Significant donuts: error, precision and significant digits

On my significant digits activity in class last week I asked the following question

“If you are doing an experiment with a balance that only reads to the tenths place, is it better to use samples with masses of approximately 5 g or 500 g?  Explain and provide examples. “

My answer to this (the one that, you know, I think is straightforward) is that 500.0 g contains four significant digits and 5.0 g contains two significant digits.  Because more significant digits implies more precision, 500.0 g is the better mass to use in the experiment.

In the past no one has ever really pushed me on this, but four or five of my students have this year, and good for them.  They could have just memorized “my answer” but instead they want to understand.  Which is awesome.

Here’s one way to think about it.  Let’s say you buy two packages of donuts.  The first is supposed to contain 10 and cost $1 (we’re talking little donuts here).  The second box contains 100 and costs $10.  You buy your boxes and take them home, but the first box only contains 9 donuts and the second box only contains 99 donuts.

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Which one are you more upset about?  Most people (unless you over think it) will say the 9 donuts bothers you more because in 10 donuts you really miss the one donut.  In 100 donuts you don’t miss it as much.

In both cases you were cheated out of 10 cents worth of donut, but in the first box this is 10% of your investment, in the second box this is 1% of your investment.  In other words, it’s the percentage difference between the expected and the actual number of donuts that matters.

Now back to masses.  If a scale reads 5.0 g, it really means 5.0 ± 0.1 g.  The difference between the expected mass (5.0 g) and the real mass (which could be between 5.05 or 4.95) is 0.1/5.0 = 2%.  For the 500.0 g mass it’s 0.1/500.0 = 0.02%.  As the mass of the object increases, the uncertainly in our measurements decreases as a percentage of the mass.

All of this gets at what we really mean when we say “significant digits”.  The more significant digits a value has, the more precisely it is known relative to the error that comes from the instrument used to obtain it.  If you use a beaker to measure 10 mL of water you have about a 20% error, so we would call that 10 mL.  You might have 12 or your might have 8 mL.  If you use a pipet, you get a ± 0.02 mL error, so we record that 10.00 mL.

Here’s the point: The rules of significant digits give us the freedom to not have to think about the exact error in every measurement. Why? Because we know that all of the error fits inside the last digit in our value and we leave it at that.

Thoughts?  Discussion?  Please comment below.