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# Combining Uncertainties in Advanced Higher Investigation watch

1. How do you go about combining a reading uncertainty and a random uncertainty?

I'm trying to calculate the uncertainty in an average acceleration, taken from five measurements of an acceleration.
For most of my results, the random uncertainty is zero or 0.002, while the reading uncertainty is 0.01, so I figure I can ignore it as it's less than a third. But I have the odd occurence when the random is 0.004 or 0.006, so I can't ignore it

So far, where the random can be ignored, I'm take the error as:
________________________
√ 5 * (Reading Uncertainty)^2

Since the average is calculated from adding five readings (with the same uncertainty) and dividing by five...But how would I incorporate the random into this too?

I was thinking maybe using percentage errors..? But that might get messy
2. Reading uncertainties are also random uncertainties.

either way you need to use the rule:

Var(Sum) = Sum(Var)

You want the varience in the Sum, and then since the mean = Sum/N, and N has no uncertainty, the error in the mean will just be the error in the Sum (root(Var(Sum))).

So you need to sum the variances. If the "Reading uncertainty" was the only error, then each reading varience is (Reading uncertainty)^2. And so you multiply by 5 and square root....exactly as you have done.

To incorporate the "random error", for any given reading, the total error will be Reading Error + Random Error. So for a given reading, the variance is (Reading Error + Random Error)^2

So to find the error in the mean acceleration, just do:

Root(Sum((Reading Error + Random Error)^2))

if you really want to just ignore all but one of the random errors then do:

Root(4*(Reading Error)^2 + (Reading Error + Random Error)^2)

hope that helps?

keep in mind that you are taking all these errors are "percentage errors"

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Updated: April 14, 2006
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