ASTR 3130, Majewski [SPRING 2015]. Lecture Notes
ASTR 3130 (Majewski) Lecture Notes
ERROR ANALYSIS, PART II: CENTRAL LIMIT THEREOM AND COMBINATION OF ERRORS
REFERENCE: Lyons Chapter 1.51.12.
We have discussed earlier the issue of experimental errors, including
the concepts of random and systematic errors and their connection to
precision and accuracy.
In this lecture I want to give some important practical aspects of dealing
with errors, including the concepts of Gaussian distributions and the
important Central Limit Theorem, and proper ways to both combine and
propagate errors.
Much of this lecture is taken directly out of Lyons' book, A Practical Guide for
Physical Science Students, for which you have already been asked to read Chapter 1.
 The Gaussian distribution is central to any discussion of the
treatment of errors.
Gaussian distribution
 The general form of the Gaussian distribution in one variable
x is:
 The curve of y as a function of x is symmetric about the
value of x = μ, at which point y
has its maximum value.
 The parameter σ characterizes the
width of the distribution.
 The factor ensures that the
distribution is normalized to have unit area underneath the whole
curve, i.e.,
 The parameter μ is the mean
of the distribution, while σ has the
following properties:
 The mean square deviation of the distribution
from μ is
σ^{2},
called the variance.
(The reason that the curious
factor of 2 appears within the exponent in the
equation for y above is to make sure that σ
is the RMS deviation.
Otherwise the root mean square
deviation from the mean would have been
σ/(2^{1/2}),
which is unaesthetic.)
 σ is known as the standard deviation.
 The height of the curve at x=
μ ± σ is
e^{1/2}
of the maximum value. Since
σ is very
roughly the half width at half height of the
distribution.
 In fact, one finds that the FullWidthHalfMaximum is
equal to
FWHM = 2.354 σ
So that HWHM = 1.177 σ. (Prove this to yourself!).
 The fractional area underneath the curve in the range
(i.e., within ± σ of the mean μ)
is 0.68.
This is a very important thing to keep in mind: When you repeat an experiment
with random errors, about 2/3 of the results will be within
1σ (one standard deviation)
of μ and
1/3 of your results will be beyond 1σ.
Note that 95% of the results will be within 2σ, and 99.7%
are within 3σ. (See Table A6.1 for more of these
values.)
Image and caption from Wikipedia:
Dark blue is less than one standard deviation from the mean.
For the normal distribution, this accounts for 68.27 % of the set;
while two standard deviations from the mean (medium and dark blue)
account for 95.45 %; and three standard deviations (light, medium,
and dark blue) account for 99.73 %.
 The height of the distribution at its maximum
is . As σ decreases the
distribution becomes
narrower, and hence, to maintain the normalization
condition, also higher at the peak.
 By a suitable change of variable to
any normal distribution can be transformed into a standardized
form
with mean zero and unit variance
The Meaning of σ
 It is customary to quote σ, the standard deviation,
as the accuracy of a measurement.
 Since σ is not the maximum possible
error, we should not get too upset if our measurement is more
than σ away from the
expected value. Indeed, we should expect this to happen with
about 1/3 of our experimental results.
Since, however, the
fractional areas beyond ± 2σ and beyond ±
3σ are only
4.6% and 0.3% respectively, we should expect such deviations to
occur much less frequently.
 To see this, consult the following figure, which is a graph showing the
fractional area under the Gaussian curve with
where
i.e., it gives (on the right hand vertical scale) the area in the
tails of the Gaussian beyond any value r of the parameter
f, which is plotted on the horizontal axis.
Central Limit Theorem
A feature that helps to make the Gaussian distribution of
such widespread relevance is the Central Limit Theorem. One
statement of this is as follows.
 Consider a set of n independent variables
x_{i}, taken at random from a
population with mean μ and
variance σ^{2}, and then calculate
the mean () of these n values.
 If we repeat this procedure many times, since the individual
x_{i} are random, then the calculated means
will have some distribution.
 The surprising fact is that, for large n, the
distribution of tends to a Gaussian (of mean
μ and variance σ^{2} / n). The
actual SHAPE of the
distribution of the x_{i} is actually
irrelevant.
 The only important feature is that the variance σ^{2} should be finite.
 Thus: If the x_{i} are already
Gaussian distributed, then the distribution of
is also Gaussian
for all values of n from 1 upwards.
But even if the x_{i} have some other distribution
 say, for example, a uniform
distribution over a finite range  then the distribution of the sum or
average of a few
x_{i} will already look Gaussian.
 Thus, whatever the original distribution, a linear
combination of a few representatives from the distribution
almost always approximates to a
Gaussian distribution.
Regardless of the shape of the parent population, the distribution of the
means calculated from samples quickly approaches the normal distribution
as shown below for four very different parent populations (left to right)
and by doing averages of an increasing number of independent "draws" from
the parent population.
Regardless of the shape of the parent population, the distribution of the means calculated from samples drawn from the parent population quickly
approaches the normal distribution as the number of averaged samples, n,
increases. Image from http://flylib.com/books/en/2.528.1.68/1/.
Some practical rules of thumb (from http://flylib.com/books/en/2.528.1.68/1/):
 If the parent population is normal,
will always be normal for any sample size.
 If the population is at least symmetric,
sample sizes of n = 5 to 20 should be OK (Gaussianlike).
 Worstcase scenario: Sample sizes of 30 should be sufficient
to make approximately normal (i.e.,
Gaussian) no matter how far the parent population is from being
normal.
 If making a distribution of , use
a standard subgroup size (e.g., all subgroups with averaging
5 observations, or 30 observations).
 An important aspect of the Central Limit Thereom (what makes it
important for empirical science) is that if we adopt the Gaussian
distribution of the to represent the error in
our knowledge of some mean measurement,
, and we adopt the width of that
distribution as the 1σ_{}
error of the derived mean value ,
then we find that
σ_{}^{2} = σ^{2} / n
σ_{} = σ / n^{1/2}
We call σ_{} the standard
deviation of the mean or the error in the mean.
The above implies the the error in the mean, σ_{}, is smaller than the error in each individual measure,
σ, by the square root of n.
The reason why the error in the mean (σ_{})
is smaller than the error in individual measurements (σ) for
n > 1 is that when we have more than one measure we have more than one estimate
of the actual value μ. With a distribution of
measures, you always have a better idea of where the mean might lie than
with only one measure, and intuitively you know that the more measures you
have (i.e., as n gets large) the better is your estimate of the mean value.
 Be clear about the difference between
σ and σ_{}:
σ is the spread in the distribution of x values, and
represents the standard deviation of a single measure. It is always the same
no matter how many times you repeat the experiment with the same measurement
apparatus.
σ_{} is the standard distribution of
your estimates of , and both
and σ_{}
can always be improved by increasing the number
of measures n, with the latter improving
as σ / n^{1/2}.
We will derive this latter equation in another way shortly.
 Imagine the parent distribution being the number of people in
a particular class of of different heights:
Image from http://scienceblogs.com/builtonfacts/2009/02/05/thecentrallimittheoremmade/ .
I can estimate the mean height of people in the class by drawing out subsamples
of n students in the class and averaging the results. If I do this
many times the answers I get for the mean height will form a
Gaussian distribution, and the
error in my calculated mean for any particular subsample is just given
by the intrinsic spread, σ, of the parent population, divided by the
square root of the number of people I used to estimate the mean
height.
Propagation/Combination of Errors
We are frequently confronted with a situation where the result of
an experiment is given in terms of two (or more) measurements (of
either the same or different types).
We want to know what is the error on the final answer in
terms of the errors on the individual measurements.
We first consider in detail the case where the answer is a linear
combination of the measurements. Then we go on to consider
products and quotients and the general
case of combining errors.
 As a very simple illustration, consider:
 Provided that the errors on b and c are
uncorrelated, the rule is that we add the contributions of
error in b and c
in quadrature (because we have seen above that in dealling with errors it is most sensible
to consider RMS deviations):
The errors in two measurements are uncorrelated when the measurement of
one variable has no bearing on the measurement of another  they are independent.
 Formal proof of simple quadrature summation of errors:
In the line just above, the first two terms are
and respectively.
The last term depends on whether the errors on
b and c are correlated. In most situations that we
shall be considering, such correlations are absent. In that
case, whether b is measured as being above or below its
average value is independent of whether
c is larger
than or less than . The result of this is
that the term
will average to zero. Thus for uncorrelated errors of b
and c, the formal proof equation reduces to:
 THOUGHT PROBLEM: What is the error in a for a combination that goes as
a = b + c ?
The General Case
Lyons derives the general propagation of errors formula
but I will not in lecture. Please read the Lyons treatment of this.
 Let
which defines our answer f in terms of measured quantities
x_{i} each with its own error
σ_{i}. Again
we assume the errors on the x_{i}
are uncorrelated.
 Then
This gives us the total error σ_{f} in terms of the
known measurement errors σ_{i} .
 Special Case: Averaging results with equal errors
σ_{i} (derivation of the
error in the mean):
This is the result already discussed as part of the Central Limit Theorem above.
 The functions
and
are so common that it is worth writing the combination of error formula for
them explicitly:
 THOUGHT PROBLEM: You should, however, make sure you can derive the
above equation for the errors of products and divisions using the
generalized equation for sigma given above.
Combining Errors: More complicated example
 A certain result z is determined in terms of
independent measured quantities a, b, and c
by the formula
 To determine the error on z in terms of those on
a, b, and c, we first differentiate
partially with respect to each variable:
Then we use the equation from the general case,
to obtain
Combining Results of Experiments Having Differing Errors
 When several experiments measure the same physical quantity
and give a set of answers a_{i}
with different errors σ_{i}, then the best
estimates of a and its accuracy σ are
given by:
and
Thus each experiment is to be weighted by a factor of . In
some sense, gives a measure
of the information content of that particular experiment.
 The weighting to get the mean a above makes intuitive sense:
We want contributions from experiments with worse errors to be less than
the contributions from experiments with better errors.
 The simplest case is when all the errors
σ_{i} are
equal. Then the best combined value a from the above
equation becomes the ordinary average of the individual
measurements of a_{i}, and the
error σ on a is , where N
is the number of measurements.
Example Problems
Appendix 1 (Page 73) of Lyons summarizes the important results and equations
described on this webpage. These equations are key guidelines that
every empirical scientist should live by.
Test problems for this material can be found at the end of Chapter
1 of Lyons.
Click here for a set of practice problems on this material, including some specifically related to astronomical contexts. The file
that comes up will be in pdf format.
All material from A Practical Guide to Data Analysis for
Physical Science Students by Louis Lyons, Cambridge
University Press: Cambridge, 1991. These notes are intended for
the private, noncommercial use of students enrolled in Astronomy
313 and Astronomy 3130 at the University of Virginia.
