# Square Root of N Law of Random Counts

1. Sep 18, 2008

### tony873004

"Square Root of N" Law of Random Counts

According to my lab manual, "Theory says that radioactive decay obeys statistics for which the standard deviation of the counts per time interval is equal to the square root of the mean number of counts for that interval, for cases when the mean is a large number."

I think they mean for cases where N is a large number.

In lab, taking counts from the geiger counter, I got an average of 616.56 counts per minute with a standard deviation (computed with Excel: stdev(my column of counts) of 30.74 from 25 sets of individual 1-minute counts.

But the sqrt(616.56) = 24.83, not 30.74. Thinking that it would reach about 30.74 if I had a larger N, I set up a Monte Carlo simulation on my computer. I generated N random gauissian numbers with a mean of 616.50, and a standard deviation of 30.74. But no matter how high I make N, even 1 million, I always get an answer of about 24.8 when I square the average of my random numbers.

Additionally, Googling for "Square Root of N Law of Random Counts" turns up nothing. Did the author of my lab manual just make this up?

Here's how I modeled it with the computer
Code (Text):

totalcounts = 0
n = 1000000

For k = 1 To n
counts = Grnd(616.56, 30.74)
totalcounts = totalcounts + counts
Next k

meancounts = totalcounts / n
Label1.Caption = Sqr(meancounts)
End Sub

Function Grnd(Mean As Double, sd As Double) As Double
Dim x1 As Double, x2 As Double, w As Double, y1 As Double, y2 As Double
Do
x1 = 2 * Rnd(1) - 1
x2 = 2 * Rnd(1) - 1
w = x1 * x1 + x2 * x2
Loop Until w < 1

w = Sqr((-2 * Log(w)) / w)
y1 = x1 * w
y2 = x2 * w
Grnd = Mean + y1 * sd
End Function

2. Sep 19, 2008

### Hurkyl

Staff Emeritus
Re: "Square Root of N" Law of Random Counts

The manual is right; your counts should be given by a Poisson distribution. Remember that you aren't computing the true standard deviation; you're merely estimating it from a sample, and all of the various sorts of errors you're familiar with can apply.

3. Sep 19, 2008

### D H

Staff Emeritus
Re: "Square Root of N" Law of Random Counts

Radioactive decay is not a Guassian process. Think about it this way: Suppose you create a sample of some radioactive substance such that at the moment of creation you have a pure sample. If radioactive decay was a Gaussian process, some of that sample would have before you created it!

Radioactive decay, as Hurkyl mentioned, is a Poisson process. The mean and variance of a Poisson distribution are equal to one another. The standard deviation is the square root of the variance, and hence your lab book's statement that "radioactive decay obeys statistics for which the standard deviation of the counts per time interval is equal to the square root of the mean number of counts for that interval." The qualification "for cases when the mean is a large number" pertains to your ability to estimate the mean, which improves (relatively speaking) as the count increases.

4. Sep 19, 2008

### tony873004

Re: "Square Root of N" Law of Random Counts

That's interesting, because under "Objectives" in our lab manual it says "Observe some properties of the Gaussian distribution".

Thanks for the replies. I'll have more questions on this in a few hours when I work on this in my lunch break.

5. Sep 19, 2008

### CRGreathouse

Re: "Square Root of N" Law of Random Counts

Well, with the help of the Central Limit Theorem, anything can help you observe the properties of the Gaussian distribution...

6. Sep 19, 2008

### tony873004

Re: "Square Root of N" Law of Random Counts

I've turned in the lab, but I'm still interested in knowing this.

Another student asked the teacher in office hours and came away with the impression that if our N's were simply larger, that
"standard deviation of the counts per time interval is equal to the square root of the mean number of counts for that interval"
would be true. That was my first guess, prior to making the computer model, because as I plotted standard deviation computed after each run vs. sqr(mean counts) after each run, the lines did see to be converging towards each other, with slopes that gave me the impression I'd need about N=200 before they met.

But I find it strange that the number I computed with our lab data, and the number I generated using the N=1 million computer model gave virtually identical results (24.823 vs 24.830) for sqr(mean counts). Is this just a coincidence?

I still don't understand what a Poisson distribution means. The Wiki article doesn't quite do it for me.

So in my numerical experiment, was my mistake to use random Gaussian numbers, rather than Poisson numbers? What would I need to do in my numerical experiment to verify that stdev = sqrt(mean count)? Can I write a computer function to generate a Poisson distribution, similar to the "GRnd" function that I wrote (see 1st post) which generates random Guassian numbers?

-Thanks

7. Sep 19, 2008

### tony873004

Re: "Square Root of N" Law of Random Counts

From the Wiki article:
So it seems I computed standard deviation wrong when I used $$\sigma = \sqrt {\frac{{\sum\limits_1^N {\left( {x_k - \bar x} \right)^2 } }}{{N - 1}}}$$.

But in this case, I don't see the point of this part of the lab. Of course if I compute my standard deviation as $$\sqrt\lambda$$ then it will be the same as the square root of my mean counts since lambda is my mean counts. It's as if they want me to show that $$\sqrt \lambda = \sqrt \lambda$$. So how can I go wrong?

Last edited: Sep 19, 2008