distribution is NOT normal.)
Suppose we take many samples of size N = 50.
The sampling distribution of the mean will be normal.
This is due to the Central Limit Theorem.

13. Given the distribution is normal, we use properties of normal distribution to do interesting things.
Various proportions of scores fall within certain limits of the mean (i.e. 68% fall within the range of the mean +/- 1 standard deviation; 95% within +/- 2 standard deviation, etc.).

14. Quick reminder about z-scores
With a z-score, we can represent a given score in terms of how different it is from the mean of a distribution of scores.
Recall z-scores from last term
This is again the height example: mean = 63 inches; SD = 2
Shown here: This is NOT a sampling distribution (This IS population distribution).
μ = 63
Xi = 64
Calculating z-score

15. Relationship of a sample mean to the population mean:
μ = 63 64
We can do the same with sample means:
(a) we obtain a particular sample mean;
(b) we can represent this in terms of how different it is from the mean of its parent population.
This IS a sampling distribution.

16. If we obtain a sample mean that is much higher or lower than the population mean, there are two possible reasons:
(a) our sample mean is a rare "fluke" (a quirk of sampling variation);
(b) our sample has not come from the population we thought it did, but from some other, different, population.
The greater the difference between the sample and population means, the more plausible (b) becomes.

17. Take another example:
The human population IQ mean is 100.
A random sample of people has a mean IQ of 170.
There are two explanations:
(a) the sample is a fluke: by chance our random sample contained a large number of highly intelligent people.
(b) the sample does not come from the population we thought it did: our sample was actually from a different population - e.g., aliens masquerading as humans. Or, more likely, it was taken from the Mensa society members.

18. Relationship between population mean and sample mean:
high
frequency of sample means
low
population
mean IQ = 100
sample
mean IQ = 170

19. A common experimental design
This logic can be extended to the difference between two samples from the same population:
A common experimental design
We compare two groups of people:
- One group get the "wolfman" drug (Experimental group).
- The second group get a placebo (Control group).
At the start of the experiment, they are two samples from the same population ("humans").

20. At the end of the experiment, are they:
(a) still two samples from the same population (i.e., still two samples of "humans" – i.e. our experimental treatment has left them unchanged). OR
(b) now samples from two different populations - one from the "population of humans" and one from the "population of wolfmen"?

21. We can decide between these alternatives as follows:
The differences between any two sample means from the same population are normally distributed, around a mean difference of zero.
Most differences will be relatively small, since the Central Limit Theorem tells us that most samples will have similar means to the population mean (and hence similar means to each other).
If we obtain a very large difference between our sample means, it could have occurred by chance, but this is very unlikely - it is more likely that the two samples come from different populations.

22. Possible differences between two sample means:
a big difference between mean of sample A and mean of sample B:
high
frequency of raw scores
low
mean of sample A
mean of sample B
low
high
sample means

23. Possible differences between two sample means (cont.):
a small difference between mean of sample A and mean of sample B:
high
frequency of raw scores
low
mean of sample A
mean of sample B
low
high
sample means

24. Frequency distribution of differences between sample means:
high
frequency of differences between sample means
low
mean A smaller than mean B
mean A bigger than mean B
no difference
size of difference between sample means
( "sample mean A" minus "sample mean B")

25. Frequency distribution of differences between sample means:
A small difference between sample means (quite likely to occur by chance)
high
frequency of differences between sample means
low
mean A bigger than mean B
mean A smaller than mean B
no difference
size of difference between sample means

26. Frequency distribution of differences between sample means:
high
A large difference between sample means (unlikely to occur by chance)
frequency of differences between sample means
low
mean A bigger than mean B
mean A smaller than mean B
no difference
size of difference between sample means