Review
• 49% of all people in the world are male. Interested
whether physics majors are more likely to be male than
the general population, you survey 10 of your friends in
physics. Here are the probabilities from a binomial
distribution with n = 10 and q = .49. How many of your
friends need to be male to support your hypothesis?
0
1
2
3
4
5
6
7
8
9
10
.001
.011
.494
.127
.213
.246
.197
.108
.041
.008
.001
Sampling Distributions
9/29
Same Thought Experiment
•
•
•
•
•
Known population
Sample n members
Compute some statistic
What is probability distribution of the statistic?
Replication
– Doing exactly the same experiment but with a new sample
– Sampling variability means each replication will result in different
value of statistic
Sampling Distributions
• Sampling distribution
– The probability distribution of some statistic over
repeated replication of an experiment
– Distribution of sample means: the probability
distribution for M
Population
Sample(s)
Sampling
Distribution
X = [ -.70
1.10
-.36
-.68
-.08], M = -.144
X = [ .09
-.88
1.16
-1.72
.40], M = -.019
X = [ -.99
.47
.65
1.52
.20], M = .370
X = [ -.84 -2.06
1.06
-.24
2.49], M = .082
.38
.16
.85], M = .768
X = [1.88
.57
Reliability of the Sample Mean
• How close is M to m?
– Tells how much we can rely on M as estimator of m
• Standard Error (SE or sM)
– Typical distance from M to m
– Standard deviation of p(M)
• Estimating m from M
– If we know M, we can assume m is within about 2 SEs
– If m were further, we probably wouldn’t have gotten this value of M
• SE determines reliability
– Low SE  high reliability; high SE  low reliability
– Depends on sample size and variability of individual scores
• Law of Large Numbers
2s
2s
– The larger the sample, the closer MMwill be toM m
– Formally: as n goesmto infinity, SE goes
m Mto 0 m
– Implication: more data means more reliability
Central Limit Theorem
• Characterizes distribution of sample mean
– Deep mathematical result
– Works for any population distribution
• Three properties of p(M)
– Mean: The mean of p(M) always equals m
– Standard error: The standard deviation of p(M), sM, equals
s2
• Variance equals
n
s
n
– Shape: As n gets large, p(M) approaches a Normal distribution
• All in one equation:
(
p(M ) » Normal m,
• Only Normal if n is large enough!
– Rule of thumb: Normal if n  30
s
n
)
Distribution of Sample Variances
• Same story for s as for M
– Probability distribution for s over repeated replication
• Chi-square distribution (c2)
– Probability distribution for sample variance
– Positive skew; variance sensitive to outliers
– Mean equals s2, because s2 is unbiased
2
(
X

M
)
• Recall: s 2  
n 1
2
(
X

M
)

• Distribution of
?
n