MC-1a  Measurement and Error

OBJECTIVES:
The major objectives of this first, short (est. 1 hr) computer lab are to begin to develop an understanding of what it means to make an experimental measurement and provide a methodology for assessing random and systematic errors in this measurement process. In addition this lab will also give you a minimal framework in which to introduce you to the PASCO© (page ) interface hardware and software.

THEORY:
By now you should have had numerous opportunities to become familiar with time and the concept of a time interval. The increment of one second will be used as an intuitive reference point. In this lab you will test your ability to internalize this one second time interval by making and recording a repetitive flicking motion with your finger. By flicking your finger back and forth you will move it though an infrared beam sensor (i.e. the PASCO photogate) and each full cycle (back and forth, approximately 2 seconds) will be simultaneously recorded, plotted and tabulated by the PASCO interface software.

Your goal in this experiment is to assess the size of systematic and random errors in your data set and learn a simple methodology for distinguishing between the two.
SYSTEMATIC ERRORS: These are errors which affect the accuracy of a measurement. Typically they are reproducible so that they always affect the data in the same way. For instance if a clock runs too slow you will make a time measurement which is less than the actual reading.
RANDOM ERRORS: These are errors which affect the precision of a measurement. A process itself may have a random component (as in radioactive decay) or the measurement itself may be technique limited which causes the reading to fluctuate. If many measurements are made then a statistical analysis of this data may be made and these errors minimized.
NOTE: In precision measurements it can be very difficult to isolate an eliminate errors, especially systematic errors.

APPARATUS:

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Computer with monitor, keyboard and mouse.

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A PASCO photogate and stand: This device emits a narrow infrared beam in the gap and occluding the beam prevents it from reaching a photodetector. When the beam is interrupted the red LED should become lit. (Plugged into DIGITAL CHANNEL #3.)

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A PASCO Signal Interface (CI-700) monitors the photodetector output vs time and can be configured to tabulate, plot and analyze this data.

PROCEDURE:
To configure the experiment you should refer to Fig. 1 below. Adjust the photogate so that one member can easily and repetitively flick his/her finger through the gap. The phone-jack cable from the photogate should be plugged into the DIGITAL CHANNEL #3 socket. Ignore the other sensors which may already be plugged into other sockets. It is important PASCO interface is turned on before the computer. If not the computer will not recognize it and, therefore, it must be rebooted to properly communicate with the PASCO module.

 

Figure 1: A schematic of the M0C components.

To initiate the PASCO (page ) interface software you will need to click the computer mouse when centered on the telescope icon in the ``toolkit'' area below. The bitmap image below gives a good idea of how the display should appear. Note that, while you are able to reconfigure the display parameters, the default values that are specified on start-up will allow you to do most of this experiment without necessitating any major changes.

You will note that a ``dummy'' first data set already exists on start-up showing a typical data run. In the table you can view all 35 data points and the statistical analysis, including mean and standard deviation. In addition there should be a plot of this data and a histogram.

 

Figure 2: The PASCO SCIENCE WORKSHOP display window

SUGGESTED PROCEDURE:

1.

Start the preliminaries by CLICKing on the MON icon and practice ``flicking'' a finger back and forth so that a two second interval appears in the window. CLICK on the STOP icon when done. (The upper left window is configured like a tape player). The same person need not perform both operations.

2.

Start the data run CLICK on the REC icon and stop it by CLICKing on the STOP icon. Each run gets its own data set in the ``Data'' display window. (If there are any data sets in existence you will not be able to reconfigure the interface parameters or sensor inputs.)

3.

With the REC option on, cycle a finger back and forth fifty times and STOP the data acquisition. DO NOT watch the time display while you do this.

4.

What is the mean time per cycle? What is the standard deviation? The mean $\overline{t}$ and standard deviation $\sigma$ are given by:

$$\overline{t}= \sum_{i=1}^N t_i/N \mbox{~~~and~~~~} \sigma = \sqrt{{\sum_{i=1}^N (t_i-\overline{t})^2}/[N-1]}~~.$$

Questions to consider:
 I. Is your mean suggestive of a systematic error?
II. Does your data qualitatively give the appearance of a normal distribution (i.e. a Gaussian bell curve.)

5.

For analyzing and quantifying the random errors one interested in assessing how a data set is distributed about the mean. The standard deviation $\sigma$ is one common calculation that does this. In the case of a normal distribution approximately 70% of the data points fall within $\pm 1 \sigma$ of the mean (90% within $\pm 2 \sigma$). Is your data consistent with this attribute?

6.

Assessing the possibility of systematic behavior is somewhat more subtle. In general $\sigma$ is a measure of how much a single measurement fluctuates from the mean. In this run you have made fifty presumed identical measurements. A better estimate of how well you have really determined the mean is to calculate the standard deviation of the mean $\overline{\sigma}= \sigma/\sqrt{N}$. After recording $\overline{\sigma}$ in you lab book, can you now observe any evidence that there is a systemic error in your data? Answer this same question with respect to the first ``sample'' data set.

7.

OPTIONAL: Systematic errors can sometimes drift over time. In the best-case scenario they drift up and down so that they hopefully average out to zero. (Clearly it would be better if they could be eliminated entirely.) With respect to $\overline{t}$ and $\overline{\sigma}$ for the first 25 and second 25 cycles do you observe any systematic trends. Use the ``region of interest'' feature of the statistical analysis software by using the mouse and highlighting (in black) data subset (through a CLICK and drag motion) over the rows of interest.