How Do Uncertainties Combine in Measurements?

[UrbanXrisis]

Measurments taken from two different values have to be added to get a final result. The two values are 5.7+/-0.3cm and 6.8+/-0.4cm

the uncertainty in the final result would be?

Do I just add up the uncertainty to get the final uncertainty?
What if I was aked to multiply them?

 

[HallsofIvy]

Do you know what "uncertainty" means? Saying that a value is "5.7+/- 0.3 cm" means that it may be as low as 5.7- 0.3= 5.4 cm. or as high as 5.7+ 0.3= 6.0.
Similarly, 6.8+/- 0.4 cm could be as low as 6.8- 0.4= 6.4 cm or as high as 6.8+0.4= 7.2 cm.

Suppose you added those two largest values? Do you see that adding any other possible values would be less than that? Suppose you added the smallest values. Do you see that adding any other possible values would be more than that? What is the range of possible sums?

Actually you are correct that you can just add the "uncertainties".

If one is "a+/- x" and the other "b+/- y", then the largest the first number can be is a+x and the largest the second number can be is b+y. Adding those, we get a+ x+ b+ y= (a+b)+ (x+y) as the largest possible sum. The smallest the first number can be is a-x and the smallest the second number can be is b- y. Adding those, we get a-x+ b- y= (a+b)- (x+y). In other words, all possible sums lie between (a+b)-(x+y) and (a+b)+(x+y) which we can write as (a+b) +/- (x+y). The uncertainty of the sum is the sum of the uncertainties.

Multiplying is more complicated. Taking the same situation as above, (a+x)(b+y)=
ab+ bx+ay+ xy and (a-x)(b-y)= ab- bx- ay+ xy. Because of the xy term, that cannot be written as "ab +/- something"- ab is not in the middle. However, normally the "uncertainties", x and y, are small so xy is even smaller. If we simply ignore that (very small) number, we have ab +/- (bx+ay). If we divide that "uncetainty", bx+ay, by ab itself, we have (bx+ ay)/ab= x/a+ y/b.

The uncertainty divided by the number itself is called the "relative uncertainty". That's were we get the engineer's "rule of thumb": "If measurements are added, the uncertainties are added. If measurements are multiplied, the relative uncertainties are added". (The term "error" is more often used for your "uncertainty").

That's a "rule of thumb" because it is not exactly true- it's ignoring that "xy" term.

 

[M98Ranger]

In this case, the uncertainty in the final result would be the sum of the individual uncertainties. So the final result would be 12.5+/-0.7cm. This is because when adding or subtracting measurements, the uncertainties are also added.

If you were asked to multiply the two values, the uncertainty in the final result would be calculated using the relative uncertainties. The formula for this is: (uncertainty in A/A) + (uncertainty in B/B), where A and B are the two values being multiplied. So in this case, the final result would be (5.7/5.7)+(6.8/6.8)= 12.5+/-0.8cm. It is important to note that when multiplying or dividing measurements, the relative uncertainties are added.