Measurement error—Page 1
Measurement Error
Definitions
.
For two variables, X and Y, the following hold:
Parameter
Explanation
x
i
N
X
X
E
)
(
Expectation, or Mean, of X
V X
E
X
x
x
(
)
[(
) ]
2
2
Variance of X
SD X
V X
x
(
)
(
)
Standard Deviation of X
COV X Y
E
X
Y
x
y
xy
(
, )
[(
)(
)]
Covariance of X and Y
CORR X Y
r
r
xy
x
y
xy
yx
(
,
)
Correlation of X and Y
yx
xy
x
2
Slope coefficient for the Bivariate regression of Y
on X (Y dependent)
Question
:
Suppose X suffers from random measurement error  that is, the values of X that we observe
differ randomly from the true values that we are interested in. For example, we might be interested in
income. Since people do not remember their income exactly, reported income will sometimes be higher
and sometimes be lower than true income. In such a case, how does random measurement error affect the
various statistical measures we are typically interested in? That is, how does unreliability affect our
statistical measures and conclusions?
Revised Question
:
Let us put the question more formally. Let X = X
t
+
, where
is a random error
term (i.e. has mean 0 and variance s²
). That is, X
t
is the “true” value of the variable, and X is the flawed
measure of the variable that is observed. We want to see how the statistics for the observed variable, X,
differ from the statistics for the true variable, X
t
. When thinking about this question, keep in mind that,
because
is a random error term, it is independent from all other variables (except itself), e.g.
COV(X,
) = COV(Y,
) = 0.
Definition of Reliability:
The reliability of a variable is defined as:
REL(X) =
x
x
t
2
2
= r
2
XtX
The first equality says reliability is true variance divided by total variance. The second equality says the
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 Spring '11
 RichardWilliams
 Variance, Richard Williams, Random measurement error

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