Easy Ways to Calculate Error in Addition & Multiplication

[SS2006]

as good as i am in physics, i just didnt try to understand error calculation
can someone give me 2 easy ways to calculate
error in addition

for example

5 +- 0.2 + 6 +- 0.1

and in mulitplication, like my newest lab
i have f = m tiems g
and the mass is +- 0.2 and gravity is +- 0.1 for example
mas is 25 g let's say, and gravity 9.8


so you easiest way to calculate error in multi, and additio please :)
thanks

 

[Ouabache]

This is getting into statistical analysis...
(i) What are your minimum values in this addition? (hint:[5-0.2]+[6-0.1], add those together) and you get a minimum solution value.
(ii) Similarly what are your maximum values? (add those together) and you get a maximum solution value.
(iii) the sum without including error is just 5+6 = 11

Now,
(a) take the sol'n from (iii) and substract your minimum sol'n.
(b) take your maximum sol'n (from ii) and subtract sol'n from (iii).

In your example, the absolute value of the sol'n in (a) equals the absolute value of sol'n of (b). So the error of your final sum is +/- that value.

For your multiplication question: follow a similar method, multiply minimum values, maximum values and compare to the product of values that did not include your initial errors. So how much did your minimum product differ from the product without including errors? How much did the maximum product differ? You may find these two "differences" are not equal. But you can still make a valid statement about how much +/- error you get in the product.
How much error would you say there is?

 

[SS2006]

ok, sot he final asnwer would be 11 +- 0.3 ?

how bout mulitplication that's the harde rone i need this b4 tomorrow!

lets say 4 +- 0.2 X 5 +- 0.3
thanks

 

[Ouabache]

ok, sot he final asnwer would be 11 +- 0.3 ?

You're doing fine ! Try the same method on multiplication question.

 

[SS2006]

ok I am quite confused bout the mulitplication one

lets say 4 X 4 and 0.2 is the +/- error on both of them, can you just solve that one and show what u did

 

[SS2006]

hmm is 4 +- 0.2 X 4 +- 0.2
16 +- 1.6?

if it is I am aliright, just let me know

 

[Ouabache]

(i) minimum product [4-0.2]*[4-0.2]=14.44
(ii) maximum product [4+0.2]*[4+0.2]= 17.64
(iii) product without error 4*4 = 16

(a) (iii)-minimum 16-14.44 = 1.56
(b) max - (iii) = 17.64-16 = 1.64

Error in product is +/- what? well +/- 1.64 does include the value of your lower error (1.56), but actually overestimates it. If this is experimental data, it is valid to state there is +/- 1.64 (units) error in the result (or 16 +/- 1.64)

 

[Ouabache]

hmm is 4 +- 0.2 X 4 +- 0.2
16 +- 1.6?

As you can see from my last note, you're sol'n is correct (with one decimal place precision ).

 

[SS2006]

very much appreciated brother :)

 

[andrevdh]

Error analysis is quite an advance topic. Therefore one usually get the formulae to calculate the error in a value (say Z) as a result of the errors in the values it depends on (say A and B). In the following formulae the errors in the values are indicated by deltas. Firstly the error in a value obtained from a sum or difference calculation are given by:
$$(\Delta Z)^2=(\Delta A)^2+(\Delta B)^2$$
The error in you first example will therefore be 0.2
In values obtained from multiplication and division calculations one sees the relative error - that is what fraction is the error of the value. The formula for calculating the error in a value obtained from multiplication/division is:
$$\left(\frac{\Delta Z}{Z}\right)^2=\left(\frac{\Delta A}{A}\right)^2+\left(\frac{\Delta B}{B}\right)^2$$
If your mass is 25 kg then the resulting force is 245 +/- 3 N.

 

[HallsofIvy]

A perfectly valid way to do this is to add the largest possible value for each and then add the smallest possible value for each.
In this case, one number is 5 +- 0.2 so the largest it could be is 5.02 and the smallest 4.98. The other is 6 +- 0.1 so the largest it could be is 6.1 and the smallest is 5.9. 5.02+ 6.1 is the largest the sum could possibly be and 4.98+ 5.9 is the smallest it could possibly be. Take the midpoint of those two numbers and calculate the possible errror from those.

Of course, what andrevdh suggested will give you (approximately) the same answer. It's an old engineer's "rule of thumb"- "when adding two measurements, the errors add; when multiplying two measurements, the relative errors add".