# Definition of significant figures

Normally when we are taught how to add numbers with regards to significant figures, we are told to round the result to the rightmost place of the least precise digit. $3.55 + 4 = 7.55$, for example, would be rounded off to $8$. But for my argument I will be considering the addition of two numbers $9$ and $2$, both of which are significant to the ones digit.

Following the usual rule for addition, $9 + 2 = 11$ and now we have two significant figures. However, whenever I think about significant figures I would intuitively think of these two definitions:

1 - Digits to the right of the last significant digit, can be any number between 0 and 9.

2 - Any digit that has the possibility of being more than one numerical value, is not a significant digit.

Using the first definition, we can put $9$ and $2$ in these forms:

$9.A$

$2.B$

where $A$ and $B$ are arbitrary digits ranging from $0$ to $9$. Thus, when we add these two numbers, we would end up with:

$$9.A + 2.B$$

$$= 9 + 2 + 0.A + 0.B$$

$$= 11 + 0.A + 0.B$$

Finally when we assign some meaningful values to $A$ and $B$ such as $(A , B) = (0 , 0)$ or $(A , B) = (9 , 9)$, we can observe something very interesting:

• For $(A , B) = (0 , 0)$:

$$11 + 0.0 + 0.0 = 11$$

• For $(A , B) = (9 , 9)$:

$$11 + 0.9 + 0.9 = 12.8$$

In the set of all possible numbers that can result from adding $9.A$ and $2.B$, both of which are significant to the ones digit, the minimum happens to be $11$ and the maximum happens to be $12.8$ (the set becomes even greater if we add more uncertain digits to the right of $A$ and $B$, and in that case the maximum resulting value starts to approach $13$ as you add more and more 9's to both $9$ and $2$).

Lastly, we can see that tens digit is always $1$, and using my second definition of significant digits, we can claim that the tens digit is significant. On the other hand, because ones digit can either be $1$ or $2$, the ones digit is not significant. Thus, using my two definitions of significant figures, we can claim that the addition between $9$ and $2$ results in one significant figure (at the tens place), rather than two significant figures one would get from the usual rule of thumb.

So my question is, are my definitions of significant digits correct? If yes, then I can say the general rules of thumb for adding numbers with significant figures is wrong and is being blindly taught and learned in high schools and universities. If not, then can you show me why not and if you have any credible sources.

The standard significant digits rules assume that the values you working with are rounded versions of the true values. Thus you cannot take (A, B) = (9,9) in your example, because 9.9 would have been rounded to 10, not 9.

Note also that 8.9 is rounded to 9, so you cannot assume that the true value has the form "9.A".

• Even if that were the case, 9 could have been rounded up from 8.51 and 2 up from 1.51 or 9 down from 9.49 and 2 down from 2.49. Doing addition for both sets would give you 10.02 and 11.98, which still presents similar problem as that in my original post. Perhaps I am confused because I do not understand the rules for rounding and the greater context surrounding it. Can you comment on that? – FrustratedCollegeBoi Jan 16 '17 at 6:37
• I don't understand the part where you say "Thus, using my two definitions of significant figures, we can claim that the addition between 9 and 2 results in one significant figure (at the tens place), rather than two significant figures one would get from the usual rule of thumb." The usual rule of thumb only says that you get one significant figure, not two, because both 9 and 2 only have one significant figure. – Ted Jan 16 '17 at 7:13

There is some inconsistency in usage. Some people count significant digits from the decimal point, which works well enough when the possible range of values is reasonably small. For the rule you cite (and others related to error analysis and propagation) to be applied consistency, one must instead start counting significant digits from the first non-zero digit from the left. This is equivalent to writing all numbers in scientific notation and simply counting digits starting on the left.

In your example, then, you’re actually adding $9\times10^0$ and $2\times10^0$. Both exponents are the same so no adjustment is necessary before performing the addition. Their sum is $1.1\times10^1$, but this has too many significant digits, so it gets rounded to $1\times 10^1$, or $10$.

There is an ambiguity that you have to deal with, though: how many significant digits does an integer that ends with a string of zeros have? A number like $1000$ can cause the precision of a calculation to collapse very quickly if not handled correctly. This is another advantage of scientific notation in this context: there’s no such ambiguity. Trailing zeros, if any are significant digits.

There are two issues here. Number of significant digits, and how much rounding is appropriate. Both are slippery concepts with apparently different definitions and advice in different fields. Here I will focus mainly on rounding. The crucial point is not to throw away information that may be useful. Here are some examples to think about.

$1.$ Distinctions between individuals and averages. If one person loses a pound during a month, that is not usually a significant event. In the process of eating, drinking, exercising, or elimination a person can temporarily gain or lose a pound during a day. However, if there is an average gain or loss of one pound for 100 subjects during a month long weight loss program, that may be a significant if not startling change. So for most purposes, it is probably appropriate to round weights of individuals to the nearest pound and to round weights of groups to the nearest tenth of a pound.

$2.$ Premature rounding. In a famous study of boarding school boys in India, it was established that students were about one $cm$ shorter in the evening than in the morning. Each student was measured twice by each of two investigators in the morning, and again in the evening. Even though it is not really feasible to measure height to the nearest millimeter, best attempts were made to do so for all eight measurements on each child. If all the original measurements had been rounded to the nearest centimeter, the "1$cm$" difference would not have been as clear. Also, it turned out that one investigator systematically measured students as a little taller than the other, and that finding might have been totally obscured by rounding to the nearest centimeter.

$3.$ Rounding before statistical tests. Certain kinds of rank-based statistical tests do not work well if there are many ties in theoretically-continuous data. So, in performing such tests, one should be careful about rounding measurements before testing.

Example: A brief simulation shows the effect of rounding. Suppose a certain measurement in population can be modeled for practical purposes as normal with mean 400 and standard deviation 60. As a practical matter, such measurements will usually span an interval such as $(260,540).$ If we simulate 100 such measurements and record them to the nearest 0.1, then typically there will be few if any ties; specifically, one tie in the instance shown below:

x = rnorm(100, 400, 60)
length(unique(round(x,1)))
## 99

But if we round to the nearest integer, it is not unusual to get as many as 10 or 15 ties (enough to make rank-based analysis problematic).

length(unique(round(x)))
## 87

$4.$ Authoritative standards. My opinion is that the rounding rules in mathematics texts are generally not appropriate in statistical practice. Governments and industries often adopt standards of practice for rounding that focus on rounding that simplifies without sacrificing useful informaion. This report by the US National Institute for Science and Technology is one example. I am not claiming that these standards are always ideal, but they are based on thoughtful consideration by people who handle real-world data.