# Appropriate degree of accuracy

Discussion in 'Mathematics' started by ic3g1rl, Mar 11, 2012.

1. ### ic3g1rl

I've taken on a tutee doing AS S1 (edexcel). His marked homework is always pointing out 'rounding errors' and he's very confused as he thinks he's rounding his answers correctly, so do I! However there is another problem to do with which numbers should be used. On the exam papers it says 'Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy.' The problem is I'm unclear what this 'degree of accuracy' should be.
As an example I'll go through my confusion based on May 2006 S1 paper, question 3, looking at their answers and comments.
The question gives a table of data consisting of figures to 2 decimal places.
a) Calculate Sxy and Sxx.(Given sum x^2= 15965.01 and sum xy = 757.467)
Answer : Sxy = 71.4685 (awrt 71.5)
Sxx = 1760.45875 (awrt 1760)
Anyone care to tell me how these answers are 'an appropriate degree of accuracy?' You have 2 decimal places provided for most of the data, and one with 3 decimal places, neither answer matches this. Is it supposed to?
b) Find the form of the regression line of y on x in the form y = a + bx. (I've skipped the coding bit as I just wanted to ask about the decimal places.)
c = 0.324364 (awrt 0.324)
y = 0.324 + 0.0406x (3 sf or better ...)
Now the answer is in significant figures, rather than decimals. Why? How are the students supposed to know this? '3 sf or better' Better than what? They have managed to give an answer to 4 decimal places which surely can't be 'an appropriate degree of accuracy!'
I'm missing something and I've looked through their textbooks and they don't make clear what the 'appropriate degree of accuracy' actually is. Can anyone enlighten me? This student is loosing marks for something that isn't really based on the statistics and he is being told he has 'rounding errors'. What he really has is confusion about what 'an appropriate degree of accuracy' is and so do I.

2. ### ic3g1rl

I've taken on a tutee doing AS S1 (edexcel). His marked homework is always pointing out 'rounding errors' and he's very confused as he thinks he's rounding his answers correctly, so do I! However there is another problem to do with which numbers should be used. On the exam papers it says 'Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy.' The problem is I'm unclear what this 'degree of accuracy' should be.
As an example I'll go through my confusion based on May 2006 S1 paper, question 3, looking at their answers and comments.
The question gives a table of data consisting of figures to 2 decimal places.
a) Calculate Sxy and Sxx.(Given sum x^2= 15965.01 and sum xy = 757.467)
Answer : Sxy = 71.4685 (awrt 71.5)
Sxx = 1760.45875 (awrt 1760)
Anyone care to tell me how these answers are 'an appropriate degree of accuracy?' You have 2 decimal places provided for most of the data, and one with 3 decimal places, neither answer matches this. Is it supposed to?
b) Find the form of the regression line of y on x in the form y = a + bx. (I've skipped the coding bit as I just wanted to ask about the decimal places.)
c = 0.324364 (awrt 0.324)
y = 0.324 + 0.0406x (3 sf or better ...)
Now the answer is in significant figures, rather than decimals. Why? How are the students supposed to know this? '3 sf or better' Better than what? They have managed to give an answer to 4 decimal places which surely can't be 'an appropriate degree of accuracy!'
I'm missing something and I've looked through their textbooks and they don't make clear what the 'appropriate degree of accuracy' actually is. Can anyone enlighten me? This student is loosing marks for something that isn't really based on the statistics and he is being told he has 'rounding errors'. What he really has is confusion about what 'an appropriate degree of accuracy' is and so do I.

3. ### maths126New commenter

I would guess that the data given is to 3 sig fig or more.
This means that since answers can not reliably be quoted to greater accuracy than the original data, the mark scheme is looking for rounding to 3 sf.

4. ### lizziec

So what values was your student getting? Exactly the same as the mark scheme and then rounding?(I assume he is showing his original calculated value and his rounded answer)
For part a) the two values are the exact ones obtained if you do no rounding and use all of the original data in full. In that case, he should be giving an "answer which rounds to" the specified value, and getting the full marks.
If on the other hand, he is rounding at an earlier stage in the calculation (eg after calculating total x or total y)... he might still get the same answer to the number of decimal places required for this question, but the teacher may be trying to train him to work with all the digits provided...
As a teacher I have sometimes written "rounding error" to indicate that a student has rounded too early - but I can now see this might be misinterpreted by the student as meaning an incorrect rounding at the end - thanks for alerting me to this possibility.
Liz

5. ### bobboots

The "rounding error" probably means they are using a rounded answer in a calculation which means that they significantly out in the answer
A simple example to make the point: evaluate to 3 s.f. (pi - 1.34)^2
pi - 1.34 = 1.80 (3 s.f)
1.80^2 = 3.24 (3 s.f)
correct solution:
pi - 1.34 = 1.801592.......
ANS^2 = 3.25 (3 s f)
Hence there is a rounding error in the first example.

6. ### Anna-Luise

To cut down on the work and guarantee effortless accuracy you can make good use of a calculator.

I typed in the two columns of figures and was able to then write down all relevant values except Sxx and Sxy which can be obtained simply enough.

You can also compare your Sigma x^2 and Sigma xy to the values given to be sure that you have entered everything correctly.

Having recorded all values in your working very accurately you can then write your regression equation giving the coefficients correct to three significant figures. [x and y were given correct to 3 s.f.]

It's impossible to say what's going on with your student without seeing their work.

7. ### ic3g1rl

All the answers I've given are official edexcel answers. I had thought the answers may be based on the idea of 3 significant figures but if you look at the answers I've quoted this doesn't hold true. I worked on your idea of 'answers can not reliably be quoted to greater accuracy than the original data' then the answers should be 2 decimal places since that is the accuracy of the original data. Edexcel have not done this; so again I ask can anyone explain what is going on?

8. ### KYPNew commenter

My understanding of 'Mark schemes' is that they are to guide the official markers as to what is an acceptable answer, and what numbers they should be looking for. So I would assume the long numbers are what the calculator/tables will give if used correctly, whereas 'AWRT' means that any number which rounds to the 3sf value given is acceptable. This is in contrast to 'CAO' - correct answer only - which means you only get the mark for exactly what it says. I believe rounding the final answer to 3sf is usually OK, unless the question says otherwise. But premature rounding is always risky. Teaching students to think about what accuracy is appropriate to suit the given data is good mathematics teaching; allowing students to get away with poor technique runs the risk that they may lose marks in the exam. There is always a warning at the front of the mark scheme that just because something is accepted on one exam doesn't mean it will always be OK - a prime example is missing off the constant of integration, which is sometimes condoned, but not always.

9. ### Anna-Luise

Rounding to a certain number of decimal places is, in general, a silly idea. Rounding to a certain number of significant figures makes more sense.

As I said above, x and y were given correct to 3 s.f..

I'm not sure what you mean by 'what is going on?'

10. ### Anna-Luise

Compare, June 2007, question 3. Values given correct to 2s.f., coefficients rounded to 2 sf.

11. ### mmmmmathsNew commenter

t in the question is given to 3sf.
As a previous poster has mentioned. The mark scheme will give the answer to lots of decimal places to help the marker identify that the candidate has done the correct calculation before they have rounded it appropriately.
Always use sf and not dp when rounding off answers.

12. ### mmmmmathsNew commenter

The answers given are all clearly to 3sf

13. ### bobboots

Reading again the OP it seems very clear. 3 s.f. is expected in S1 unless otherwise specified.

On a different issue - why hasn't the student asked the teacher if (s)he is unsure what it means - I would be annoyed if I went to the effort of repeatingly pointing out a mistake and one of my students didn't ask for clarification if they were unsure what it meant.

14. ### ic3g1rl

I could accept this but the rest of the answers don't match:
c) Estimate the length of the rod at 40 degrees C.
Answer: l = 2461.948 (awrt 2461.95)
d) Find the equation of the regression line of l on t.
Answer: l = 2460.324 + 0.0406t ( awrt 2460.32, f.t. their 0.0406, l and t)
e) Estimate the length of the rod at 90 degrees C.
Answer: l = 2463.978 (awrt 2464)
Only one of these is 3 sf the rest are not!

15. ### KarvolOccasional commenter

There should be an explanation on the accepted degree of accuracy either on the syllabus for the course or on the exam paper itself.
If there is nothing there, then I suggest - as have others - that your student gets in touch with the teacher.

16. ### mmmmmathsNew commenter

Because l in the question is given correct to 6sf

17. ### frustumStar commenter

c/d/e above clearly have to break the 3sf rule of thumb, as if we only use 3sf, we're not going to detect any change in the length of the rod at different temperatures.