Descriptive statistics, such as average and standard deviation, are important measures in many experiments. They are also of great value when analyzing data from repeated observations that can arise from calibration experiments. But even with the power of descriptive statistics, there are questions that cannot be answered, questions that force us to consider the difference between philosophy and science.
In Table 1 [99] Freedman, Pisani, and Purvis report the results of 100 calibrations of NB 10, a standard 10 gram sample sent to the US in accordance with the Treaty of the Meter of 1875. The recorded observations reflect the number of micrograms under 10 grams, and the individual observations can be retrieved by subtracting these values from 10. The variation in observations is the result of bias and/or chance error: Individual observations = Exact values + Bias + Chance error.
Using only this table of data, what is the best guess at the expected value of the chance error in the measurements? One intuitive response would be to find the average of the recorded observations and change the sign (since all the readings are below 10 grams). Another intuitive response is that the expected chance error should be zero. These responses may both be reasonable; but which is best?
In making a choice, it is important to note the assumptions that come with each option. In the first option, the basic assumption is that the exact value of NB 10 is 10 grams. With this, the observed data does represent error measures, and the average of these would give the best guess at the expected error. More precisely, using the changes of scale [Freedman 92]
| Average (Errors) | = Average (Individual observations - Exact value) |
| = Average (Individual observations) - 10 | |
| = Average (- (Recorded observations)) | |
| = - Average (Recorded observations). |
Note that "Errors" here means "Bias + Chance errors" and there is no way to distinguish these further without knowledge of the exact amount of bias.
In the second option, the working assumption is that we have no way of knowing the exact value of NB 10. Our best guess at such would be the average of the individual observations (which can be obtained from Table 1). With this,
| Average (Errors) | = Average (Individual observations - Exact value) |
| = Average (Individual observations) - Exact value | |
| = Average (Individual observations) - Average (Individual observations) | |
| = 0. |
This option means that we expect no bias to be present (imagine it has been absorbed into the exact value) and "Error" measures only "Chance error." The value "0" signifies that chance errors can add or detract from the exact value, but on average they cancel out. This approach is an accordance with the prevailing paradigm of western science: we can describe only what can be measured.
Although the last sentence suggests a strong preference, shouldn't there be only one solution to the question of expected value of the errors? No -- and Plato (yes, Plato!) presented a good argument for why this duality exists. By imagining observers fixed in place and looking at shadows projected on the wall of a cave, Plato introduces us to the idea that what we "see" is not reality, but only the shadows of things as they really are [Plato 316-320 or sections 515-517]. Are the cave dwellers living in the world of shadows or the world of light? Plato points out that a cave dweller can't tell unless he frees himself from his fixed frame of reference, looks behind, and observes the real forms. This dilemma is manifest to us: do we know the true mass of NB 10 or only the reality we can observe by measurement? Are we philosophers or scientists? Are we looking at the light or fixed within the land of shadows?
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