But in many (all?) cases, the individual data point itself can be considered to have been drawn from a random distribution. For example, suppose I measure a voltage as 8.00 V using a specific voltmeter. The manufacturer tells me that the accuracy of the meter is 1 digit +/- 0.5%. In effect, the manufacturer is telling me that if I measured the SAME voltage with a set of their instruments, then the readings of those meters would not all be the same and they should range within 0.05 V of the "correct voltage". That single point is a single instance of a value drawn from a random distribution. As such, the error bars are telling us about the distribution of values for that specific point, whereas the parameters of the curve itself relate to a different random distribution.
Or if I estimate the number of cars traveling on a street by counting the cars for 5 minutes each minute, the number of cars counted is exact, but the estimate of the hourly count has a random distribution of its own. If I try to draw a fit thru the 168 points collected in a week, the parameters of that fit would have their OWN distribution and uncertainty.
From: Phys-l [firstname.lastname@example.org] on behalf of John Denker [email@example.com]
Sent: Sunday, October 21, 2012 2:37 PM
Subject: [Phys-L] points don't have error bars (distributions do)
I've been thinking more about the idea that there's no such
thing as a random number. You can have a random /distribution/
over numbers, in which case the randomness in the distribution,
not in any number drawn from the distribution.
This has important implications for how we think about data,
including raw data that we take in the lab. The individual
points don't have error bars. The error bars pertain to the
/distribution/ from which the points were drawn.
Tangential remark: This can be seen as reason #437 why sig
figs are a bad idea, i.e. unhelpful in practical terms, wrong
in conceptual terms, and just plain bad in every way. In this
case, sig figs force you to attribute error bars to every data
point you write down, even though that's conceptually wrong.
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