Precision and accuracy

[Kurt Roeckx]

On Fri, Jan 20, 2017 at 12:44:38PM -0800, Gary E. Miller wrote:
> Imagine in front of you are two handheld voltmeters, and a super
> precision voltage source.

So let me correct all those terms to what people normally use them
for.

You have a voltmeter those shows volt with 3 digits behind the
decimal place, so for example 1.234. The resolution is 1 mV.
(That's millivolt) It does not say anything about the precision or
the accuracy of that meter.

Suppose you have a reference voltage that has a voltage of
exactly 1.000000000 V. And you measure that voltage with your
voltmeter, and it shows you 0.995, 0.994, 0.993, 0.994, 0.992,
0.994, 0.995, and so on. Let's assume the average value is 0.994
V, that it's a normal distribution and that the standard deviation
is 1 mV. People could say that that 1 mV is the precision, and 6
mV is the accuracy.

Suppose you take a different voltmeter, and that one always shows
1.002 V. People might claim the precision is 0, but a better
number would be halve the resolution, so 0.5 mV. An even better
number could be 0.43 mV. That's sqrt(3) / 2 * halve the resolution. I has
at least a rectangular distribution, and I think that's the correct
formule to match the coverage of 1 standard deviation.

But the 2nd example already shows why the first's precision is
actually wrong, and they both need to get added together, and the
first would be 1.1 mV (sqrt(1^2+0.43^2)).

It's also possible that that 2nd meter is averaging over a longer
period of time than that first one, and in that way improve it's
precision.

And if you really start to care about putting numbers of what the
accuracy and precision is, you get into troubles. Which is why
they're never talked about like that. You don't start from a
measurement and then come up with what the precision and accuracy
is. Because you're probably forgetting all kinds of errors like
the drift. But you can ussually find values for some of those in
the datasheets of your device, and they're ussually not even clear
what they really mean.

If you go and look at the calibration sheet of your reference
voltage, or one of those 8 digit volt meters, you're not going to
find a mention of the accuracy and precision. Instead they're
going to talk about the uncertainty, and say your reference voltage
gives a voltage of 1.000000001 +/- 3 nV, where the 3 nV is an
expanded uncertainty, maybe that it's a normal distribution,
and that it has a k factor of 2 to have a 95% coverage.

For your voltmeter they're going to measure multiple values and
each time give you the reference voltage they, the uncertainty
of the measurment, and the value your device is showing, and if
that's within specification or not.

> Now plug in the Fluke meter.  I feed in the same 1.000000000 Volts
> and I read on the meter 1.000 Volts.  How accurate, NIST traceable,
> would you say the Fluke meter is?  You guess 0.1%?  Maybe, maybe not.
> 
> Now I set the calibrator to 1.000490000 and the meter still reads
> 1.000.  I set the meter to 1.00050000 and the meter changes to 
> read 1.001.  How accurate is the meter?  I say the meter is accurate
> to 0.001%

That's some very nice fluke meter you got there. The best I could
find claims an accuracy of 0.05% (I assume of reading, but could
also be of range) + 1 digit, which would be at least 1.5 mV. All
the others are worse with most doing 0.5% + 2 digits.

> The Fluke meter can in fact reliably, repeatably, NIST traceably,
> differentiate between two inputs that are only 0.001% apart!  When that
> Fluke meter said a voltage was 1.000, you know it was between 0.99951
> Volts and 1.00049 Volts.  You could in fact use it to accurately,
> repeatedly, NIST traceably tune a voltage divider to 1.000 Volts
> +/-0.001%

That error would still be 0.5 mV, or 0.05%.

> Compare that to the cheap meter that is only 5%.  And yet they both have
> a precision of 0.1%.

That cheap meter probably also does 0.5%. (And they both have the
same resolution, not precision.)


Kurt