Lecture Notes - Error Analysis
No measurement is perfectly exact or accurate.
Instrumental, physical, and human error
True Value.
The value if we were able to eliminate all error.
We can never hope to measure the "true value".
Usually we don’t know the true value
We still can estimate the error
Reliability of a measurement
Depends on estimating the error / uncertainty in data to obtain it
Indeterminate errors
Causes: operator errors or biases, fluctuating experimental conditions,
varying environmental conditions, inherent variability of measuring
instruments
Effects can be reduced by averaging the results
 A single measurement is not sufficient
Systematic errors
The error has the same size and same sign for every measurement
A bias in the observer or the instrument
Experimental blunder
Can be more serious than Indeterminate errors:
There is no sure way to discover and identify them
Their effects cannot be reduced by averaging results
Effects may be corrected when re-done by another experimenter
 A single measurer is not sufficient
Small indeterminate error  high precision
Small indeterminate error and small Systematic error  accurate
Standard Methods for Expressing Error
Absolute measures e.g. 34.0 g  0.7 g
All measurements are within 0.7 g
Relative uncertainty
absolute error / size of the measurement
e.g. 0.7/34 = 0.02 or 2%
e.g. Termperature measurement, instrument reliable to  0.5 degree?
Relative or absolute measurement?
Relative:
0.5% in measuring boiling point (100 degrees)
10% in measuring cold water at 5 degrees
"Undefined" in measuring freezing point 0 degrees
Nonsensical
 Relative measurement is important to characterize our labs
 Common sense and good judgment must be used
Data Error propagate through the calculations to produce errors ion the
results.
 It is the size of the data errors' effects on the results which is most
important.
E.g. Ave velocity = displacement / time
Suppose time is 8.3 s. Suppose displacement is 1.000 0.0001 m.
hw is velocity constrained? (Upper and lower limit)
Suppose time is 8.3  .1 s. How is velocity constrained? (THEY
TRY) We must assume a worst-case combination of signs.
Sum and difference rule: add absolute error
Product and quotient rule: Add relative errors
Power rule: Multiply the relative error by the power. Holds also for
fractional powers.