Experiment 3A. Measurement of Resistance – Statistical Error
NOTE: In order to fully understand this, read the discussion of experimental error
provided as a separate section.
Prior to Lab: Solve the following problem to hand in at the beginning of lab: A simple
pendulum is constructed from a mass and a string. The pendulum is
determined to be 30.0 cm long ± 0.50 cm.
a. If the acceleration of gravity is 980.0 cm/s2 ± 0.10 cm/s2, what is the
measurement uncertainty in the period of the pendulum?
b. In lab the period of the pendulum is determined to be 1.11 s, 1.09 s, 1.11 s,
1.10 s, 1.10 s, 1.14 s, 1.11 s, 1.08 s, 1.10 s, 1.10 s in 10 trials. Determine
the standard deviation for this data.
c. How does the measurement uncertainty found in part a compare with the
statistical uncertainty in part b?
Object: To determine the accuracy of resistors made by a manufacturer and apply
statistical error analysis to the result.
Apparatus: DMM and one hundred 100 Ω, 5% tolerance resistors.
NOTE: In order to fully understand this lab and the following lab, read the
discussion of experimental error provided as a separate section.
Discussion: Normally the time spent in physics lab is too limited to take repeated
measurements and investigate the statistical behavior of random error, but this week
we will take time to do a simple experiment which will demonstrate the statistical
nature of random error.
We will measure the resistance of 100 resistors with a DMM and look at the
statistical nature of the set of measured values. We will attempt to answer questions
related to a relatively small sample (10) and a larger sample (100) of measurements.
The most common methods of reporting the measurements are through the
mean (average) and the standard deviation from the mean which includes roughly 64%
of the measurements (See your handout on Experimental Error.)
Procedure:
1. With the DMM turned on and switched to the 200 Ω range, measure the resistance
of 100 resistors.
2. Record this data in a column in the order that you measure the resistances. Use the
spreadsheet program (Excel).
3. Select the first 10 resistors and duplicate these measurements in a second column.
Analysis:
Do the following for each column of data.
1. Using the function AVERAGE() determine the mean for each of these columns of
data.
2. Using the function STDEV() determine the standard deviation from the mean for
each of these columns of data.
3. Count the number of times that each value of resistance occurs (for example,
98.1 Ω occurs 5 times, 98.2 Ω occurs 4 times, 98.3 Ω occurs 7 times, etc.). Plot
a vertical bar graph of the number of occurrences (y-axis) versus the value of
each resistance (x-axis). (This chart is a histogram.)
Report: Answer the following questions in your report.
1. Are the histograms approximately Gaussian (a bell shaped curve)?
2. Which set of data is more likely to be Gaussian, or doesn’t it matter?
3. What is the mean and standard deviation for each data set?
4. Are 64% of the measurements within the range of plus or minus one standard
deviation from the mean?
5. Is the manufacturer’s value equal to the mean? Is it within the range of values
delineated by one standard deviation away from the mean?
6. What do you suppose the manufacturer’s stated tolerance of 5% means?
A question for each of you to think about: If the size of a class is 15 students, do
you suppose that a randomness of distribution of grades would apply? What if the
class size is 50? 100? 1000? DO NOT ANSWER THIS QUESTION IN YOUR LAB
REPORT!
EXPERIMENT 3B: MEASURING POTENTIAL DIFFERENCE WITH A DIGITAL
MULTIMETER
AN EXERCISE IN SYSTEMATIC ERROR
Object: To determine the potential difference across a resistor, and determine which
scale provides the most accurate reading.
Discussion: Instrumentation is a necessity for making any kind of measurement in
experimental science. However, the measurement is only as good as the
accuracy and sensitivity of the instrument. In any measurement there are two
sources of possible error. First is the accuracy of the instrument itself. This
accuracy is determined by the calibration of the instrument which is done either
by the manufacturer or by a calibration lab. In the case of general physics lab,
we use the manufacturer’s calibration. The second source of error is the experimenter’s ability to read the instrument (or data).
A digital multimeter (DMM) is a good instrument to use to explore this aspect of
measurement because of the availability of several scales of sensitivity. In this
lab, we will consider the accuracy of this common electrical measuring
instrument with relation to choice of scale, and ability to read the instrument.
Both these aspects are important to obtaining the most accurate measurement.
We will use the DCV ranges of the multimeter to obtain potential difference
readings across two resisters supplied with a nominal 3 volts from a battery
eliminator. The manufacturer claims an accuracy for the DMM of ±0.5% of
the reading for the DMM. In addition we don’t know whether the meter is
rounding up or down the last decimal position, so our reading error is ±0.5 digit
in the last decimal place for both our zero reading and our voltage reading for a
combined reading error of ±1 digit.
Figure 1. A wiring diagram for the experimental apparatus. The 3
volts is generated by the battery eliminator with the switch on it set
for 3 v. The zig-zag lines represent resistors, the rectangle marked
DMM is the digital multimeter, and the other lines represent wires
connecting the various elements of the apparatus.
Figure 2. The apparatus and wiring
arrangement for the experiment. The DMM is
connected to the resistor board to measure the
potential difference (voltage) 180 Ω resistor.
Figure 3. The same apparatus as Figure 1,
however, the DMM is now properly connected
to measure the potential difference (voltage)
across the 1000 Ω resistor.
Experimental Procedure:
1. Attach the battery eliminator to the end terminals of the board containing the
1000 Ω and 180 Ω resistors.
2. After checking with your instructor, plug in the battery eliminator.
3. Measure the potential difference across each resistor on the 200 V, 20 V, and
2 V ranges as necessary to fill in the following table. Begin measurement
with the least sensitive scale (the scale with the largest range). Do not
use a scale that is smaller than the measured potential difference
determined on a less sensitive scale.
4. Determine the uncertainty of each measurement and record it. (Instrumental
uncertainty plus reading uncertainty: ΔV = 0.5% × reading + 1 unit in the
smallest displayed decimal position.) For example if your reading is 2.46 v,
your uncertainty is (0.005 × 2.46 v + 0.01v = 0.02 v). Your value for the
voltage is then 2.46 v ± 0.02 v.
5. Determine the fractional uncertainty of each measurement.
Range
Volts
V1000
(volts)
ΔV1000
(volts)
ΔV1000
V1000
(%)
V180
(volts)
ΔV180
(volts)
200 volt
20 volt
2 volt
Report: What do you conclude about the use of the DMM to measure voltage (potential
difference)? What can you conclude about systematic error?
What can you conclude about the voltages across the resistors?
ΔV180
V180
(%)