Why do we divide by uncertainty value to get the minimum possible outcome?

Question

I’ll explain it with an example, say there is a cylinder whose diameter is known with a +/- 5 percent uncertainty, so this means the actual value could be at maximum 5% higher than the nominal value and the min 5% lower. What I’ve seen in a textbook is that in order to get the minimum possible value, they divided the nominal value (let’s call it d) by 1+ 5 percent : min=d/1.05

Why can’t they just multiply the nominal value by 0.95 just like how we multiply d by 1.05 to get the max value?

In: 2

Answer 1

We know that any measurement we take has a possible error of plus or minus 5%. In other words, the value we get as a measurement is somewhere between 95% of the true value and 105% of the true value.

If we want to figure out the range of true diameters that could possibly result in this measured diameter, we know that we have

(True Diameter) * 105% = (Measured Diameter)

And

(True Diameter) * 95% = (Measured Diameter)

As our bounds. The first equation is the lowest possible true diameter, and the second is the largest possible true diameter, because we know that the measured diameter can be no more than 105% of the true diameter and no less than 95% of the true diameter.

This is why we divide the measured diameter by 1.05 to get the smallest possible true diameter. If we assume a diameter any smaller than that, adding 5% in the positive direction still doesn’t get us up to our measurement.

This is a situation where it might actually be easier to assume your measurement capability is worse. Imagine that, instead of having a 5% margin of error on your measurement, it’s 50%.

Further imagine that the measurement you get is diameter = 1.

The smallest possible value of the true diameter is given by the case where your measurement error was + 50%, as in the measurement was 50% higher than the true value. This would be a true diameter of 0.667, because 1 is 50% more than 0.667. The largest possible value of the true diameter would be the case where your measurement was actually – 50%: assuming your measurement is 50% less than the true diameter means the true diameter is 2.

The reason the bounds are different is that if you start at a large number, and you move down by 50%, that’s a bigger difference than if you start at a small number and move up by 50%.