# Estimating standard deviation of radioactive decay [closed]

I recently did a laboratory exercise where we had an amount of silver containing two isotopes. Silver with 108 and 110 neutrons, respectively. Using sensors and software, we measured the amount of emitted electrons from the silver in question. We made a measurement every 5 seconds for 900 seconds. By plotting emissions against time, we got a plot looking as follows:

But the problem is that we need to have error bars on our measurements, and it's been a year since I studied a statistics course, so I'm more than a little rusty. The error bars should be ± 1 standard deviation, I believe.

As far as I can recall, since this is about radioactive decay, generally speaking, a Poisson Distribution is a pretty good fit. The Poisson distribution is pretty neat in that the variance of the distribution is the same as the mean. But of course, I don't know the distribution so I have to estimate it from my sample.

The total amount of emissions measured is about 6500, and the measured time is 900 seconds. My first thought was, perhaps naively, that the sample mean using the maximum likelihood approach, would be 6500/900 $$\approx$$ 7. As mentioned in this post:

https://math.stackexchange.com/questions/780198/how-do-you-estimate-the-mean-of-a-poisson-distribution-from-data

This sounds easy and comfortable and would mean that the standard deviation would be around 2.6. But it would also mean that the error bars would be of constant length, which I feel doesn't really make sense if I look at my graph, where the oscillations are much greater near the end of the graph. I feel that the error bars there should be larger. Am I wrong in this?

Any help is appreciated, thank you.

EDIT: Since I was unclear in the post, according to the instructions for the laboration exercise, I'm supposed to have error bars on the raw data. To quote the instructions: "Determine the statistical error for each data point and show the errors in the plot."

We are also supposed to give the statistical and systematic error for the half-lives themselves, but that's another thing.

## closed as off-topic by Emilio Pisanty, Jon Custer, Aaron Stevens, ZeroTheHero, ChairNov 13 '18 at 14:33

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• Some statements in your question confused me. Please clarify: You're asking for the error bars on the raw data points shown on your plot, and not for the error bars on the value of the half-life of the sample, which I presume was the final goal of the experiment? – Samuel Weir Nov 10 '18 at 18:31
• Ah, my apologies. As I understood the exercise, since they specifically tell me to first use the the function errorbar in matlab to plot the error bars, I assume they mean error bars on the actual raw data points. It does say that further into the exercise they also want the statistical and systematic errors associated with the half-lives. But it is not stated in such a way that they necessarily want an interval there. To answer your question: on the raw data, as I understand. – Jirwe Nov 10 '18 at 18:44
• The estimated error on $N$ counts is the square root of $N$ if the number of counts is large enough. – Pieter Nov 10 '18 at 21:20
• Frankly, you'll have to ask your instructor to find exactly what they're looking for. – Emilio Pisanty Nov 10 '18 at 23:23

## 1 Answer

If there are a large number of radioactive Ag atoms in the sample, then you can expect that only a very small fraction of them will decay in any 5 second interval. Also, the decay of any one atom is independent of the decay of any other atom. So a Poisson distribution is the appropriate one for describing the variance of the decay count in any such 5 second interval. So for the first data point on the plot which looks like it has a y-value of 375, the variance is 375 and the standard deviation is $$\sqrt{375}$$ ≈ 19.36 . So using one-sigma error bars, this data point would be described by a y-value of 375 +/- 19. Just use the same procedure for all the other data points.

• So the errors do vary across the plot, as I thought! So if I understand correctly, in this case the sample mean for every 5 second period is simply the measured value. Is that from the maximum likelihood estimation? And from that, since for a Poisson distribution the variance is the same parameter as the mean, the standard deviation is simply the square root of that? It almost feels too straightforward. Does this approach still hold when we get further to the right in the graph? Because at that point the measured electrons are relatively few in number. Closer to 10 than the 300 from the start. – Jirwe Nov 10 '18 at 21:36
• Yeah, it crossed my mind that there may not be many radioactive Ag atoms left out towards the right end of the plot and so the argument for using a Poisson distribution might break down there. But, I wouldn't worry about it because (1) that portion of the plot contains relatively little information and wouldn't be the focus for determining the half-life and (2) using the Poisson distribution there results in reasonably sized error bars, except for the case when the count is 0. So I would use error bars which are either equal to sqrt(count) or 1, whichever is greater. – Samuel Weir Nov 10 '18 at 21:58
• No, it is a Poisson distribution when the counts are small it is just that using a simple $\pm \sqrt{n}$ error bar that becomes suspect. – Rob Jeffries Nov 10 '18 at 23:23
• Thank you! I think that answers my question. I went ahead and marked this as the accepted answer. But I'm curious. Since this question was only about getting error bars on the raw data, how would you go about getting a statistical error for the half-lives if you were to roughly approximate it by looking at the graph? Since all the fluctuations are on the raw data, I mean. – Jirwe Nov 11 '18 at 9:23
• To get the half-life, fit the data on the plot to the model, an exponential decay function. There are many data analysis programs out there that can do least-squares fitting of functions to data points with different errors or weights (e.g., Igor Pro, Mathematica). You may want to restrict the fitting to times less than about 300 sec since the data after that may be dominated by noise because it's so close to zero. Alternatively, you could re-plot the data and error bars on a semi-log plot and the data should fall on a straight line and maybe you can estimate the half-life error by eye. – Samuel Weir Nov 11 '18 at 18:21