EXPERIMENTAL ERROR
“Absolute certainty is a privilege of uneducated minds - and fanatics. It is, for scientific folk, an
unattainable ideal.” Cassius Kuyser
Physics is primarily a science of measurement and the explanation of the meaning of that
measurement. In the real world of measurement, it is impossible to be exact. There are too
many ways that uncertainty can creep into the measurement. Some of the possibilities include:
(1) random variation of the measured quantity from one reading to another (measurements
differ from one another in a random manner due only to the inherent nature of
measurement)
Example: measuring the length of the edge of a book yields 25.2 cm, 24.9 cm, 25.3 cm, 25.5 cm, and
24.9 cm,
(2) calibration error of the measuring instrument (all instruments are built to a limited
precision)
Example: The manufacturer of the ruler (our meter sticks) states that his methods of manufacture
make a ruler which is accurate to 2 mm of the true length for 1.000 m,
(3) inaccuracy of the instrument (a worn or heavily used measuring instrument will
sometimes develop a bias or consistent inaccuracy)
Example: The ruler is made of wood which has dried out over time and shrunk so it always show a
value that is too large,
(4) bias on the part of the measurer (the measurer will read the instrument in various
manners which may contribute to random variation or may contribute to always making
the measurement too small or too large)
Example: The measurer was told the edge of the book is 25.0 cm so the numbers are read to be
close to that value.
The definition of experimental error () is the difference between the actual or true value of
measurement (ztrue) and the measured value (zmeas).

 = |ztrue - zmeas|
Error is therefore an indication of the uncertainty of the accuracy of the measurement.
Types Of Error.
The above examples of error can be divided into two types — indeterminate or random and
determinate or systematic.
Random Error
Example number 1 is an example of random error. The best method of dealing with this type is
by taking many measurements and apply statistical methods. If the number of measurements
is large, then the measurements will approach that of a normal distribution (Gaussian curve),
and the determination of averages, deviations, and the standard deviation provide useful
information about the accuracy of the measurement.
The average or the mean is the most likely value of this statistical model.
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For the example of the length of the edge of the book the mean is
(25.2 cm + 24.9 cm + 25.3 cm + 25.5 cm + 24.9 cm)/5 = (125.8)/5 = 25.16 cm
The deviation from the mean is the absolute value of the difference between a measured value
and the mean. The mean of the deviation (average deviation) is an indicator of the precision of
the measurement. The smaller the average deviation, the more precise the measurement is
likely to be.
The average deviation of the mean is
(|25.2 - 25.16| + |24.9 - 25.16| + |25.3 - 25.16| + |25.5 - 25.16| + |24.9 - 25.16|)/5
=(0.0 cm + 0.3 cm + 0.1 cm + 0.2 cm + 0.3 cm)/5 = 0.2 cm.
NOTE: Be sure to apply the rules for significant figures (see your text).
The standard deviation is the most common method for determining the range of values which
is most likely to include the true value. The formula for the standard deviation is
  
2

i
n 1
where  is the standard deviation, i is the deviation from the mean, and n is the number of
trials or measurements.
The standard deviation of the measurement of the edge of the book is

  
i
n 1
2

2
2
2
.   0.2  0.3
0.0 2  0.3 2  01
5 1


0.2
 0.2cm 
4
Applying the rules for significant figures requires that  = 0.2 cm. The length of the edge of the book is
now written as 25.2 cm ± 0.2 cm.

Statistical theory tells us that for there is a 64% chance that the true value lies within ± 0.2 cm
of the measured value of 25.2 cm.
Systematic Error
Most of the experiments in introductory physics are limited by the time of the laboratory period.
Because of this time limit, it is usually not possible to make several measurements of the same
quantity. Under these conditions statistical methods of analysis are of little value.
Examples 2 through 4 are all of the instrumental or systematic type of error which in principle
can be accounted for. Bias (Example 4) can be avoided by careful measurement methods.
Instrumental bias (Example 3) can be avoided by comparing the instrument to standards.
Calibration uncertainty (Example 2) is considered in addition to statistical error or in the case
of the experiments in introductory physics as the sole source of measurement uncertainty.
It is the mathematical ways to handle the systematic error due to calibration uncertainty which
will concern the rest of this introduction to error. The principle assumption will be that all
possible sources of error must be included in the quoted uncertainty of measurement. For
example if the edge of another book is quoted at 25.1 ± 0.3 cm, it is assumed that calibration
error and reading uncertainty (a type of human calibration error) are included. The absolute
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values of the errors are combined to provide the largest possible range of values. It is always
assumed that all uncertainties combine to provide the largest possible uncertainty of
measurement.
The derivation of the formulae for determining instrumental error depends on the theory of
partial differentials. However the results can be summed up in a few simple statements. For
the purposes of this discussion we will refer to absolute uncertainty and fractional
uncertainty.
Absolute uncertainty is the uncertainty of a measurement expressed in the same units as the
measurement. For example the absolute uncertainty of a current expressed as 1.22 A ± 0.03 A
is 0.03 A.
Fractional uncertainty is the uncertainty of a measurement expressed as a fractional value or
as a percentage. For example, the fractional uncertainty of the current (1.02 A) in the above
paragraph is 0.03 A/ 1.2 A = 0.02 or 2%.
1. If the measurement requires adding or subtracting values, the uncertainty of reading
each value is added to the calibration uncertainty. All values must be expressed as
absolute uncertainty.
For example, in reading the ruler for measuring the edge of the book, it is best to use a reference other
than the edge of the ruler. Thus a reading at one end of the ruler might be 2.0 cm and at the other end
26.9 cm. The length of the edge of the book is the difference of these two measurements or
26.9 cm - 2.0 cm = 24.9 cm.
Suppose now that each mark can be determined to ± 0.3 mm because of the rounding of the edge of the
book and the thickness of the tick marks indicating values of distance on the ruler. The reading
uncertainty is then 0.3 mm at the 2.0 cm mark and 0.3 mm at the 26.9 cm mark for a reading uncertainty
of ±0.6 mm.
Furthermore, the calibration error of the ruler was stated earlier to be 2 mm for each 100 cm or a fractional
calibration error of 0.2 %. 0.2 % of 24.9 cm is 0.05 cm
From the above the grand total of the instrumental error would be ±1.1 mm.
In terms of a general formula, the uncertainty is
  r1  r2  s
where is the uncertainty in the length measurement, r1 is the reading uncertainty in the
first reading of the meter stick, r2 if the reading uncertainty in the second reading of the meter
stick, and s is the calculated uncertainty due to calibration uncertainty of the meter stick.
Note that even though the readings are subtracted to determine the length, the uncertainties
are added.
2. If measurements must be multiplied or divided a fractional uncertainty must first be
calculated by dividing the uncertainty by the measured value. The fractional
uncertainties are then added.
The fractional uncertainty for the edge of the book is
1.1 mm ÷ 24.9 cm = 0.0044 or 0.44 %.
If another dimension (width) is determined to be 10.2 cm ± 0.7 mm, the area of the cover of the book
would be
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A = 24.9 cm  10.2 cm = 254 cm2
with the uncertainty being the sum of the fractional uncertainties or
0.0044 + 0.0069 = 0.0113 or 1.13%.
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The uncertainty in cm2 would then be found by
.0113  254 = 2.9 cm2.
In terms of a general formula, the uncertainty is
A  w


A

w
where each term (fraction) in the equation represents the fractional uncertainty for area A,
length , and width w. Note: As in multiplication of two measurements, division also requires
that fractional uncertainties be added.
3. Exponents and roots may be dealt with by multiplying the measurement(s) by a
constant representing the absolute value of the exponent or the exponent representing
the root.
4. Multiplicative constants such as  or 2 have no uncertainty, they are exact.
EXAMPLE: A simple pendulum is measured to be 15.0 cm using a meter stick that can be read to an
accuracy of 1.0 mm at the bob (weight) and to 0.2 mm at the pivot or support end and has a calibration
error of 0.2%, and the acceleration of gravity is known to be 9.80 m/s2 ± 0.01 m/s2. Determine the
uncertainty of the period of oscillation in seconds.
The equation for the period of a simple pendulum is
T  2

g
First determine the uncertainty of the length of the pendulum () by adding the reading
uncertainty at the left and right of the meter stick to the calibration error of the meter stick.
(rule 1)
  10
. mm  02mm  0.002  150mm
  15
. mm
Since the formula for a simple pendulum is complicated by multiplicative constants and the
square root sign, we need to analyze it in terms of rules 2, 3, and 4. First rewrite the equation
using exponents in place of the square root.
1
  2
T  2 
 g
or
T  2
1
2
1
g2
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Now every factor is multiplied, divided or written with an exponent, and the rules should be
clearer. According to rule 2 we will work with fractional uncertainties for T,  and g, and
according to rule 3 we will multiply  and g by ½. Also according to rule 4, 2 and  have no
fractional uncertainty and are dropped from the expression for uncertainty. Thus
T 1  1 g


T
2 
2 g
Substituting
T 1 015
. cm 1


T
2 15.0cm 2
0.01
9.80
m
s2
m
s2
T 1
1
 0.010  0.001
T
2
2
T
 0.0055  0.55%
T
To find the absolute uncertainty of the period in seconds, we must first calculate the period of
oscillation of the pendulum. Using the equation for the period of a pendulum given above, we
get
T  2
0150
.
m
 0.777s
m
9.80 2
s
The absolute uncertainty of the period is then
T  0.0055T  0.0055  0.777s
T  0.0043s
and
T= 0.777 s ± 0.004 s
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