http://www.fusor.net/board/download_thread.php?site=fusor&bn=fusor_historynews&thread=1277302500

Hi Mark,

Your calculation looks good. I’m glad you got help from Rob Noulty; he is certainly a good source of information.

Statistical uncertainty is what Rob means when he says you have a 50% error (or equivalently, the example I made in a post upthread of 400 +/- 200 n/s). There are two ways to derive the uncertainty–experimentally, by making many trials of the same experiment and calculating the variance, which is impractical under the circumstances; or by application of the appropriate statistical model to calculate the “standard deviation”. Because the bubble detector is binary–when a neutron interacts you either get a count (bubble) or you get no counts–the appropriate statistical model is the binomial distribution, whose standard deviation is

sigma = SQRT(x * [1 – p])

where x is the experimentally-measured value, e.g. 4 bubs, and p is the probability of detecting a neutron incident on the BTI. Upon the further simplification that p is small, which it is since on the order of one in a million neutrons incident will form a bubble, the standard deviation simplifies to

sigma = SQRT(x) = SQRT(4 bubs) = 2 bubs

This practically means if you were to repeat this same experiment many times, you would expect various results (as calculated from the Poisson CDF, see Wikipedia’s Poisson distribution article):

0 bubs: once per 56 experiments

1 bubs: once per 14 experiments

2 bubs: once per 7 experiments

3 bubs: once per 5 experiments

4 bubs: once per 5 experiments

5 bubs: once per 6 experiments

6 bubs: once per 10 experiments

[etc.]

Quantities that are derived from the measurement, e.g. neutron source rate, total fusion rate, carry uncertainty propagated through the calculation from all the constituent experimental values. If there is a lot of uncertainty in the detector’s effective position (and there will be, because it is close to the source and has considerable geometric extent), that will enter the calculation too. Calculating the total number of fusions by multiplying the neutron source rate by two (to account for the aneutronic H-2(d,p) reaction) is only a very coarse approximation.

Convention is to report values rounded to the nearest statistically-significant figure. If the neutron source rate is 364.2 n/s and the uncertainty is 182 n/s, the reported value is “400 +/- 200 n / s.”

-Carl

BTW, Carl’s blog is awesome: http://carlwillis.wordpress.com/

]]>Also, you may want to consider the factor of what your sensor is measuring – presumably you put it 7 cm away from the reactor, but how did you orient your sensor, (radially, tangentially) etc. Although again, this stuff will all fall in the error of your number of bubbles detected.

]]>Ah, I see, it’s from the standard deviation: just 1/sqrt(N) (sigma_mean = sigma/sqrt(N), I assume sigma == 1 here)

http://en.wikipedia.org/wiki/Standard_deviation#Relationship_between_standard_deviation_and_mean

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