PHY 445/6 / PHY 515/6

Error analysis

A single measurement

Assume you are supposed to measure the voltage between two leads. You connect your voltmeter, and it reads 9.7908789V. What do you know about this voltage? As long as you have only one measurement, you do not know much. A single measurement by itself is pretty much useless. You have no idea about the error of the value (the fact that the voltmeter shows so many digits, does not mean that the value is reproducible).

Of course, in real life you would leave the voltmeter on, and watch the stream of numbers you can read on the screen. You may observe something like the sequnce of numbers in the data file you have just downloaded. We will use Excel to analyze the numbers.

Random voltages

Highlight the first column in Excel. Click on "File", "Open" and select "Files of type" as indicated in the screen shot. Navigate the directories until you find the file you have downloaded. Open the data file in Excel.
At this point you have to save the file in your directory as an Excel (.xls) document. Use the "File", "Save as..." command
Let us investigate the nature of randomness by creating a distribution or bin plot. We want to count what fraction of the numbers fall within a given intervall, let's say between 9.1 and 9.3. Excel can do this, and more, automatically! Type in the cells C1 and C2 the numbers 9.1 and 9.3 , then highlight the two. You should see this:
Place the cursor to the lower right corner, and pull it down. You get this:
Click on "Tools", "Data Analysis...", "Histogram", and fill out the form like this:

When you click OK, the window disappears, and the result of the process appears in column C and D (see next figure).

Now we are ready to make a graph. Highlight the meaningful data range. Use the "chart wizard" (a little icon on the top). The first screen is like this:
Highlight "XY(Scatter)". Follow the steps. Always label the axes!!
You can edit the appearance of the graph by double cilcking on the scales, texts, etc. Finally, you have your graph.
What do we see on this graph? The voltage seems to fluctuate around the average of 10.1V. It has an (approximately) equal probability of being anywhere between 9.6V and 10.6V. In retrospect, after collecting so much data, we can tell that the 9.7908789V we measured in the first instance is not that accurate after all. In fact the best way to report that voltage is U=10.1 +/- 0.5 V (The symbol "+/-" stands for "plus-minus")

Of course, if the measurement is repeated so many times, we can use all that information to improve the accuracy of the voltage. What is the "best" value of the voltage? The average of all the voltages is a good guess. But how accurate is the average? Move on to the next page to see the answer. Save your workbook before you move on.

This page was created by Laszlo Mihaly. Las updated 1/8/98.