Some hints for the Radioactive Decay lab

Transcription

1 Some hints for the Radioactive Decay lab Edward Stokan, March 7, 2011 Plotting a histogram using Microsoft Excel The way I make histograms in Excel is to put the bounds of the bin on the top row beside the column of data, like: Now, the goal is to make is so that the bin is 1 if the counts number falls inside the bin. Looking at the first row for the data, row 4, the number of counts is 2, and the bin with 2 shows one, which is good. The other bins for that trial show zero, which is also good. To get this, you'll need a formula that uses an AND() statement inside an IF() statement. Here's what I used: =IF(AND($C4>=E$3,$C4<F$3),1,0) What this is saying is that, if C4 is >= E3 AND C4 < F3, then the result is 1, otherwise the result is zero. Also note the placement of $ to ensure that the same formula can be used for all bins and all trials. It's easy to check if you've done this right by looking to see if the correct bins have a one for each trial. When your bins are populated correctly, sum each column to plot as a vertical bar graph to produce your histogram: (1)

2 From here, you can add axis labels, a title, change the width of the bars, most importantly, change the label for each bin on your histogram. Note that the first bin is always labelled 1, which is definitely wrong. You can change this by right-clicking one of the bars on the histogram, selecting Select Data and changing the Horizontal (Category) Axis Labels on the right: This is what it should look like when you're done. Blank Sample Radioactivity Counts Number [0,1) [1,2) [2,3) [3,4) [4,5) [5,6) [6,7) [7,8) Radioactive Counts Recorded Plotting distributions on the same graph First, start by finding the mean and standard deviation for the recorded data using AVERAGE and STDEV. Next, make a column for all possible value of counts recorded. I went from zero to eight in steps of 0.1 for the blank sample using the Fill function. The Poisson distribution related to each counts value can be found by using the POISSON function, while the normal distribution can be found using the NORMDIST function. Some examples are given on the next page. (2)

3 So: =POISSON(B58,$C$52,FALSE) =NORMDIST(B58,$C$52,$C$53,FALSE) The first number in each function call refer to the counts number, zero for B58. The second number sent to the function is the mean of the observed data, 2.22 stored at C52. The third value for the NORMDIST function is the standard deviation of the observed data, calculated using STDEV, and stored at C53. Finally, both function calls say FALSE because you do not want the cumulative distribution. If you've called the functions correctly, they'll produce numbers smaller than one for each count value. This is because the distributions are normalized such that if you integrate their entire domain, they'll add to one. This is clearly not the case with the data. The area under the data histogram is equal to the sum of the bin widths multiplied by the bin heights. Thus, the area under the histogram is 45. The correct distribution values can then be found by multiplying each distribution value by the area under the graph, 45, stored at F53 in the spreadsheet above: =C58*$F$53 =D58*$F$53 One final step before plotting anything is to add a second counts column beside your normalized Poisson and normal distribution columns. I've done that with column F in the chart above, we'll require this to plot the data correctly, since the axis for the bar graph will be translated horizontally compared to the distributions. Now for plotting, right click your bars on the histogram and click Select Data. Click add and for series values, highlight the column of normalized Poisson distribution values, column G in the screenshot above. Set the name to "Poisson" or something to not be confused, later. At first, the chart will look like crap since it'll try and plot it like a histogram, so right click the red or Poisson bars, select Change Series Chart Type, and set it to some sort of X Y (Scatter) plot. It'll still look like crap because the horizontal axes of the graphs don't match at all. To fix that, right click your Poisson data again, select Select Data, edit (3)

4 your Poisson series, and set the x-values to the duplicated counts value, i.e. column F. It should now look better, but still offset by a bit, so this can be fixed by adding a small offset to the duplicated counts values in column F, like 0.5 or so. The chart should then look good! This is the progression of how the graph changes if the steps are followed: The normal distribution is then easy to add. Right click one of the Poisson bars and select Select Data, go to add a series. For the x-values, select the offset counts values (i.e. column F) and for the y-values, select the scaled normal distribution (i.e. column H). It should look correct from there without any modifications. From here, a legend may be added, and the type of graph, markers, or lines can be played with to get something nice: Blank Sample Radioactivity Counts Number [0,1) [1,2) [2,3) [3,4) [4,5) [5,6) [6,7) [7,8) Radioactive Counts Recorded Measured Poisson Gauss (4)

5 Other stuff The rest of the lab should be relatively straightforward. For b) Radioactive Source counts, question 3., when estimating the probability, explain how you do this. One way might be to determine the area of the bars between n and n + 50, then dividing that by the total area of the histogram, which will give a probability, but you can also take fractional areas, or other methods. Make sure to explain this. When figuring out the same thing with the normal distribution, you can use the normal distribution table, or calculate it numerically using Maple, MATLAB, or even Excel. To figure it out using Excel, you'll have to use NORMDIST again, but with the FALSE option turned to TRUE to give the cumulative distribution: =NORMDIST([mean + 50],[mean],[standard deviation],true) - NORMDIST([mean],[mean],[standard deviation],true) I think that's what the expression may look like, but I haven't checked to see if it's correct, so make sure it makes sense (i.e. no negative numbers!) before you hand it in. Also, this is a long-format lab, even though I said it wasn't... sorry about that! (5)