CLIFFORD KONOLD AND ALEXANDER POLLATSEK
One aspect of this situation that makes using a mean of the observations
particularly compelling is that, conceptually, we can separate the signal from the
noise. Because we regard an object as having some unknown but precise weight, it
is not a conceptual leap to associate the mean of several weighings with this actual
weight, while attributing the trial-by-trial variations to a distinctly different thing:
chance error produced by inaccuracies of the measurement instrument and by the
process of reading values from it. Indeed, we can also regard each individual
weighing as having two components—a fixed component determined by the actual
weight of the nugget and a variable component attributable to the imperfect
measurement process.
The relative clarity of this example hinges on our perception that the weight of
the nugget is a real property of the nugget. A few philosophers might regard it
(possibly along with the nugget itself) as a convenient fiction. But to most of us, the
weight is something real that the mean weight is approximating closely and that
individual weighings are approximating somewhat less closely. Another reason that
the idea of central tendency is compelling in repeated measurement situations is that
we can easily relate the mean to the individual observations as well. To help clarify
why this is so, we will make some of our assumptions explicit.
We have been assuming that the person doing the weighing is careful and that
the scale is unbiased and reasonably accurate. Given these assumptions, we expect
that the variability of the weighings would be small and that the frequency
histogram of observations would be single-peaked and approximately symmetric. If
instead we knew that the person had placed the nugget on different parts of the
balance pan, read the dial from different angles, or made errors in transcribing the
observations, we would be reluctant to treat the mean of these numbers as a central
tendency of the process. We would also be hesitant to accept the mean as a central
tendency if the standard deviation was extremely large or if the histogram of weights
was bimodal. In the ideal case, most observations would be close to the mean or
median and the distribution would peak at the average, a fact that would be more
apparent with a larger data set because the histogram would be smoother. In this
case, we could easily interpret the sample average as a good approximation to a
signal or a central tendency and view the variability around it as the result of random
error.
These assumptions about the procedure and the resulting data may be critical to
accepting the mean of the weighings as a central tendency, but they are not the only
things making that interpretation compelling. As indicated earlier, we maintain that
the key reason the mean observation in this example is relatively easy to accept as a
central tendency is that we can view it as representing a property of the object while
viewing the variability as a property of a distinctly independent measurement
process. That interpretation is much harder to hold when—rather than repeatedly
measuring an attribute of a single object—we measure an attribute of many different
objects, taking one measurement for each object and averaging the measurements.
CLIFFORD KONOLD AND ALEXANDER POLLATSEK
One aspect of this situation that makes using a mean of the observations
particularly compelling is that, conceptually, we can separate the signal from the
noise. Because we regard an object as having some unknown but precise weight, it
is not a conceptual leap to associate the mean of several weighings with this actual
weight, while attributing the trial-by-trial variations to a distinctly different thing:
chance error produced by inaccuracies of the measurement instrument and by the
process of reading values from it. Indeed, we can also regard each individual
weighing as having two components—a fixed component determined by the actual
weight of the nugget and a variable component attributable to the imperfect
measurement process.
The relative clarity of this example hinges on our perception that the weight of
the nugget is a real property of the nugget. A few philosophers might regard it
(possibly along with the nugget itself) as a convenient fiction. But to most of us, the
weight is something real that the mean weight is approximating closely and that
individual weighings are approximating somewhat less closely. Another reason that
the idea of central tendency is compelling in repeated measurement situations is that
we can easily relate the mean to the individual observations as well. To help clarify
why this is so, we will make some of our assumptions explicit.
We have been assuming that the person doing the weighing is careful and that
the scale is unbiased and reasonably accurate. Given these assumptions, we expect
that the variability of the weighings would be small and that the frequency
histogram of observations would be single-peaked and approximately symmetric. If
instead we knew that the person had placed the nugget on different parts of the
balance pan, read the dial from different angles, or made errors in transcribing the
observations, we would be reluctant to treat the mean of these numbers as a central
tendency of the process. We would also be hesitant to accept the mean as a central
tendency if the standard deviation was extremely large or if the histogram of weights
was bimodal. In the ideal case, most observations would be close to the mean or
median and the distribution would peak at the average, a fact that would be more
apparent with a larger data set because the histogram would be smoother. In this
case, we could easily interpret the sample average as a good approximation to a
signal or a central tendency and view the variability around it as the result of random
error.
These assumptions about the procedure and the resulting data may be critical to
accepting the mean of the weighings as a central tendency, but they are not the only
things making that interpretation compelling. As indicated earlier, we maintain that
the key reason the mean observation in this example is relatively easy to accept as a
central tendency is that we can view it as representing a property of the object while
viewing the variability as a property of a distinctly independent measurement
process. That interpretation is much harder to hold when—rather than repeatedly
measuring an attribute of a single object—we measure an attribute of many different
objects, taking one measurement for each object and averaging the measurements.
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