Solving Sig Fig Challenges: Is Rounding Necessary?

[emberjelly]

Homework Statement

Hi,

I am having a lot of trouble with significant figures, for no reason.

I will give an example of the kind of thing I can't understand. My teacher insists that the number of sig figs in an answer must equal the number of sig figs in the data with the least number of sig figs.

But what if I have a number (no specific question) which has data with 1 sig fig. And an answer that is, say, 150. Must I round this answer up to 200, thus achieving 1 sig fig, but at the cost of 25% of the accuracy of my calculation? This is the worst case I have encountered so far, but I see no mention in my "sig fig" guide of exceptions to the rules.

This is very important to me, as I have been frequently losing marks because I just can't bear to round some of my calculations so savagely.

Thank you!

Homework Equations

The Attempt at a Solution

 

[PeterO]

Homework Statement

Hi,

I am having a lot of trouble with significant figures, for no reason.

I will give an example of the kind of thing I can't understand. My teacher insists that the number of sig figs in an answer must equal the number of sig figs in the data with the least number of sig figs.

But what if I have a number (no specific question) which has data with 1 sig fig. And an answer that is, say, 150. Must I round this answer up to 200, thus achieving 1 sig fig, but at the cost of 25% of the accuracy of my calculation? This is the worst case I have encountered so far, but I see no mention in my "sig fig" guide of exceptions to the rules.

This is very important to me, as I have been frequently losing marks because I just can't bear to round some of my calculations so savagely.

Thank you!

Homework Equations

The Attempt at a Solution

That answer should not be given as 200, but as 2 x 10².
It is then clear that you have rounded an answer off to express with the appropriate number of significant figures.
And besides, you didn't know the answer was 150, you only knew it was somewhere around that value - and condensing the answer to 2 x 10² is telling the world that you really didn't know anything exactly.
I also shows why we like to use instruments that will give at least 2 significant figure precision.

 

[NoPoke]

This is not as obvious as it seems. What you are being asked to do is present an answer without the false levels of precision that might be implied by a multi-digit answer.

Suppose that you have two measurements from an experiment represented by 50 and 3 with the 50 actually only having a single digit of precision.

The ranges for each measurement would thus be 45-55 and 2.5-3.5

Simply multiplying 50 x3 gives 150 as an answer but has no information about he precision of the measurements. To see what this might be take the worst case minimum values and the worst case maximum values and end up with an answer of 150 with a range of 112.5 - 192.5 I've kept the full calculated answer just to avoid confusion at this point. Rounding the 150 answer either up or down will actually mis-represent the result though it will preserve the precision of the orginal data.

So how to preserve the approximately 100 spread in calculated value that a single significant figure indicates?

A good answer you should give in my contrived case is 150 +/- 50

=========

Try it with say 8 and 6 as measured values and the calculated result of 8x6= 48 . What is the range of values given that the measurements are only one significant digit? Then try 3 and 2.

=========

As an aside you have to be very careful when you subtract two measurements. Errors are magnified by subtraction.

PeterO is absolutely right about using measurements that give readings to two or more significant figures. 1 sig fig is going to cause problems.

 

[emberjelly]

That answer should not be given as 200, but as 2 x 10².
It is then clear that you have rounded an answer off to express with the appropriate number of significant figures.
And besides, you didn't know the answer was 150, you only knew it was somewhere around that value - and condensing the answer to 2 x 10² is telling the world that you really didn't know anything exactly.
I also shows why we like to use instruments that will give at least 2 significant figure precision.

This doesn't make any sense to me at all, why do we know nothing? If I am given 3 objects approximately 50cm, and I put them in a row, I estimate 150, I don't say hey, let's call it 200... I could say 180, and be more likely to be accurate, or 130, 120. Anything basically, 200 is as far from the truth as possible.
And I don't believe it is required to say 2* 10^2 because 0's before a decimal point but after significant digits are not considered significant anyway, at least this is what I gather from what I have read.

NoPoke, your answer of 150 +- 50 makes a lot more sense, but I am pretty sure sure my teacher has never mentioned given answers in this form, as in he always just writes one number and leaves it. Would I get away with putting answers in this form in the future?, although I guess only my teacher could really aswer that anyway hey. I might ask him, perhaps it is allowed to just give to 2 sig figs sometimes. It sounds like everyone has their own rules on it anyway.

Thanks for the replies guys,
emberjelly

 

[PeterO]

This doesn't make any sense to me at all, why do we know nothing? If I am given 3 objects approximately 50cm, and I put them in a row, I estimate 150, I don't say hey, let's call it 200... I could say 180, and be more likely to be accurate, or 130, 120. Anything basically, 200 is as far from the truth as possible.
And I don't believe it is required to say 2* 10^2 because 0's before a decimal point but after significant digits are not considered significant anyway, at least this is what I gather from what I have read.

NoPoke, your answer of 150 +- 50 makes a lot more sense, but I am pretty sure sure my teacher has never mentioned given answers in this form, as in he always just writes one number and leaves it. Would I get away with putting answers in this form in the future?, although I guess only my teacher could really aswer that anyway hey. I might ask him, perhaps it is allowed to just give to 2 sig figs sometimes. It sounds like everyone has their own rules on it anyway.

Thanks for the replies guys,
emberjelly

If you are ADDING numbers, significant figures are handled quite differently! The number of decimal places equals the smallest number of decimal places. Similarly subtraction.

51.35 + 122.4 + 3.456 ==> 177.206 which is then expressed a 177.2
Note how all the figures are used while calculating, it is just the final answer that is expressed correctly.

Also don't confuse accuracy with precision. Accuracy comes from making your measurements carefully. Precision comes from using a suitable measuring device. Significant figures arise from precision.

It is only when multiplying [or dividing] that the minimum number of sig. figs. idea applies.

EDIT: If you have a whole lot of measurements to great precision, the sig. fig. rules are a reminder that you should aim for similar precision with every measurement you make, or your final result will be compromised.

 

[NoPoke]

PeterO gave you the insight that a single significant figure is going to cause trouble. Not all the time, just often enough to cause confusion and misunderstanding.

The 150 +/- 50 is not an answer to two significant figures. I know it looks like it is as there are two digits in the 150 before the zero. But it isn't. The significance of the result is given by the +/-50.

When you write out any number to x significant figures that missing +/- range is what the x sig figs is telling you. Most of the time the compact form using sig figs and the value +/- range are the same. As it takes more space to include the +/- range the sig figs approach is common. Sometimes the sig figs short form just doesn't work very well.

My contrived 150 result was one such example. You want to express that the answer is 150 but that the range of this answer is such that only the first digit specifies the significance. The significant figures approach to expressing an answer just doesn't handle this particular case well.

Try the following subtract 1,000,000 from 1,000,000 with both measurements having a single digit of precision. What is the answer? Is it possible to express it using significant figures?

 

[mikelepore]

In addition to what others have already said, I think you can usually do better than to have a measurement like "approximately 50 cm." That may mean anything between 45 and 55, being rounded to 50, which means you have a possible error range that's about the width of a person's hand. Consider whether there a reason for the measurement to be that rough, for example, if you have to take a measurement while something is moving around, then "approximately 50" may be the best that a person can do. But if the zero in the 50 an actually a measured zero, because the edge of the object was observed to be near a line on a ruler, and you read off "zero", then you can write "50. cm" and have two sigfigs.

 

[emberjelly]

In addition to what others have already said, I think you can usually do better than to have a measurement like "approximately 50 cm." That may mean anything between 45 and 55, being rounded to 50, which means you have a possible error range that's about the width of a person's hand. Consider whether there a reason for the measurement to be that rough, for example, if you have to take a measurement while something is moving around, then "approximately 50" may be the best that a person can do. But if the zero in the 50 an actually a measured zero, because the edge of the object was observed to be near a line on a ruler, and you read off "zero", then you can write "50. cm" and have two sigfigs.

This is a very good point. I did speak to my teacher, and he said in cases as extreme as these, it is ok to use one more sig fig than is given in the data.