Calc Errors: Calculating Tolerance in Physics

[Jimmy87]

Homework Statement

Hi, please could someone give me some guidance on how tolerance should be calculated in physics.

If given the following statement: (x-t) is proportional to p and then asked to show whether or not your measurements of these values support this statement within a tolerance of 15% how should this be done as I have thought of several ways to do this and didn't know which one is considered the best for tolerance. This task involved me taking two separate measurements of x, t and p. Here are my results:

x1 = 22.1, t1 = 5.3, p1 = 8.3
x2 = 41.5, t2 = 5.3, p2 = 18.7

Homework Equations

The Attempt at a Solution

If we know that (x-t) is proportional to p then we can work out the constant of proportionality (k) for both. So we have (x-t)/p = k. For the first set of measurements this gives (22.1-5.3)/8.3 = 2.02. For the second set of measurements we would have (41.5-5.3)/18.7 = 1.94.

Therefore, k1 = 2.02 and k2 = 1.94
To see if these are within 15% tolerance I don't know what to do as I can think of several ways of doing it.

First, you could find the percentage difference, e.g. (2.02-1.94)/2.02 x 100 = 3.96% but then how do you know whether to use k1 or k2 on the denominator?

Second, you could use k1 to find an expected value in the second set of measurements and compare this to the actual measurement. Example - Use k1 to find an expected value for p2:

p2exp. = (x2-t2) / k1 = (41.5-5.3) / 2.02 = 17.9

We know the measured value of p2 is 18.7 so we can use the following equation:

(measured value - expected value)/ expected value x 100

(18.7-17.9) / 17.9 x 100 = 4.5%

Third, you increase the lower valued constant by 15% and see if this is within the constant with the higher value. Example for k2 - 1.94 x 1.1 = 2.13. Since 2.13 is larger than k2 we can say that the tolerance is within 15%.

These are three different ways I got from a combination of my own thinking about it and looking online. I just wanted to know if all three methods are valid and which one is considered the best for calculating tolerance in physics.

 

[mfb]

All three methods are fine to make some rough estimate, and none of them would be used for an actual publication.