07 May 2004

Tales of a Home Lab: How 'Good' Is Your Measurement?

James Firmiss, MAST

I finally cleared up a corner of the room near my kitchen and have turned it into the beginnings of a lab bench.  I've got a lab space, a place for notebooks and reference books, and access to electrical outlets. So far there is no equipment aside from some rulers and measuring cups -- something I plan to remedy.

Uncertainties

One thing every scientist needs to be aware of is that all measurements have some degree of uncertainty.  This is unavoidable.  Uncertainties crop up due to a number of factors but can mostly be categorized into two groups: instrumental uncertainty and statistical uncertainty.

Instrumental uncertainty results from physical limitations of the measuring instrument.  How sensitive is the reading?  Is the instrument properly calibrated?  Is the instrument being read correctly?

Statistical uncertainty results from natural variations in the item being measured.

For the most part I'll be talking about instrumental uncertainties.

Accuracy and Precision

With care, instrumental uncertainties can be minimized, but never fully eliminated.  Therefore, a scientist should have some knowledge as to how 'good' a measurement can be.  But how do you quantify such uncertainty?

A full discussion of this would be far too long for this column. So let's focus on two key terms, accuracy and precision, that describe how "good" measurements are.  People often carelessly use these terms interchangeably, but their meanings are quite different.

Accuracy is the degree to which a measured value represents the true value.

Precision is the degree to which multiple measurements of the same value agree.

You can have one without the other.  For instance, let's say you had a rock that you know weighs 12.203 kg.  If you weighed this rock on a digital bathroom scale, you might end up with a value of 12 kilograms.  While this measurement is accurate, the low resolution of the scale provides only a rounded or truncated value that prevents you from obtaining a more precise measurement.

Now lets say you have a higher resolution scale, such as one used at a deli counter.  You measure the rock several times and get 12.62 kg, 12.61 kg, 12.62 kg, 12.63 kg, and 12.61 kg.  These values are far more precise, with each measurement varying only by about 0.01 kg. But, if you're certain the rock weighs 12.203 kg, then this scale is not very accurate.

In summary, precision relates to the smallest distinction you can make between two measured values and accuracy relates to how closely those values represent the actual value of the property being measured.

Calibrating a Home Balance

I cobbled together a balance for my lab desk out of things around the house.

Last week I mentioned that a U.S. nickel coin has a mass of 5.000g according to a U.S. Mint fact sheet. I planned to use this fact to measure the mass of one square inch of graph paper.  With such a measurement I could, in turn, use cut out sections of graph paper to weigh out specific small quantities, such as 150 milligrams of a reagent.

My paper mass set uses "10 divisions per inch" graph paper.  I cut out several 4 squares, a convenient size for my balance. I also cut 1 , 0.5 , and 0.1 areas. This is easy to do with graph paper, since 1 squares with 0.01 subdivisions have already been marked.

But how accurate and precise is such a seemingly simple setup?

My first goal was to measure the precision of this balance.  I put one nickel on one mass pan and added paper squares to the other pan until a balance was achieved.  I then removed the weights, re-zeroed the balance, and repeated the procedure four more times with the same nickel. Here are the results:

Trial        Area (1)
1             102.7
2             103.1
3             102.8
4             102.9
5             103.2

1. Area of graph paper section ( )

The average paper area needed to balance the nickel is 102.94 . The maximum deviation from this average is 0.26 (trial 5).  I could have tried cutting the paper into its smallest squares of 0.01 , but if measurements of the same thing in my setup can vary up to 0.26 there is no point in trying to measure differences much smaller than this.

I can now say that the instrumental uncertainty of this balance is plus or minus the mass of 0.3 of graph paper. (I rounded up the 0.26 number to play it safe.)  Assuming the nickel has a mass of 5.000 g means 1 square inch of it would be,

Multiply this by our instrument uncertainty to represent this error in grams.  The result is 0.0147 grams, or about 15 milligrams.

Now that I have an idea of the balance's precision, how accurate are these results?

Unfortunately, I have no really good way of measuring the accuracy of this balance without a well-established standard mass.  A nickel is a good estimate of 5 grams, but there's no guarantee that a particular nickel weighs exactly 5.000 g.  I tried more nickels and obtained the following results:

Nickel    Area (1)
1            102.9
2            104.8
3            104.4
4            105.4
5            102.6
6            105.2

1. Area of graph paper section ( )

The measured masses of the different nickels varied quite a bit.  I chose nickels in reasonably good condition, with few scratches and little visible wear.  Nevertheless, the heaviest required 2.8 more graph paper than the lightest -- far more than the measured precision of the balance. This is a good indication that this variation is real and not merely instrumental uncertainty.  This is also a good example of a statistical uncertainty.

The range of these values is 102.6 to 105.4 with 104.2 being the average.

Using these extremes we can calculate possible values of g/ of paper as

to

Another way we can represent this range is by a single mid-range value with a plus or minus error, as in 0.04805 0.0065 g/ (note if we add or subtract this error of 0.0065 from the value 0.04805 we'd get the two extreme values shown previously).

Lastly, if we convert this value into masses, based on an average of 104.2 needed to balance out 5.000 g, we'd be able to express our measurement as 5.00 ± 0.68 g.

Think about what happens if we were to use this as our measuring standard.  If we measure out what we'd think should be 5.00 g, there would be an error of 0.68 g in our measurement.  If we try to measure out 10.00 grams, the error would be twice as much.  If we only measured 2.5 grams, the error would be halved.  The error scales with sample size.  A better way to represent it would be a fractional error.

In our case it would be 0.68 g /5.00 g * 100% = 13.6%

We can round this to 14%.  This would be placed on any measurement.  For example, a measurement of 2.0 g on this balance would carry this 14% error (2.0 g * 14% = error of about ±0.3 g).

The overall error of this balance should add the 15 mg instrumental error.But that value would probably be negligible compared to the 14% statistical error for all but the smallest measurements.

Conclusion

Despite a promising precision of 15 milligrams, without a better mass standard to calibrate this balance, its overall accuracy is quite low. It probably shouldn't be used for anything more than rudimentary measurements.

I am probably better off buying a calibrated weight set (www.Ebay.com has these for about $10 and up), or even a digital scale (about $25 and up).

There is far more about the topic of error analysis than there is space to cover here.  Look up "error analysis" in the introductory chapters of chemistry and physics textbooks or on a web search engine.

Created by Mathematica  (April 16, 2004).