# Estimating standard deviation of radioactive decay

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.

New contributor
Jirwe is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
• 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 2 days ago
• 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 2 days ago
• The estimated error on $N$ counts is the square root of $N$ if the number of counts is large enough. – Pieter 2 days ago
• Frankly, you'll have to ask your instructor to find exactly what they're looking for. – Emilio Pisanty 2 days ago

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.
• 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 2 days ago