🚀 go-pugleaf

On Fri, 12 May 1995, Steve Luzader wrote:

> 2. The formula for calculating %error is
>         (measured value - accepted value)/ accepted value    x100%
>    They had to calculate the total torque on a mobile (made of wooden blocks
>    and fishing line). The total torque on each block should be zero for
>    it to be in equilibrium, BUT how do you calculate the %error when the
>    accepted value is zero?

Aarghhh! One of my pet peeves, and responses which drive me up the wall.

So many elementary books, particular high school and freshman college lab
manual, perpetuate this notion that the 'goodness' of your experimental
result is judged by comparing it to some 'book value' or 'accepted
value'. The formula you give above is completely out of line with
accepted practice in the sciences. If you submitted a paper to the
physical review in which you used that, your paper would probably be
rejected with extreme prejudice. So *why* do we encourage students in
practices which are so misleading and useless? This is one of the 'bad
habits' we college physics teachers must encourage our students to 'unlearn'.

The `goodness' of a result, expressed as a +/- *uncertainty* is found
through careful analysis of the experimental data and procedure, and
sometimes by statistical analysis involving repeated measurements and
expressing their amount of scatter. In the process, an algebraic
'propagation of error' analysis is done (this is called 'error
analysis'). The experimental uncertainty of a result can (and should) be
determined EVEN IF THERE IS NO 'ACCEPTED VALUE' to compare with! As I say
to my students: "An answer is no good unless you know HOW good it is."

An acceptable synonym for 'experimental uncertainty' is 'experimental error'.

When there *is* an authoritative value to compare with, it's appropriate to
compare your result with that, and that difference is called an
'experimental discrepancy'. In a well designed experiment, the
discrepancy should be smaller than the maximum uncertainty, and on the
order of the probable uncertainty, or the uncertainty expressed as a
standard deviation. If it isn't smaller, one suspects an unnoticed
determinate error, or perhaps a 'blunder'.

Reference: D. C. Baird. _Experimentation, An Introduction to Measurement
Theory and Experimental Design_, Prentice Hall (I don't have the date of
the second edition at hand, but it is around 1990.) I'll dig up some
other references and post them later. I'm not near my library.

*Sometimes* (not always) it's appropriate to express an error or a
discrepancy, as a percent. (I despise the inappropriate use of
'percentage'. 'Percentage' is used when one is not expressing it
quantitatively, 'percent' is used for quantitative measures.")

An uncertainty in a temperature reading in Celsius of Fahrenheit should
NOT be expressed as a percent. If the uncertainty was 1 degree in a
measurement of 100 degrees, you'd calculate 1%. If the uncertainty were
the same amount, 1 degree, in a reading of 10 degrees, you'd get 10%.
Think how much 'better' the result would *seem* if you used degrees Kelvin!
Ridiculous, isn't it. One must examine the nature of the thing being
measured to determined which is meaningful, absolute discrepancy or
percent discrepancy. That means one must engage the brain and think about it.
One must also examine the nature of the measuring instrument, and ask
which is constant over the range of the instrument: Absolute error?
Percent error? Niether?

Percent uncertainty is *never* appropriate for a measurable whose
measurement scale is chosen arbitrarily, such as Celsius and Fahrenheit
temperature. Percent uncertainty might (in some cases) be meaningful if
the temperature was expressed in absolute units (degrees Kelvin). Absolute
measure, +/- form is *always* appropriate and meaningful.

Where *do* students get that silly notion that it's *always* 'better' to
express something as a percent? Where do they get the notion that results
are 'better' if you take data with a computer? Why do they think that
errors expressed as 'standard deviations' are better than 'average
deviations' or even limits of error? If sample size is, say, 10 or less,
standard deviations *aren't even appropriate* as measure or error, even
with the Bessel small-sample correction, for you have no idea whether the
distribution is gaussian. Many distributions which come up in elementary
physics aren't gaussian. Note that if the distribution of x is gaussian,
the distribution of 1/x is not, nor is the distribution of x^2, etc.
Fortunately, if the errors are small, one is saved by the mathematically
'robust' nature of the 'gaussian hypothesis'. It's another example of
students who learn rules and procedures and apply them blindly and
thoughtlessly.

Other examples where percent uncertainty is INappropriate are: refractive
index, dielectric constant, specific gravity.

One experiment we have used to get across ideas related to scatter of
repeated measurements, is a measurement of resistors with an ohmmeter.
Years ago we acquired (surplus) a huge bag of resistors all marked 15K
with a stripe indicating limits of error of 10%. We divided them into
bags of 10, and had students measure them with ohmmeter, NEVER MEASURING
THE SAME RESISTOR TWICE. Then they took new boxes and measured series
combinations of two. Then parallel combinations of two. They then were
asked whether the parallel and series results were consistent with the
algebraic rules for error propagation.

One interesting thing. The resistors in the bag (the parent distribution)
were all mis-marked. The student measurements gave an average value of
14.4K, well outside the error limits marked on the resistors, and well
outside of the average deviation the students calculated. We wanted to
see if they'd have the courage to say that *their* value was good, the
*authoritative* value (marked on the resistors) was *wrong*. Want to
guess the responses?

Most students didn't notice the discrepancy and didn't comment on it.
Some thought they'd blundered in the calculation.
A very few volunteered that the resistors were mis-marked.

Be very, very concerned!

        -- Donald