# Significant Figures

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This tyrannosaurus is 70,000,003 years old. How many significant figures should I quote?

Big Ideas:
• Numbers mean very different things in different circumstances. Context is all important.

When an attendant at a natural history museum was asked the age of the Tyrannosaurus in the collection, he replied "Seventy million and three years old". When queried about the extraordinary accuracy of his answer, he said: "Well it was seventy milllion years old when I started working here, and that was three years ago"1

"How many significant figures should I use?"

Does any aspect of the presentation of data cause more angst and confusion than the thorny question of significant figures? Why do those of us who read work from physics students at all levels, still find things like:

P = 348.95746542 W

or

y = 0.6249 ± 0.1 m       ?

Leaving aside the possibility of simple carelessness, here are some real issues underlying this problem:

(a) Numbers mean different things to different people.

Much of how we view numbers is formed by (a) math teachers and (b) accountants.

Mathematicians have a profound reverence for numbers; physicists take a much more pragmatic view. Mathematicians get all steamed up about the irrationality of ; physicists don't really care. In 30-odd years as a scientist I have never needed to know to better than four significant figures, because I have never measured anything of consequence to four significant figures; few people have (unless they work in a standards lab).

Accountants have a profound reverence for numbers too. That's because oddities in the human psyche cause us to worry more about being short-changed a few cents than if our pension plan loses $1,000 a day in a stock-market tumble. But as a physicist, if you read for example that the energy content of a particular type of coal is, say, 28.57 MJ/kg, you will realize that the most significant number here is 6 - the implicit 106 in "MJ". The next most significant number is "3". (b) The Illusion of Accuracy and Competence. If you give lots of supposedly significant figures, people will think you really know what you are doing. This approach is taken by the media, governments, and boosters and spin-doctors of all kinds. The estimated cost of a 100 million dollar project is given to the nearest dollar (9 significant figures) and then the project overruns its initial budget by a factor of 3 or 5 (i.e. even the first significant figure was wrong). One the other hand, giving numbers to one significant figure is no guarantee that even that one can be relied upon. BP's original estimate of the flow of oil into the Gulf of Mexico during the great spill of 2010 was 1,000 barrels a day. Later estimates ranged as high as 100,000 barrels a day2. (c) Not all numbers are created equal. If a botanist tells me that a certain plant grows to a height of 1 m, I naturally expect the plant grow to something between, say, 0.5 and 1.5 m. That is, the accuracy of "1 m" is ± 50%. If, however I'm told the plant will grow to 9 m in height, I will assume a final height of something between 8.5 and 9.5 m, an accuracy of ± 4.5% (I'll leave you to decide whether to round that up to 5%). Thus "9 m" is implicitly much more accurate than "1 m", because if the botanist had meant 8 m or 10 m, she wouldn't have said "9 m". Thus the expressions "1 m" and "9 m" both have the same number of significant figures, but very different implied accuracies. Note: if we used binary notation for numbers, we wouldn't have this problem, but let's not go there. (d) Is "0" a significant figure? Context is everything. If I enquire from a friend about the cost of, say, a night in a certain hotel, and I get the answer "$100", I assume I'm not going to get away with much change out of $150. If on the other hand I am checking out of the said hotel and the clerk says "That will be a hundred dollars even, Sir", I know the cost is$100.00. In one case only one digit (and not the zeros) is significant; in the other case, all five digits are significant, zeros included.

Similarly, if I read in a scientific paper that a certain piece of manufactured apparatus was 1 m long, I would assume that it was made to be 100 cm long, probably not 99 or 101 cm, and certainly not 90 or 110 cm. That is, my assumption of accuracy is much greater than the one digit given would imply.

The issue of the significance of "0" also arises in the quoting of uncertainties. If a distance is given as "(600 ± 100) km", then its clear the zeros are not significant. You should not lose marks for giving an uncertainty to three significant figures, especially because writing the result this way is much comprehensible than "(6 ± 1) x 105 m". See my page on Scientific Notation3

(e) There are no rules.

Or worse, there are conflicting rules. Students complain to me, legitimately, that they get different and mutually incompatible rules about significant figures from different people in biology, chemistry and physics labs, and sometimes from two different TAs in the same lab. Much worse, they can lose marks for doing what one TA wants, and then having their lab marked by the other guy. This pushes the students into a vortex of angst that detracts from what we are really trying to teach them (unless you are trying to teach them to follow arbitrary rules).

My "rules"

If students push me for a hard rule, here's one that works almost all the time: two or three significant figures. If you quote any more or less you need a very good reason. Points to watch:

- look at the last "significant" figure, the one on the right. Does it really mean anything?

- use one significant figure for really vaguely known quantities, or quantities that are naturally variable (e.g. the heat of combustion of coal4).

- don't round off before the end of the calculation; you'll run the risk of throwing the baby out with the bathwater.

- on the other hand, don't bother writing down all the significant figures for intermediate steps; it looks amateurish. Spreadsheets5 are useful here; you can hide most of the digits used in the calculation and not show them anywhere. "Right click on cells -> Format Cells -> Number" or use action on the toolbar.

- use more than three if you are considering small differences between large numbers (like accountants do).

- however many significant figures you use, the least significant digit of the number MUST be the same as the least significant digit of the error (unlike the bad example given at the start of this article). "y = 0.6249 ± 0.1 m" is bad (what's the point of the "249" if the "6" may be "5" or "7"?); y = 0.6249 ± 0.0021 m is OK (assuming you can understand your system well enough to estimate the error to 2 sig figs).

- advice to TAs: only knock marks off where really ludicrous numbers of digits are given, or uncertainties quoted with non-agreeing significant figures, as per the examples at the start of this article. If you would have used two and the student quotes three or even four, hang loose, chill out and let it pass. You'll get lots more opportunities to wield the red pen before the night's marking is out.

Example

Have a look at my "Clean Coal" calculation6. Here several numbers are know only vaguely, and I quote "typical" values, often to one significant figure, e.g 30 MJ/kg, or η =0.4. I am also missing many sources of CO2, like mining or transportation. On the other hand, I can calculate the number of seconds in a year to as many significant digits as I wish.

I end up with a 8.9 Mtonnes CO2 per GWe per year. I could round this off to "9", but that looks too accurate. Do I know whether it is 7 or 11? No. So let's leave it at 10.