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Sampling Error
think of sampling as benefiting from room for movement
Sampling Error
think of sampling as benefiting from room for movement
Population
µ
Sample
_
X
_
The population mean is µ. The sample mean is X.
Population
µ

Sample
_
X
s
_
The population mean is µ. The sample mean is X.
The population standard deviation is , the sample sd is s.
Sample
_ C
XC
Sample
_ D
XD
Population
Sample
_ B
µ
Sample
_ E
XE
XB
Sample
_ A
XA
In reality, the sample mean is just one of many possible sample
means drawn from the population, and is rarely equal to µ.
Sample
_ C
XC sc
Sample
_ D
XD sd
Population
Sample
_ B
µ 
Sample
_ E
XE se
XB sb
Sample
_ A
XA sa
In reality, the sample mean is just one of many possible sample
means drawn from the population, and is rarely equal to µ.
Sampling error = Statistic - Parameter
_
Sampling error for the mean = X - µ
Sampling error for the standard deviation = s - 
Unbiased and Biased Estimates
An unbiased estimate is one for which the mean sampling
error is 0. An unbiased statistic tends to be neither larger
nor smaller, on the average, than the parameter it estimates.
_
The mean X is an unbiased estimate of µ.
The estimates for the variance s2 and standard deviation s
have denominators of N-1 (rather than N) in order to be
unbiased.
2
SS
=
N
s2
SS
=
(N - 1)
s
SS
=
(N - 1)
Correlations describe co-variation
on two variables.
What’s the correlation?
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60
80
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60
80
80
60
RDG
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20
80
60
MATH
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80
60
SCI
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20
80
60
CIV
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20
2
0
CONCPT
-2
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60
80
20
40
60
80
-2
0
2
What’s the correlation?
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60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
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80
60
SCI
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20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
.69
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
.68
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
.60
40
CIV
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
.13
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
.65
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
.51
40
CIV
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
.16
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
.51
40
CIV
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
.15
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
What’s the correlation?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
.06
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
Rounded, the mean reading score is 52 and sd=10 and the mean math score
is also 52 and sd=10. If a given student’s reading score is 67, then what is
your best estimate of her math score?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
.69
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
In predicting someone’s math score, if you could have just one piece of
information, and it is either (a) her reading score, or (b) her self concept
score, which would you rather have? What’s it worth to you?
40
60
80
20
40
60
80
80
60
RDG
40
20
80
.69
60
.16
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
Explain to a non-statistician what it means to say “reading and math scores
are correlated r=.69 in this population”.
40
60
80
20
40
60
80
80
60
RDG
40
20
80
.69
60
MATH
40
80
60
SCI
40
20
80
60
CIV
40
20
2
0
CONCPT
-2
20
40
60
80
20
40
60
80
-2
0
2
A key to understanding r
zy = zx
When X and Y are perfectly correlated
We can say that zx perfectly predicts zy
zy’ = zx
Or
^
zy = zx
When they are imperfectly correlated, i.e.,
rxy ≠ 1 or -1
zy’ = rxyzx
r is the slope of the predicted line, with a zero-intercept of z’y=0
Explain to a non-statistician what it means to say “reading and
math scores are correlated r=.69 in this population”.
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