INTRODUCTORY PHYSICS LABORATORY
A NALYSIS
OF
E XPERIMENTAL E RRORS
All physical measurements in the verification of theories and concepts are subject to
uncertainties that depend on the measuring instruments used and the conditions under which the
experiment was performed. Any uncertainty in a single measurement will result in an uncertainty
of the final experimental result. These uncertainties are often referred to as EXPERIMENTAL
ERRORS! This aim of this treatise is to enable students to become aware of the potential sources of
error and bias that are inherent in any process of measurement. A student should be able to apply
the concepts of arithmetic mean, standard deviation, and standard error in analyzing errors likely
to occur when making precision measurements in the laboratory.
TYPES
OF
E X P E R I M E N TA L E R R O R S
There are three fundamental types of experimental errors:
Systematic errors
Personal errors or Mistakes
Random Errors
SYSTEMATIC ERRORS
Systematic errors usually cause the results of a measurement to be consistently too high or
too low below the true (or actual) value. These errors may be reduced by having well designed
experiments and good equipment. Generally, the errors are due to:
a. Faulty instruments – usually due to poor calibration, adjustments of the instruments, or
slight imperfections in the construction or design of the instrument. Consequently, the
readings or measurements are “off” by the same amount each time. Checking the zerooffset before measurements and applying the appropriate “adjustments” to the readings
reduces the systematic errors.
b. Theoretical over-simplification of the system – when all the variables that affect a system
are not taken into account. An example is when frictional effects are neglected in the
theoretical problem. Consequently, there is always a significant discrepancy between the
expected theoretical results and the experimental results obtained when these frictional
effects cannot be ignored.
c. The environmental conditions during measurements may differ from the standard
conditions used for the calibration of the instrument scales. These variations have to be
compensated for in order to minimize the systematic errors.
Any experimenter will strive to identify, minimize, and eliminate any obvious systematic
errors as much as possible. Good experimental designs may utilize the advantages due to
symmetry or repetition of measurements in reverse order to minimize subtle systematic errors.
Unfortunately, it is very difficult to reliably identify and estimate systematic errors.
PERSONAL ERRORS (OR MISTAKES)
Personal errors can be completely eliminated if the experimenter exercises utmost care,
caution, and skepticism while reading the data (observational mistake) and while performing the
needed calculations (arithmetic mistake). If the measurement values are read incorrectly or if the
calculations are wrongly carried out, the entire result will be wrong! Therefore, the experimenter
is strongly encouraged to cross-check the data and calculations. In a group, a one-time careless
mistake in measurement by one partner can affect the entire result. Therefore, each partner should
independently read the data and check any calculations for accuracy.
RANDOM ERRORS
Random errors are usually due to unknown variations in the experimental conditions. The
sources of these random errors cannot always be identified and can never be totally eliminated in
any measurement. This class of errors usually causes about half of the measurements to be too
high and the other half of the measurements to be too low. These random errors may be:
a. Observational - inconsistency of an observer in estimating the last digit when reading the
scale of a measuring device between the smallest division.
b. Environmental - physical variations that may affect the equipment or the experiment
setup such as fluctuations in the line voltage, temperature changes, or mechanical
vibrations.
Fortunately, random errors may be minimized by taking several careful observations of the
same quantity and can be determined by statistical analysis. These random errors are sometimes
referred to as statistical errors.
ESTIMATION OF STATISTICAL ERRORS IN MEASUREMENT
We shall now discuss methods of finding the error in a measurement. The first method is to
estimate the random errors in measurements in advance – “a priori method”. The second method
involves the use of the random spread in the measurements, also known as the “scatter”, in
estimating the errors. It is important to state the precision or the estimated error in a
measurement. For a measured value, x, that has an estimated error, δx, the measurement would be
reported as x ± δ x . We now focus on different methods for estimating measurement errors.
Estimation in advance (a priori method)
If several measurements of a given object yield the same result, then there no random errors
present in the measurements. In this case, the smallest division on the measurement scale (or least
count) is used as the error estimate.
2002 Martin Okafor
General Physics – Error Analysis
page 2
Least Count
In all physical measurements, every measuring instrument must have a measurement scale
and that scale must have a least count. The least count is the smallest marked scale division on
that instrument. The uncertainty (estimated error) introduced in a single measurement is known to
be approximately equal to the value of the least count of the measurement scale. For example, if
all the readings of the length of a rod, using a meter stick, are equal to 156 mm (no random
errors), then the quoted length of the rod may be (156 ± 1 mm). This length indicates that the
least count of the meter stick (1 mm) is the error estimate of the measurements. Sometimes, by
visual “calibration” of the interval between the smallest scale division – interpolation – some
experimenters confidently estimate the measurement error in terms of a fraction of the least count.
Arithmetic Mean or Average ( x )
Having obtained several repeated measurements of a certain single quantity (x) without any
mistakes and other systematic errors and with only random errors present, the readings will
cluster about their average value ( x ). The average of several measurements is called the
arithmetic mean or mean. This Mean (or average) is the most valid representation of the quantity
being measured when only random errors are present. If N measurements of the quantity x are
made, the Mean is given by the equation
n
x=
∑x
i
i =1
N
(1.1)
or
x=
x1 + x2 + ⋅⋅⋅ + xn
N
(1.2)
Average Deviation (a.d.)
When a few measurements are made, the “scatter” or “dispersion” or “spread” of the
measurements about the mean value can be used to find the random error of the measurement. In
one or more of the laboratory experiments, you will be required to take N (where N ≥ 3) specific
measurements of a given object. The mean value of these measurements will be considered as the
best estimate of the true value of the mesurement. A simple and easy method of estimating the
error in these measurements is the average deviation. To calculate this average deviation, first
find the difference between each measurement and the mean, ( x − x ) . This yields the deviation
of each measurement from the mean. Next, calculate the average of the absolute values of these
deviations from the mean -- counting all the deviations as positive. This is the average deviation
(or average deviation from the mean). The average deviation is thus considered as an estimate of
the measurement error. The Average deviation (a.d.) is defined as:
a.d . =
2002 Martin Okafor
∑ x−x
N
General Physics – Error Analysis
(1.3)
page 3
Standard Deviation (σN or σ )
If single measurements (N) are made with only random errors present, a useful measure for
determining the degree of dispersion or scatter of the data about the mean is the Standard
Deviation (also known as root-mean-square (r.m.s) deviation). Thus, the observed standard
deviation indicates a measure of the deviations of the measurements from the mean and is given
by the equation:
N
∑(x − x )
σN =σ =
2
i
i =1
(1.4)
N
This equation can also be expressed in two equivalent forms as shown below:
σ=
∑x
2
(∑ x)
−
2
∑ x −(x)
N
2
N
=
N
2
(1.5)
For large numbers of measurements, a random sample of the data is often considered in
statistical analysis. The standard deviation for the entire population of measurements is the
observed standard deviation (σ) while the standard deviation for the sample is known as the true
or estimated or unbiased or sample standard deviation (σN-1), where N is the sample size. These
observed and sample values of the standard deviation are slightly different due to the difference
in values of the sample mean and the observed mean. The sample standard deviation is given by
N
σ N −1 =
∑( x − x )
i =1
i
N −1
2
=
∑x
2
(∑ x)
−
N −1
N
2
(1.6)
Standard Error (α)
If several series of measurements are made with only random errors present, a measure of the
scatter of the mean values of each series around the true mean value is described by the standard
error of the mean (or standard error). The theoretically estimated standard error is defined as:
α=
σ
N −1
(1.7)
We apply the standard error in stating the estimated uncertainty in the mean value. This
standard error indicates the number of significant figures in the mean value of the measured
quantity. Thus the final result is expressed as x ± α . The first digit of the standard error, after
rounding to one figure, indicates the decimal place in which the uncertainty in the mean exists.
Therefore, the last digit retained in the mean should be in the same decimal place as the first digit
of the standard error.
2002 Martin Okafor
General Physics – Error Analysis
page 4
For example, if, for a given set of measurements, the calculated mean x = 12.3456, and the
standard error, α = 0.0013, then the standard error α = 0.001 (rounded to one significant figure).
The standard error indicates that the uncertainty in the measurement exists in the third decimal
place. Therefore the result is stated as: 12.346 ± 0.001.
The resultant number of significant figures in the reported mean indicates the precision of the
experiment.
The following example will illustrate the concepts of stating the measured quantity in terms
of the mean value with the precision indicated.
EXAMPLE
Nine repeated measurements of the length (in centimeters) of a wooden board are given as
follows: 21.25, 21.42, 21.50, 21.72, 21.84, 21.00, 21.32, 21.68, 21.45.
(a)
Calculate the mean:
N
x=
(b)
∑x
i =1
= 21.46444 cm
Find the standard error:
α=
(c)
N
i
σ
= 0.09 cm (rounded to one significant
N −1
figure)
State the length of the block.
x = 21.46 ± 0.09 cm
(The standard error predicts that the uncertainty in the mean exists in the second decimal
place)
2002 Martin Okafor
General Physics – Error Analysis
page 5
Normal Distribution or Normal Error curve
For a large number of measurements, the range of the results could be grouped into small
(sample) “classes”. With only random errors present, a plot of the frequency distribution of the
means of individual classes is a characteristic bell-shaped curve that is symmetric about its center
line described as a normal distribution curve. This center line coincides with the mean value of
the measurements. The presence of variations caused by systematic errors, would cause a normal
distribution curve to be “skewed” (causing a shift of the peak of the curve) to the left or right of
the the mean.
Frequency
Normal Error Curve
-5
-4
-3
-2
-1
0x
1
2
3
4(x)
σ5
From statistical analysis, for large number of measurements with only random errors present,
approximately 68% of the measurements will lie within ± 1σ, [i.e. ( x − σ , x + σ ) or one
“STDDEV” of the mean]. Approximately 95 % of all the measurements will be within ±2σ.
Essentially, all measurements should fall within ±3σ of the mean. Any measurement that is not
within ±3σ of the mean is likely a reading/measurement mistake and should be discarded. If the
data points are scattered over a narrow range of values, σ will be small and the distribution curve
would be narrow, clustering around the mean. Otherwise, if the scatter of the data is large, σ will
be large and the distribution curve will be broad, spreading over a wider range of values on both
sides of the mean.
Precision and Accuracy
Based on the definition of standard deviation, for a set of measurements, a low value of the
standard deviation indicates a high precision – i.e. the data points are closely clustered, with low
scatter. Hence, the smaller the standard error, the more precise are the set of measurements.
Scientists engaged in original research usually seek high precision of their data. Precision of a
measurement indicates the degree of reliability or repeatability of the results. Thus, a high
precision implies that a series of measurements would be successfully repeated under similar
conditions to yield results which are in close agreement.
2002 Martin Okafor
General Physics – Error Analysis
page 6
However, for most physical science experiments conducted in the freshman college
laboratories, the objective is often to perform the experiments with great care in order to obtain
reasonably accurate values of an established quantity or physical constant such as the acceleration
due to gravity, or verify a known physical law or principle. The accuracy of a measurement refers
to how closely a measurement compares with a known “standard” or “accepted” or “theoretical”
value. Sometimes, measurements with a high precision may cluster very closely around an
inaccurate mean value which has a large deviation from the expected theoretical value. This is
usually due to the presence of systematic errors. Accuracy is greatly affected by the amount of
systematic errors present and thus measures the correctness of the measurement. So, high
precision measurements do not necessarily imply high accuracy. On the other hand, a
measurement may appear highly accurate but with very poor precision. Unless a measurement has
a high precision, its accuracy cannot be considered as realistic.
Estimated uncertainty of a measurement usually reflects both the precision of the instrument
and the accuracy of the measurement. Accuracy of a measurement is determined by calculating
the Percentage Error.
Percentage Error
The percentage error (or percent error) is calculated from the following equation:
Percent Error =
Experimental − Accepted
× 100
Accepted
(1.8)
Percentage Difference
For a quick comparison of two measurements, it is often necessary to calculate their
percentage difference. In this case, none of the measurements being compared is considered an
“accepted” value. Instead the percentage difference (or percent difference) indicates how close
the measurements compare to each other relative to their average value. The percent difference is
calculated as:
Percent Difference =
2002 Martin Okafor
value1 − value2
× 100
average ( value1, value2 )
General Physics – Error Analysis
(1.9)
page 7
PROPAGATION OF ERRORS
As previously noted, most calculations performed in the laboratory work involve the
derivation of an unknown quantity by using two or more measured values that have some error
associated with them. Propagation of errors is a method to determine how the error in the derived
quantity relates to the errors in the measured values. Let us consider two circumstances:
1. If there are no repeated measurements for the measured values and if there is insufficient
knowledge of the estimated errors in the measured values, an approximate way of estimating the
error in a calculated quantity is by applying the rules for significant figures. For any measured
value, the reliably known digits are reported and the last digit reported is estimated by
interpolating between the smallest marked scale division. Do not attempt to interpolate to more
than one estimated place. Correctly stated, measured values indicate the appropriate number of
significant figures that are applied to subsequent calculations. It is, therefore, very important that
the proper significant figures be taken into account for all calculations performed in the
laboratory! Review the Rules for Significant Figures.
2. The second circumstance involves a situation where two or more measured values, each
with known estimated errors, yield a calculated or derived quantity. It can be shown by methods
of statistical analysis that certain rules can be applied to combine the errors in each measured
quantity to determine the error in the derived quantity – known as propagation of errors! These
rules are reviewed in the following section.
Addition and Subtraction of more than one measurement
When the estimated errors in measurements are much less than the measured values, a better
approximation of the error in the result of an addition or subtraction is given by the square root
of the sum of the squares of the estimated errors.
Consider an example where a quantity L is calculated from the measurements x and y where
x ± α x and y ± α y
When the calculated quantity, L(x,y), is defined as a function of x and y:
i.e. for addition: L = x + y , or subtraction: L = x - y
using advanced statistical methods, the error in L, is given by
αL =
(α x ) + (α y )
2
2
and the result is stated as L ± αL.
Generally, for any linear combination such as
If z = ax + by, or z = ax – by, then
2002 Martin Okafor
General Physics – Error Analysis
page 8
σ z = a 2σ x 2 + b 2σ y 2 --------------------------(1.10)
Multiplication and Division of multiple measurements
When measurements are multiplied or divided, the fractional error in the result is the square
root of the sum of the squares of the fractional errors of the measured values.
Consider the equations:
for multiplication: L = x ⋅ y or division: L =
y
x
the fractional error in L is given by:
2
αL
α  αy 
=  x  +   ----------------------- (1.11)
L
 x   y 
2
and the result is again stated as L ± αL.
Generally, for an expression which is a product of powers:
If z = axnym, then
The fractional error is given by
2
σy 
σz
σ 
= n 2  x  + m 2   ---------------(1.12)
z
 x 
 y 
2
and the percentage error is given by
% Error in z = n 2 ( % error in x ) + m 2 ( % error in y )
2
2
(1.13)
Advanced discussion of error analysis may be required for certain experiments. However, a
solid background of the basics of error analysis discussed should serve as a guide to the student in
computing experimental errors in the investigation of basic physical phenomena.
2002 Martin Okafor
General Physics – Error Analysis
page 9
REFERENCES
Lichten, William (1999) Data and Error Analysis -- 2nd ed., Prentice-Hall, Upper Saddle River,
NJ.
McClave, James T., and Dietrich, Frank H. (1992) A First Course in Statistics – 4th ed.,
Macmillan Publishing Company, New York, NY.
Nolan, Peter J., and Bigliani, Raymond E. (1995) Experiments in Physics -- 2nd ed., Wm. C.
Brown Publishers, Dubuque, IA.
2002 Martin Okafor
General Physics – Error Analysis
page 10