Percent Error and Error Propagation

[Bigjoemonger]

I have a radiation source.
I record a dose rate measurement of that source.

1 year later I correct the previous measurement accounting for the radioactive decay of the source.
I take a new measurement and verify it is within +/-5% of the decay corrected value.
Let's say i take a measurement and it has a percent error of +2%. This is within +/- 5% so I accept the measured value as the new value.

1 year later I decay correct the previous accepted value.
I take a new measurement and verify it is within +/-5% of the decay corrected value.
Lets say the new measurement has an error of +4%. This is within +/-5% so I accept it.

What equation do I use to propagate the total error from the original measurement to the current measurement?

Do I just add the percentages, so it'd be a total error of +6%. Or do I do something like sqrt(2^2 + 4^2) for a total error of 4.47%. Or something different?

 

[tkhunny]

Addition will not serve you well. If it helps, treat it like a compound interest problem with a varying interest rate.

1 <== Beginning, EXACT measurement.

1 * (1 + 0.02) = 1.02

1.02 * 1.04 = 1.0608 <== This models the 2nd year error as partly from he base measurement and partly from the error on the error.
In other words:
----- 1) 2% from last year.
----- 2) 4% from this year.
----- 3) 0.08% which is 4% of last year's 2%

sqrt(1.0608) = 1.02995 <== Average 2.995% error / year
Are we concerned that the total error is over 5%? 6.08% > 5%.

 

[Bigjoemonger]

Correct. I am concerned that by decay correcting and doing +/-5% and then decay correcting that and doing +/-5% again then it's not accurately representing the error of the measurement.

Currently I'm thinking it would be better to use the first measurement as the baseline for all successive years.
So after 1 year you decay correct the original measurement 1 year.
Then after two years you decay correct the original measurement by two years.
Thus preventing the two year measurement from being impacted by error in the 1 year measurement.

 

[tkhunny]

The only difficulty in what you have presented is in your error going the same direction twice in a row. Two years isn't a great sample, but it leads to an important discussion.

Typically, if your theory of decay is reasonable, you should get variations above and below your expectation. If you keep getting error in only one direction, you should reexamine your theoretical model. Thus, it may be instructive to alter the base value every year, but there is also value in tracking the overall error across all study years.

 

[JeffM]

This is one of the most common errors in understanding error propagation.

[MATH]855 \le u \le 945.[/MATH]
We know u is within plus or minus 5 percent of 900.

[MATH]950 \le v \le 1050[/MATH]
We know v is within plus or minus 5 percent of 1000.

[MATH]\therefore -945 \le - u \le -855 \text { and } 950 \le v \le 1050 \implies[/MATH]
[MATH]950 - 945 \le v - u \le 1050 - 855 \implies 5 \le v - u \le 95[/MATH].

The possible error in our estimate of v - u is 95%.

Let's try it with 1% accuracy.

[MATH]891 \le u \le 909.[/MATH]
[MATH]990 \le v \le 1010[/MATH]
[MATH]81 \le v - u \le 119[/MATH]
The possible error is 19%.

To get close to accuracy of 5% on that difference, we need accuracy of approximately 0.3% in the underlying variables.

[MATH]897.3 \le u \le 902.7.[/MATH]
[MATH]997 \le v \le 1003[/MATH]
[MATH]94.3 \le v - u \le 105.7[/MATH]
The possible error is 5.7%.

That is the reason that science demands very accurate measurements and why accountants are obsessive over seemingly small amounts.