The Student Room Group

Appropriate degrees of accuracy

"Here is a solid bar made of metal. The bar is a cuboid. The height of the bar is h cm. The base of the bar is a square of side d cm. The mass of the bar is M kg.

d = 8.3 correct to 1 dp
M = 13.91 correct to 2dp
h = 84 correct to nearest integer

Find the value of the density of the metal to an appropriate degree of accuracy. Give your answer in g/cm^3. You must explain why your answer is to an appropriate degree of accuracy."

So, I know density=mass/volume.

I've found the upper bound density to be 2.44843646 g/cm^3 and the lower bound density to be 2.360159389 g/cm^3.

Would I then find the mean of these to find the density to an appropriate degree of accuracy? If not, can someone help with how I'd do it? I always struggled with these appropriate degree of accuracy questions.
Original post by Adidas02
"Here is a solid bar made of metal. The bar is a cuboid. The height of the bar is h cm. The base of the bar is a square of side d cm. The mass of the bar is M kg.

d = 8.3 correct to 1 dp
M = 13.91 correct to 2dp
h = 84 correct to nearest integer

Find the value of the density of the metal to an appropriate degree of accuracy. Give your answer in g/cm^3. You must explain why your answer is to an appropriate degree of accuracy."

So, I know density=mass/volume.

I've found the upper bound density to be 2.44843646 g/cm^3 and the lower bound density to be 2.360159389 g/cm^3.

Would I then find the mean of these to find the density to an appropriate degree of accuracy? If not, can someone help with how I'd do it? I always struggled with these appropriate degree of accuracy questions.


the calculation involved several different rounding methods. you should use the strictest one on your answer.... so 3 sig fig would win over 4 sig fig etc.
Original post by the bear
the calculation involved several different rounding methods. you should use the strictest one on your answer.... so 3 sig fig would win over 4 sig fig etc.


So would I find the mean of the upper bound density and lower bound density and then round?
Original post by Adidas02
So would I find the mean of the upper bound density and lower bound density and then round?
Yes, that will work.

In general, the "finer details" of what you do won't matter, because they'll get lost in the rounding. So you would normally get the same answer just by calculating treating the given answers as exact.

If you're unlucky, you might get a final answer very close to the "round up/down" boundary and then it gets tricky (*). (e.g. you've worked out your answer should be to 3 dp and you get 1.2349997 - very hard to be confident it should round to 1.234 v.s. up to 1.235 here).

In general, examiners will make sure this isn't likely to be an issue by choosing the initial values appropriately.

[Something it's worth realising is that if (*) occurs, in practical terms, *both* 1.234 and 1.235 are reasonable answers. One is an underestimate by .00049997 and one an overestimate by .00050003; for a teacher/examiner to argue that one is clearly better when they are drumming home to you at the same time that "you can't trust any of the values past the 3rd decimal place" would by hypocritical, to say the least].
Original post by DFranklin
Yes, that will work.

In general, the "finer details" of what you do won't matter, because they'll get lost in the rounding. So you would normally get the same answer just by calculating treating the given answers as exact.

If you're unlucky, you might get a final answer very close to the "round up/down" boundary and then it gets tricky (*). (e.g. you've worked out your answer should be to 3 dp and you get 1.2349997 - very hard to be confident it should round to 1.234 v.s. up to 1.235 here).

In general, examiners will make sure this isn't likely to be an issue by choosing the initial values appropriately.

[Something it's worth realising is that if (*) occurs, in practical terms, *both* 1.234 and 1.235 are reasonable answers. One is an underestimate by .00049997 and one an overestimate by .00050003; for a teacher/examiner to argue that one is clearly better when they are drumming home to you at the same time that "you can't trust any of the values past the 3rd decimal place" would by hypocritical, to say the least].


Okay, thank you :smile:
Reply 5
Original post by DFranklin
Yes, that will work.

How would you know what to round to after finding the mean of the UB and LB? If you chose e.g. 2.404 then that wouldn't get you the mark because the real value of the density may not round to 2.404 to 4sf.

The usual way to do this question would be to follow the bear's method.
(edited 6 years ago)
Original post by Notnek
How would you know what to round to after finding the mean of the UB and LB? If you chose e.g. 2.404 then that wouldn't get you the mark because the real value of the density may not round to 2.404 to 4sf.

The usual way to do this question would be to follow the bear's method.


So round first?
Reply 7
Original post by Adidas02
So round first?

The common way to do this question:

UB : 2.44843646
LB : 2.360159389

Start from the least accurate rounding i.e. 1sf. They both round to 2 to 1sf so it would be correct to say that the density rounds to 2 to 1sf.

Next check 2sf : they both round to 2.4 to 2sf so the real value must round to 2sf.

2.4 is more accurate than 2 so it would be more appropriate but you have to keep going until you get the best accuracy.
Original post by Adidas02
So round first?


no ....use the most accurate figures you can during the calculation. at the very end round your answer according to the strictest of the measurements ( remember that this means if you use measurements with 3 sf, 5 sf, 11 sf then you go with the 3 sf at the end )
Original post by Notnek
The common way to do this question:

UB : 2.44843646
LB : 2.360159389

Start from the least accurate rounding i.e. 1sf. They both round to 2 to 1sf so it would be correct to say that the density rounds to 2 to 1sf.

Next check 2sf : they both round to 2.4 to 2sf so the real value must round to 2sf.

2.4 is more accurate than 2 so it would be more appropriate but you have to keep going until you get the best accuracy.


Okay, that makes sense - thank you :smile:
Original post by Adidas02
I've found the upper bound density to be 2.44843646 g/cm^3 and the lower bound density to be 2.360159389 g/cm^3.


The highest accuracy there is 2dp/4sf, you shouldn't put higher (3dp/5sf). I would write 2.40+-0.04, it's the mean of two values with uncertainty rounded to 1sf.
Original post by Notnek
How would you know what to round to after finding the mean of the UB and LB? If you chose e.g. 2.404 then that wouldn't get you the mark because the real value of the density may not round to 2.404 to 4sf.The mean cannot differ from the real answer by more than (UB-LB)/2, this lets you choose the appropriate number of decimal places.

I have seen questions (syllabuses) where you are *expected* to find upper / lower bounds rather than "the standard 'round to the worst accuracy of the input values'". In those syllabuses, it's not uncommon to get formulae such as Y=ab10Y = ab^{10} where inaccuracy of one variable is unexpectedly magnified (or conversely, minimized, although an appropriate formula isn't springing to mind), and the standard method doesn't give the best results.

Of course, I have no idea what method the OP is actually supposed to use.

Edit: an obvious example where the error is minimised is period of a pendulum treated as a function of the initial amplitude. To first order, the amplitude doesn't affect the period at all.
(edited 6 years ago)

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