Significant Figures?

Question

In the problem 2.9cm x 3.5 cm x 10.0 cm = 101.5

What would the answer be with the correct number of sig figs?

Edgar and Mr. X,

So do you mean the answer is 100 cm?
or 101.5 cm

This is a multi-step problem and I don't want to mess up the first step. =)

Our teacher says the number of sig figs in the answer is the same as the least precise number in the calculation. She doesn't refer to decimal places. So it's just 100 right?

Answer 1

Short answer: The result must be given as 1.0 E2 cc ["cc" stands for "cubic centimeters" and E2 means 10^2]. (The results 101.5 and 101 (or, rather, 102) are too precise whereas, strangely, 100 is not precise enough!).

A meticulous professional scientist could also state the result as 101.5(13) cc. [See explanation below.]

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NOTE:
I have now rewritten the text posted below into permanent shorter articles (one for each concept discussed) using this particular example. You may access the whole thing at
http://www.numericana.com/answer/rounding.htm#significant
Sorry for producing the long-winded initial reply below. I hope you'll find the edited version easier to read!
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Forget "rules of thumb" if you want to get to the bottom of this... A correct analysis of rounding error is worth doing at least once in your life. Chances are that your teacher hasn't worked out the computation given below under the heading "COMPLEMENT" and does not expect you to do it either. You may skip that for now, but (please) make sure to come back to it later in your scientific life ;-)

A number like 2.9 designates a quantity which has been rounded and whose actual value is anywhere between 2.85 and 2.95. Likewise "10.0" is anywhere between 9.95 and 10.05...

So, the minimum value of your product is:
2.85 x 3.45 x 9.95 = 97.83
The maximum value is:
2.95 x 3.55 x 10.05 = 105.25
Your result really is 101.5 "take or leave 3.7".

However, you can't give the result 101.5 all by itself, since it would imply that you believe the correct value is between 101.45 and 101.55, which is an accuracy of 0.05%. That's 74 times more precise than the accuracy we trust (namely 3.7%). So, such a result would be clearly misleading.

If you give a result of 101 (or 102) you are claiming an accuracy of 0.5% which is stil too optimistic (by a factor of 7).

Now, the bad news: If you give a result of 100, the TWO trailing zeros imply that only the first digit is trusted, so the true value so stated is anywhere between 50 and 150. Now, the "impression" you are giving is too pessimistic by a factor of 14. (We did not have this problem with 10.0 because a "trailing zero" AFTER the decimal point must indicate a significant digit.)

If you give a result of 110, "standard rules" imply that you're claiming a result between 105 and 115. This does not reflect your actual belief,in spite of the 5% uncertainty, because you know that the result could be as low as 97.83.

So, what's the answer? Well, you must find a way to claim a result between 95 and 105 (the fact that it could be up to 105.25 is not really significant) by stating that your first zero is a significant digit but the second zero is not. There's no way to do that with ordinary notation, but it's easy to do with "scientific" notation where you denote a nonzero number by a number from 1 (included) to 10 (excluded) multiplied by a power of 10. The letter "E" is used for the latter part in calculators and computers (10 is 1E1, 100 is 1E2, 1000 is 1E3, etc.) In the notation 1.0E2, the "0" is necessarily asignificant digit (or else it wouldn't be there) so this does designate exactly what you want: a number between 95 and 105.

COMPLEMENT:

In most cases, the coarse rules for "significant digits" are not flexible enough to convey the known information about numerical uncertainties, even roughly.

Professionals routinely state the accuracy of their figures by giving the uncertainty expressed in units of the last figure between parentheses (see second link below for examples).

Technically, this uncertainty is the "standard deviation" of your measurement, which is about 1/3 of the bounds you figure out at a 99.73% confidence level for normal "bell curve" errors (see first link for an explanation of this).

However, the uncertainty which comes from rounding is not a "bell curve" but a square function, so the standard deviation which is to be stated between parentheses is half a unit of the last digit divided by sqrt(3) which is to say about 0.29 units of the least significant figure (trust me on this one,. unless you know how to compute standard deviations, in which case this is a nice exercise).

So, if a value like 3.5 has been obtained by rounding, then it represents with equal probability any value between 3.45 and 3.55). Therefore, we could write it "professionally" as 3.500(29) or, more coarsely 3.50(3).

The "quick and dirty" way to estimate the standard deviation of the above result is to consider that it's between 97.83 and 105.25 with (roughly) uniform probability within that interval. You would take the middle of that interval (namely 101.54) as a result and estimate the standard deviation to be 0.29 time the distance to either extreme (namely 3.71). This comes to about 1.08 and you would state your final result as 101.54(108). That's how many professional routinely do it. It's somewhat incorrect, though...

The correct approach is to work out the standard deviation of a "random variable" equal to:
(2.9 + 0.1X) (3.5 + 0.1 Y) (10.0 + 0.1 Z)
where X, Y and Z are 3 independent random variables uniformly distributed between -0.5 and +0.5

The average is precisely 101.5 (which is certainly more satisfying than the number 101.54 obtained from the above "quick and dirty" method). Unless you have a black belt in statistics, you may want to trust me for the rest: The exact standard deviation is 1.3444711...

All told, the final result can be expressed either as 101.50(134) or, more commonly, 101.5(13) since uncertainties are normally expressed with 2-digit precision.

That's it (more than what you bargained for, I am sure).

Answer 2

that's the right answer. 1 significant figure after the decimal point.

if the 10.0 was just 10, then you wouldn't have any, but the 0 makes that place significant.

Answer 3

1.0x10^2

You only have two significant figures in the first two numbers being muliplied, and so the product has two sig-figs as well.

"Our teacher says the number of sig figs in the answer is the same as the least precise number in the calculation. She doesn't refer to decimal places. So it's just 100 right?"
Yes. The decimal place has nothing to do with it. There are two significant figures. It's just 100 with the first 0 also significant, or as I said 1.0x10^2.

Answer 4

According to significant figures rules you would use the number with the most amount significant figures to begin with
2.9 x 3.5 x 10.0 = 101.5


Good souce below from Purdue, suggest sharing with your classmates!

Answer 5

You already have it: 101.5

The problem was stated with precision to one decimal place, so the answer should also be stated to the same precision.

If the problem had been 2.9 x 3.5 (which equals 10.15), then the correct answer would be 10.2 to keep the same number of significant digits.

Answer 6

the rule is that (0,1,2,3,4,) are down but the (5,6,7,8,9)
are upper value.
Here is 101.5=~102 cm^3