lab report physics

Measurements and Plots

As we discovered last semester, no measurement is without error (especially in this lab) and the nature of taking measurements can be subjective. How can we take measurements such that the numbers we obtain make sense? If you were to quote a measurement to someone else how would you determine the number? How would you determine the error? How comfortable would you be that the number and error you gave are accurate?

As an introduction to this semester's physics labs you will take a series of measurements. You will measure 3 different quantities. Whatever quantities you choose will be measured 20 times each. The first quantity will be displayed as an average and standard deviation. The second quantity will be displayed as a histogram and the third quantity will be displayed as a cumulative distribution function.

We will initially meet up in the physics lab for a chat before you go and take various measurements. You can measure whatever you want and use whatever equipment you want. In the Monday night section we will go down to the nature trail (weather permitting) and so sensible shoes are advised. (A map of the RMU 100-acre woods is provided at the back.)

2. Background

How representative is your measurements?

Imagine you wanted to measure the length of a hand. You take out your ruler and measure the length of your friends hand and determine it to be 20 cm. Are all hands 20 cm? Obviously not. So you could take the average and standard deviation. Right? This would tell you how big on average a hand is and how variable the size of peoples hands are. But women statistically have smaller hands than men. Would the average give a good representative measurement for the size of an individual hand?

Here's a series of measurements of the length of weaver ants. The bimodal distribution of sizes of weaver ant workers, shown in the figure, arises from the existence of two distinct classes of workers, namely major workers and minor workers.

Say you took several measurements and you obtained the average size of a weaver ant to be 7 mm. Obviously your standard deviation would be large, but even your average doesn't represent that many ants, with few ants having a length of 7 mm. To represent the size of the ants you really need to delve deeper into the numbers.

What if you had two variables (say, the length of an ant and its speed)? What if there were lots of peaks (a multimodal distribution)? Things could get very misleading if the statistics are misrepresented!

Always remember an experiment should tell a story. Your objective in performing these experiments is to reveal new information and convey this information to the reader. You don't want to “get the right answer” as there is no right answer, you want to present your measurements in a clear way. Does the way you present your measurements clearly depict what you are measuring?

Phys 2215 Laboratory Manual 2

3. Equipment

Something with which to make a measurement.

4. Procedure

You will measure three different quantities (could be the size of something, the time it takes for something to occur, the temperature of something,...) and present these measurements in a clear way so that the reader understands the distribution of your measurements. Weather permitting we will head outside and measure things in the natural world, as nature is inherently diverse. Choose quantities that you believe will highlight this diversity!

The first requirement is that one of the quantities you present should be represented using the average and standard deviation (described below). Measure this quantity 20 times.

The second requirement is that one of the quantities you present should be represented using a histogram (described below). Measure this quantity 20 times.

The third requirement is that one of the quantities you present should be represented using a cumulative distribution function (described below). Measure this quantity 20 times.

5. Analysis

Average and Standard Deviation

A distribution of measurements can be presented through the moments of the distribution. Usually we only care about the first and seconds moments – the average and standard deviation. Although higher moments exist such as skewness and kurtosis. In a spreadsheet these can be calculated very easily using the functions AVERAGE() and STDEV().

Histograms

As mentioned earlier, the data might not be centered around a given value but have a more complicated structure. For example, bimodal distributions where the data is centered around two different values. Therefore, histograms are a wonderful way of displaying this kind of data.

A histogram plot depicts the frequency with which certain measurements occur. In other words, the x-axis displays “bins” or ranges over which your data might fall. On the y-axis is plotted the number of times, or frequency, with which your data falls into these bins. For example, in the figure shown there is 3 times that the measured data fell in between 65 and 70, and so above this bin on the x axis is a bar of height 3.

Plotting histograms in a spreadsheet generally involves counting yourself how many of your measurements fall in to a given bin and creating a separate list of the bin range in one column and the number of times a measurement fell in to this bin (the frequency) in the adjacent column. Then you can plot this data using a bar graph. This can be cumbersome for large data