Error Analysis
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Error in this context does not mean mistake but rather refers to the
uncertainty in a measurement. All measurements in practice and even
in principle have some error associated with them; no measured quantity
can be determined with infinite precision.
Statistical Errors (also known as Random Errors)
Most measurements involve reading a scale. The fineness of the scale markings
(how close together they are) is limited and the width of the scale lines is
nonzero. In every case, the final reading must be estimated and is therefore
uncertain. This kind of scalereading error is random since we expect that
half of the time the estimate will be too small, and the other half of the
time the estimate will be too large. We expect that random errors should
cancel on average, that is, many measurements of the same quantity should
produce a more reliable estimate. Statistical errors can be controlled by
performing a sufficiently large number of measurements.
The error estimate on a single scale reading can be taken as half of the
scale width. For example, if you were measuring length with a scale marked
in millimeters, you might quote the reading as 17.0 mm ± 0.5 mm.
If you measured the same length many times, you would expect the error on
the measurement to decrease. This is indeed the case.
The best estimate of the measured quantity is the mean or average of all the
measurements. Simply add all the individual measurements together and
divide by the number of measurements.
The best estimate of the error associated with the mean value is called the
"error on the mean" and is given by (the error on a single measurement)
divided by (the square root of the number of measurements). Obviously,
this will decrease as the number of measurements increases.
The final reading for a quantity should be quoted as:
(mean) ± (error on the mean).
Error Propagation
Addition and Subtraction
If several quantities with associated random errors are given by:
x ± x, y ± y, ... , z ± z, then
the sum or difference is given by q ± q where q might be
q = x + y  z
and the error on q is propagated from the errors on x, y, ... , and z
as follows
Notice that the errors are added in quadrature, even when the quantities
are being subtracted. The error always increases when adding or subtracting
quantities.
Multiplication and Division
If several quantities with associated random errors are given by:
x ± x, y ± y, ... , z ± z, then
the product or quotient is given by q ± q where q might be
q = x*y/z
and the error on q is propagated from the errors on x, y, ... ,
and z as follows
Systematic Errors
These errors are more insidious than statistical errors. Systematic errors
are difficult to detect, and the sizes of systematic errors are difficult
to estimate. Increasing the number of measurements has no effect on
systematic errors because the error is always in the same direction (all
measurements too high, or all measurements too low). Careful instrument
calibration and understanding of the measurement being made are part of
prevention.
For example, suppose that you are using a stopwatch to time runners in the
100meter dash. You are quite adept at making the measurement, but
 unknown to you  the watch runs 5% fast. All times will be 5% too
high. There will be no immediately obvious indication of a problem.
If you happen to be familiar with the runners' normal times, you might
notice that everyone seems to be having a slow day. To prevent such
problems, one should calibrate the stopwatch with a known standard
such as the Nation Institute of Standards and Technology's standard
time service on short wave radio.
The rules are: 1) the error should have one significant figure; 2) the
number of decimal places in the measurement should be the same as the
number of decimal places in the error.
Always remember: There is no such thing as "human error". Try to find the
deeper cause for any uncertainty or variation.
Abstract
Equipment
Dcell, red lead wire, black lead wire,
analog voltmeter.
Data
Mark your Dcell so that you will be able to recognize it again.
Connect the black lead wire between the the COM (common) terminal of the
analog volt meter and the bottom of the Dcell. Connect the red lead wire
between the 3volt maximum terminal of the analog volt meter and the top
of the Dcell. Record the measured voltage with an error estimate.
Error Analysis
Calculate the fractional error, V / V * 100% (keep one or two significant figures in the
fractional error).
Connect the red lead between the 15volt maximum terminal of the
analog volt meter and the top of the Dcell. Record the measured
voltage with an error estimate.
Calculate the fractional error.
Connect the red lead between the 30volt maximum terminal of the
analog volt meter and the top of the Dcell. Record the measured
voltage with an error estimate.
Calculate the fractional error.
Exchange Dcells with the group across the aisle. Perform the same
measurements as on the previous page. For each of the three measurements
state whether or not the readings by the two groups agree, that is, do the
error estimates overlap? (See Taylor page 5 if you are confused.)
Make sure that you are comparing measurements
for the SAME Dcell. That is, you should have your readings
and their readings both for YOUR Dcell. They should have their readings
and your readings both for THEIR Dcell.

Our readings

Their readings

Overlap? (Y/N)

3 V max




15 V max




30 V max




List two sources of random (or statistical) error.
List two sources of systematic error.
Conclusion
Make a general statement about the procedure for devices with multiple
analog scales to obtain the lowest fractional error possible.
Summarize what you have learned today
(not what you have done).
A Puzzle
Using a Dcell, a flashlight bulb, and one wire, make the
bulb glow. Draw a large diagram of your method below. Make sure it is clear
enough so that a fiveyearold could look at your figure and light the bulb.
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