By Mary Pat Campbell
A group of actuaries were at a conference being held in Hartford. As part of the conference entertainment, they went to the Connecticut Science Center’s dinosaur exhibit, accompanied by a guide.
"These bones", announced the guide, "are 70,000,004 years old."
The actuaries - always impressed by precision in measurement - were astounded.
"How do you know the bones’ age so precisely?" asked one of the group.
"Easy!" came the reply. "When I first came here, they told me it was 70 million years old. I've been working here for four years now."
How often do we make this same mistake in our work, taking cost estimates out to multiple decimal places, when we’ve really only got a good estimate in the millions of dollars?
Let’s look at the basics surrounding significant figures, and how they’re used. What are Significant Digits?
Significant figures and their rules have been developed for ease of use in a variety of fields. Trying to reflect the imprecision in direct measurement of quantities, many of us first learned these in science lab classes. Many have also probably forgotten about their proper use.
The concept originates in that if we can make a measurement of, say, 4.2 centimeters, we’re really saying the length is between 4.15 cm and 4.25 cm. We’ve basically got a confidence interval as our measurement, but we’re marking 4.2 cm for convenience.
Now, many of the estimates we use don’t have an explicit ruler, but we can construct confidence intervals on items such as mortality rates. I will use examples of that sort below.
So how can you tell you’re looking at significant digits (also called significant figures or sig figs)? This is, of course, assuming that what you’re looking at was properly recorded.
- Non-zero digits are significant
- Zeroes between significant digits are significant
- If there are places past the decimal represented, all those digits are represented
Here are some examples of numbers and how many significant digits they have:
|Number||Number of Significant Figures|
The whole reason we’re determining whether digits are significant or not is so that we can appropriately treat them when transforming them with operations. Let’s look at the basic arithmetic operations, as well as some functional applications.
Rules for Arithmetical Operations
Remember, the concept of significant figures is to provide easy rules-of-thumb in how one should combine and communicate amounts. One could also take the implied confidence interval for each number (for example, the number 4,000 implies a range from 3,500 to 4,499) and apply all the possible combinations of these ranges to see how they transform.
However, this gets to be a bit ridiculous. The rules of thumb below are rather standard.
Multiplication and Division
These items are the easiest to deal with, because they come from the concept of scientific notation and appropriately follow rules there.
The rule is this: the result is rounded to the number of significant figures of the item with the least number of sig figs.
|Operation||Number of sig figs for first argument||Number of sig figs for second argument||Number of sig figs for result||“Exact” result||Result to appropriate sig figs|
|4,123 * 0.004||4||1||1||16.492||20|
|4,123 / 0.004||4||1||1||1,030,750||1,000,000|
|4,123 * 10.004||4||5||4||41246.49||41250|
|4,123 / 10.004||4||5||4||412.1351||412.1|
Now, one wants to be careful about application of this rule. Exact figures are ignored for these rules of thumb – these rules apply only to measured or estimated quantities.
For example, if you are scaling something from monthly to annual figures by multiplying by 12, that 12 is considered to have infinite precision.
Something to note here, too: this rule of thumb can make for some lopsided results. Let us consider the first example above. The implied intervals are [4122.5, 4123.5) and [0.0035, 0.0045). If I multiply the endpoints to get the result, I see: [14.42875, 18.55575). The sig fig rule result isn’t even in this interval!
The “exact” result often gives a better midpoint for the interval – the issue is that the information about the width of the confidence interval is lost if you start using more significant figures than the rules allow for. You give an impression of precision beyond what is justified.
However, if you prematurely round, you get biased estimates.
So in general, these rounding rules are not applied at each individual step. What one usually does is a series of operations without rounding (to one’s floating point precision limits, which is a subject for a different article) and then present the final result to the appropriate number of significant digits.
The next item gets to this problem more closely.
Addition and Subtraction
Here, you’re not counting the number of significant digits. What you’re doing is looking at what decimal place the last significant digit occurs.
The rule for addition and subtraction, from Wikipedia:
When adding or subtracting using significant figures rules, results are rounded to the position of the least significant digit in the most uncertain of the numbers being summed (or subtracted). That is, the result is rounded to the last digit that is significant in each of the numbers being summed. Here the position of the significant figures is important, but the quantity of significant figures is irrelevant.
|Operation||What place last sig fig occurs for first argument||Number of sig figs for second argument||Where result should have last sig fig||“Exact” result||Result to appropriate sig figs|
|4,123 - 0.004||ones||thousandths||ones||4122.996||4123|
|4,123 + 4,000||ones||thousands||thousands||8,123||8,000|
|10.004 – 0.004||thousandths||thousandths||thousandths||10||10.000|
|4,123 + 10.004||ones||thousandths||ones||4133.004||4,133|
The first example above shows the issue with the initial (rather old, geeky) joke: the age of the bones had been told to the guide only to one sig fig, and 4 years wasn’t enough to move the needle. If only they had been there for 10 million years….
This is a little more squishy, and what was really developed for use by engineers using base 10 logarithms.
The following rules are again from Wikipedia:
In a logarithm, the numbers to the right of the decimal point are called the mantissa and the number of significant figures must be the same as the number of digits in the mantissa (for example, log(3.000×104) = 4.47712125472, should be rounded to 4.4771).
When taking antilogarithms, the resulting number should have as many significant figures as the mantissa in the logarithm.
What often occurs when applying nonlinear functions is that one uses about the same number of sig figs as the original argument. It doesn’t necessarily work well for all functions, but it often does the job.
These rules on significant figures and their use are codified by ASTM International (formerly American Society for Testing and Materials) in their standard ASTM E29 – 08: Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specification. Alas, one needs to pay about $50 for a copy, and while I’m sure it’s scintillating material, I think I’d rather spend it on (more) lectures on language from John McWhorter.
While the above does detail official rules on recognizing significant figures as well as carrying them forward in operations, we do not need to be doctrinaire in using these concepts in our own work. We just need to be certain we are not misleading our customer if we report our results to more decimal places than are significant.”False precision also feed into the “numbers are maaaaaagic” attitude many of our less quantitative colleagues have, and by including too much we may actually be communicating less. So get yourself a glass of your favorite drink, and cozy up with ASOP 41 (on Actuarial Communications) next to a warm fire while pondering these issues.
Mary Pat Campbell, FSA, MAAA, is life analyst for Conning Research & Consulting, Inc. She can be reached at firstname.lastname@example.org.