Measurement and Error Analysis

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Appendix A: Measurement and Error Analysis
This appendix will go over key components required for this course
Random errors
Systematic errors
1-D statistics
Linear 2-D statistic
Error propagation
Rounding rules, and
Comparing precision versus accuracy.
Precision is how closely and reliably one can determine a quantity. It is affected by the randomness
of four broad categories:
a) the measurement device
b) the measurer
c) the measured quantity, and
d) the measurement’s environment.
Precision is how far or how close subsequent determinations tend to be from each other; Precision
is quantified with the
standard error.
Accuracy is measure of how close the experimentally obtained value compares to that quantity’s
accepted, true, value. Sometimes an accepted value might be known. In such cases testing accuracy
will not always be possible. When it is possible, accuracy analysis can reveal
systematic errors
that might have affected the experimental determination that are not properly accounted for in the
standard error.
An experimentally determined value is considered
reliable within plus or minus one standard error
of that value. This predicts a roughly 7/10
th chance that a subsequent determination of the same
quantity will fall within this range. Equivalently, this predicts a roughly 7/10
th chance that the
current determination falls within this range around the best estimate to the theoretical true value
of what is being determined. This is how you express the experimental precision of the value, the
quantification of the impact of random errors.
Broadly speaking, errors fall into two categories: random and systematic. We will thoroughly
discuss random errors first, along with how to treat them with statistics. This will provide a
valuable context when discussing
systematic errors.
Random Errors
If a measurement is only affected by random errors, it will just as likely be too big or too small
around the theoretical accepted, true, value of what is being measured. Mathematically, for true
value µ,
𝑟𝑒𝑠𝑢𝑙𝑡 = 𝜇 + 𝑟𝑎𝑛𝑑𝑜𝑚 𝑒𝑟𝑟𝑜𝑟
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with the random error having an equal chance of being positive or negative. How far the
measurement result deviates—the size of a particular instance of random error – is also random.
The standard error will be an attempt to quantify the typical, average, size of these deviations and
will be how we represent precision.
Random errors influence all real measurements due to the measurement device, the measurer, the
measured, and the measurement’s environment. The impact of random errors can be reduced with
the averaging process.
Systematic Errors
A systematic error is anything that would not be properly reflected in quantifying precision, the
influence of random errors. The influence of systematic errors may or may not affect any of the
preceding statistics, but if it does, it would not affect the statistical calculations in the way they are
designed to handle; if the statistics treats a systematic error properly, then it is actually a random
error. Yet this does not mean a systematic error will affect a statistical calculation at all or even
enough to be noticeable beyond the normal variations random errors produce. Yet, this does not
mean that all mathematics will not be useful in accounting for systematic errors. For example, by
analyzing anomalous results in statistical calculations, like a test of accuracy, this can help expose
systematic errors.
So overall, systematic errors can be a varied as nature itself. Unlike random errors, there are no set
ways of dealing with them. They may or may not be detectable and treatable mathematically. They
can never be fully removed from any real measurement. By analyzing data; by mindfully
designing, constructing, and performing experiments; by identifying and judging relevance; the
sagacious scientist can try to account for systematic errors to the ideal point where their influence
is negligible compared to the influence of random errors.
In other words, you try to make it so
you can ignore
systematic errors according to level of precision you hope to obtain.
If you cannot ignore them, meaning their impact is affecting your results (it is non-negligible) and
you cannot account for them by either adjusting your data, your experiment, or your analysis, then
you, at the very least, must make sure to properly acknowledge their effects in your experimental

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However, do not clutter your reports with trying to list systematic errors that are clearly negligible
at your precision. Judgment and forthrightness are essential qualities in experimental reporting.
Example: Consider measurement of a plastic sheet using ruler to measure the length. We know
for certain that the length is after the 2.4 cm. Now is where the uncertainty begins, we must
guess as best we can as to where in-between the divisions the length lies. L=2.46 cm? L=2.47
cm? L=2.48 cm? With what we can see, reporting more
precision by carrying any more digits
beyond this guess is misleading. In other words, without some way to improve our precision,
like using a Vernier caliper and/or a magnifying glass, there is no way we could legitimately
claim something like L=2.472 cm.
So when realizing this, if the best our eyes can
estimate is L=2.47cm, could we tell the
difference between 2.4699 or 2.4701 cm?
2.4698 or 2.4702 cm? Such
randomness is
beyond our ability to detect. We can see how
the limitations of a measurement device can
affect precision; yet for such a measurement
like this, precision is also personal. For example, can we
reliably tell the difference between
2.47±0.02cm? Some can and some cannot. The key to reliability is in repeatability. The ideal
way to quantify your precision will be based on how your measurements tend to
deviate from
each other.
There has also been an unspoken assumption
about this measurement. The reality of such a
measurement is that there are two measurements,
for we must properly
zero it first. Consider this
situation. This is very undesirable for a variety
of reasons. For the moment, consider what the
previous conversation would be like if we did
not catch that this was not properly zero’ed. Suppose we read this as L=2.52 cm, then our
discussion of randomness would be the same but shifted by this kind of non-random error.
That last range of potential randomness would look like this: “How about 2.51 or 2.53 cm?
2.50 or 2.54 cm? Can we reliably tell the difference between 2.52±0.02cm?” In just our
discussion of trying to quantify our potential randomness, we would not catch this error. If
we somehow knew the accepted, true, value for the length of this widget, perhaps from the
box the plastic piece came in, then we might catch this shift in comparing what we measured
to what is accepted—a test of accuracy.

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How to minimize experimental error: some examples

Type of
Example How to minimize it
You measure the length of the
plastic three times using the
same ruler and get slightly
different values: 2.472 cm,
2.476cm, 2.469 cm
By taking more data and averaging,
random errors can be reduced and can be
evaluated through statistical analysis.
If the cloth tape we used to
measure has worn out and had
been stretched from years of
use. This results in an error in
measuring the length of plastic
Similarly, the electronic scale
used to measure chemicals may
read too high because it is not
properly tared/zeroed
throughout your experiment.
This error is difficult to detect and cannot
be analyzed statistically since all your
measurements are off in the same
How would one can compensate for the
incorrect results when the tape is
stretched out?
How would you take care of improperly
tared scale?

An average reduces the influence of random error. If systematic errors are reduced to negligible
levels, an average becomes your best estimate to the theoretical true value you seek. And as long
as systematic errors continue to be negligible, this estimate approximates this theoretical true value
better, more reliably, as the number of measurements increases.
One-dimensional averages
If there is set of N independent determinations of X, where X = [X1, X2,…..,XN], then their average

Microsoft Excel =AVERAGE( ).
𝑋1 = ∑ 34
5 67

A single determination of X’s standard deviation in one-dimensional statistics is
𝑆3 = :;6758 (𝑋𝑁6 –𝑋11)?
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Microsoft Excel =STDEVA( ).
The standard deviation is usually symbolized with the Greek letter sigma:
Linear, two-dimensional statistics
If there is set of 𝑁 independent determinations of (𝑥, 𝑦) pairs; (𝑥8, 𝑦8), (𝑥?, 𝑦?) … , (𝑥5, 𝑦5); that
are suspected as having a linear relationship such that
𝑌 = 𝑚𝑋 + 𝑏 with slope 𝑚 and intercept 𝑏,
then the average values for this
slope (m) and this intercept (b) are the following:
𝑚 =
𝑁 ∑ 𝑥
6𝑦6 – ∑ 𝑥6 ∑ 𝑦6
𝑏 =
∑ 𝑥
6? ∑ 𝑦6 – ∑ 𝑥6 ∑ 𝑥6𝑦6
𝑤𝑖𝑡ℎ 𝐷 = 𝑁 ; 𝑥
6? – M; 𝑥6N?
Microsoft Excel =SLOPE(known_ys, known_xs).
Microsoft Excel =INTERCEPT(known_ys, known_xs)
The averages are determined in a process often called
linear regression. In addition, the best
estimate for the
standard deviation in the dependent variable y is
= :;6758 ( 𝑦6 – (𝑁𝑚-𝑥62+ 𝑏) )?
Microsoft Excel =STEYX(known_ys,known_xs)
The estimated quantification of the reliability of a calculated one-dimensional average—the
standard error of a 1D mean is
𝑆31 =
Microsoft Excel =STDEVA(A:A)/SQRT(COUNT(A:A))
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The standard error is sometimes referred to as the standard error of the mean.
Linear correlation coefficient (R2)
The linear correlation coefficient R2 is a number between zero and one and helps us interpret the
relationship between two quantities.
2 tending toward 1 implies linear dependence.
2 tending toward 0 implies independence.
2 tending toward some number in-between 0 and 1 implies dependence, but not necessarily
How reliably we know where R
2 is heading depends on the influence of random error; therefore,
this reliability improves as the number of data points increases. Another consequence of this is that
2 also implies precision; even if you have a linear relationship, R2 can still drift away from 1
depending on, and implying, the precision of your data.
𝑅? = (∑(𝑥6 – 𝑥̅)(𝑦6 – 𝑦1))?
∑(𝑥6 – 𝑥̅)? ∑(𝑦6 – 𝑦1)?
Microsoft Excel =RSQ(known_ys, known_xs)
Error Propagation
In general, you will want combine your various measurements and averages mathematically—
adding, multiplying, raising to a power, and so on—and you need a way to properly combine the
standard errors of these quantities to find the standard error of the final quantity.
You will be
expected to propagate errors throughout the class and for the final.
Alas, the way to properly
show this involves multivariable calculus. However, here is a general feel for how the process
See supporting appendix for additional examples and walkthroughs.
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You can propagate any errors you will find in this course with just these two formulas. We can
find two immediate corollaries as our first examples:
𝐴 = 𝐵 + 𝑘 for a constant 𝑘 (we know with certainty, i.e. 𝑆W = 0), then
(𝑆Y)? = (𝑆Z)? + (0)? 𝑆Y = 𝑆Z
Now let us consider if 𝐴 = 𝑘𝐵, then
[𝑆𝐴Y? = [𝑆𝐵Z? + [0 𝑘? → (𝑆Y)? = [𝐵𝐴 ∗ 𝑆Z? → (𝑆Y)? = (𝑘 ∗ 𝑆Z)? → 𝑆Y = |𝑘| ∗ 𝑆Z
Note the interesting distinction in adding (or subtracting) a constant over multiplying (or dividing)
by a constant. One has no effect on the standard error and the other scales it. Also notice the step
right before
𝑘 was substituted in, the 𝑘 was still there but contained in 𝐴. In other words, when
just using the formula, the last example would normally come out to
𝑆Y = 𝐴
We will get to propagating through the averaging formulas in a moment, but we can still use the
previously discussed cases for two quick examples.
𝐿𝑒𝑡 𝜌 = 1/𝑚 → [𝑆𝜌c? = [-1 𝑆𝑚d? + [0 1?
A nice property of propagating errors is the squaring part that automatically lets you rewrite a
negative exponent as being positive. (Analogous to the formula for adding and subtracting being
the same, the formula for multiplying and dividing is essentially the same too.)
[𝑆𝜌c? = [+1 𝑆𝑚d? + [0 1? → 𝑆c = 𝜌 𝑆𝑚d
𝐴 = 𝐵 ± 𝐶 ± 𝐷 ± ⋯
(𝑆Y)? = (𝑆Z)? + (𝑆h)? + (𝑆i)? + ⋯
Y = j(𝑆Z)? + (𝑆h)? + (𝑆i)? + ⋯
𝐴 = 𝐵k ∗ 𝐶d ∗ 𝐷l ∗ …
[𝑆𝐴Y? = [𝑙 𝑆𝐵Z? + [𝑚 𝑆𝐶h? + [𝑛 𝑆𝐷i? + ⋯
Y = 𝐴m[𝑙 𝑆𝐵Z? + [𝑚 𝑆𝐶h? + [𝑛 𝑆𝐷i? + ⋯
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For a more general example, suppose
𝑧 =
(𝑥 + 𝑦)
j𝑥 – 𝑦
p? = [3 𝑥𝑆r+sO𝑦? + [- 1 2 𝑥𝑆rtO𝑦?
[𝑆𝑧p? = [3 𝑥𝑆r+sO𝑦? + [1 2 𝑥𝑆rtO𝑦?
= 𝑆rtO = u(𝑆r)? + v𝑆Ow?
[𝑆𝑧p? =
r)? + v𝑆Ow?
𝑥 + 𝑦 ⎠ ⎞
+ ⎝ ⎛1 2 u(𝑆r𝑥)?-+𝑦v𝑆Ow?⎠ ⎞?
[𝑆𝑧p? = 9 (𝑆r()𝑥?++𝑦v)𝑆?Ow? + 1 4 (𝑆r()𝑥?-+𝑦v)𝑆?Ow?
= 𝑧m9 (𝑆r()𝑥?++𝑦v)𝑆?Ow? + 1 4 (𝑆r()𝑥?-+𝑦v)𝑆?Ow?
The same can be done with the 2D averages, propagating 𝑆O through the formulas determining the
slope and intercept from linear regression.
The estimated quantifications of the reliability of the two-dimensional averages from linear
standard error of slope, 𝑚, and the standard error of intercept, 𝑏, are
= 𝑆
Om𝑁 𝐷 𝑎𝑛𝑑 𝑆~ = 𝑆Om∑𝐷𝑥6?
𝑤𝑖𝑡ℎ 𝐷 = 𝑁 ; 𝑥6? – M; 𝑥6N?
Now we have our general set of standard errors along with a means to propagate them into any
formulas we will encounter in this course. So let us discuss their expected behavior as the number
of measurements gets larger and larger.

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Theoretical Convergences
If we have only random errors, then one-dimensional statistics predicts the following as the number
of measurements N gets larger and larger, theoretically approaching infinity.
𝑋1 and 𝑆3 are the best estimates of their theoretical true values, 𝜇 and 𝜎3, then
𝑋1 5 ÄÅÅÇ 𝜇 and 𝑆3 5 ÄÅÅÇ 𝜎3
If we have only random errors, then two-dimensional linear statistics predicts the following as the
number of measurements N gets larger and larger, theoretically approaching infinity.
m, b, and Sy are the best estimates of their theoretical true values of slope α, intercept β, and
standard error of the dependent variable σ
y, then
ÄÅÅÇ 𝛼 and 𝑆
ÄÅÅÇ 𝛽 and 𝑆~
Fractional Error
Fractional error is a ratio comparing some sort of deviation to what is being deviated from. In terms of
precision, this is a standard error compared to the quantity it is the error of,
𝐹𝐸 =
r 𝑥
In terms of accuracy, this is the deviation from what is found experimentally to what is considered the
accepted, true, value over the accepted, true, value.
𝐹𝐸 =
árà – 𝑥âää
Sometimes it is desirable to convert this to a percent fractional error,
𝑃𝐹𝐸 = 𝐹𝐸 ∗ 100
This course’s general test of accuracy is a PFE called a percent difference.
% 𝑑𝑖𝑓𝑓 = é𝑥árà𝑥- 𝑥âää
é ∗ 100
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Rounding Rules
We are now ready to discuss the formal rounding rules for this course. You always use unrounded
numbers in calculations
; however, when formally reporting a result, you want to present it as
reliably as you know it along with some indication of how accurate your value is which depends
on the
precision of your measurements. For example, Alex needs to find the circumference of a
round table (
C = 2p r) with a tape measurer and p is an irrational number that goes on forever. It
would be reasonable for Alex to report the circumference as 3.14 meters and it would be
unreasonable for Alex to report 3.141592653 meters because Alex did measure to the nanometer
Rounding Rules when formally presenting a result
Keep one significant figure in your result’s standard error and that tells you where to round the
An exception can be made if the first digit in the standard error rounds to a one, then you can keep
a second significant figure in the standard error and round the result to where that second digit is.
If your standard error happens to be larger than the value it represents, then that value is totally
unreliable. Just keep one significant figure for both the value and its standard error and put a
question mark by your value.
If you can do a test of accuracy with a percent difference, just keep 2 significant figures.
Remember, this is just how to present your experimental finding. You
always use unrounded
numbers in calculations
, which is easy when using Excel because you can just select the cell with
the unrounded number in it.

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Ideally you will not spend your life as a scientist rediscovering things that have already been
discovered where checks of accuracy will not be possible. Suppose you determine how many
moles of an unknown gas you have in a container as
𝑄 = 5.747 𝑚𝑜𝑙, 𝑆ì = 1.987 𝑚𝑜𝑙
You would formally report this as
Hopefully the situation of your standard error being much larger than the value is represents will
be quite rare. Consider trying to find the density of some metal with a foot and stone.
𝜌 = 3.34643
o , 𝑆c = 442.2365
o , % 𝑑𝑖𝑓𝑓 = 23.94185 %
You would formally report this as
𝑄 = 6 ± 2 𝑚𝑜𝑙
𝜌 = 3 (? ) ±
% 𝑑𝑖𝑓𝑓 = 24 %
Suppose when determining the height of the John Hancock skyscraper, and after repeated
measurements, you find
ℎ1 = 463.45997 𝑚, 𝑆ôö = 123.456789 𝑚, % 𝑑𝑖𝑓𝑓 = 0.97167 %
You would formally report this as
After performing Millikan’s Oil Drop experiment, you find the mass of an electron to be
á = 9.109784 𝑥 10to8 𝑘𝑔, 𝑆d
= 0.000139 𝑥 10to8 𝑘𝑔, % 𝑑𝑖𝑓𝑓 = 0.00440644 %
You would formally report this as
𝒉 = 4.6𝑥10? ± 𝟏. 𝟐 𝑥10? 𝑚
% 𝒅𝒊𝒇𝒇 = 𝟎. 𝟗𝟕 %
𝒉 = (4.6 ± 𝟏. 𝟐 )𝑥10
? 𝑚
% 𝒅𝒊𝒇𝒇 = 𝟎. 𝟗𝟕 %
á = 9.10978𝑥10to8 ± 1.4𝑥10to• 𝑘𝑔
% 𝑑𝑖𝑓𝑓 = 0.0044 %
á = (9.10978 ± 0.00014 )𝑥10-31 𝑘𝑔
% 𝑑𝑖𝑓𝑓 = 0.0044 %

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Comparing Precision with Accuracy
Compare the following two deviations. On one side, use the statistically derived best estimate of
the range of reliability from the best estimate to the accepted, true, value of
x. On the other side,
use the actual deviation of your best estimate to what is considered the accepted, true, value of
? ™𝒙𝒆𝒙𝒑 – 𝒙𝒂𝒄𝒄
Case one: 𝑺
> ™𝒙𝒆𝒙𝒑 – 𝒙𝒂𝒄𝒄
This means that whatever deviation you see in that test of accuracy is within the plus or minus
swing of reliability random errors seem to have already forced upon you. In other words, just based
on the determination of precision, whatever deviation is happening on the right hand side is already
inside what is considered reliable. This implies that
systematic errors have been reduced to
negligible influences based on your level of precision.
Case two: 𝑺
< ™𝒙𝒆𝒙𝒑 – 𝒙𝒂𝒄𝒄
This implies there are errors having a clear influence on your data that are not being properly
reflected in the standard error determinations. In other words, hunt for non-negligible
errors. This is when you want to look inside the absolute value to see if 𝑥árà > 𝑥âää or 𝑥árà <
âää , and try to better determine the nature of the inaccuracy.
Remember, these are still tests based on estimations. As you might have noticed in the rounding
rules concerning standard errors,
there will always be significant uncertainty in them based on
the number of actual measurements we can make
in the laboratory time allotted to us.
Furthermore, you should often question the accepted value itself. Often a physical quantity is
determined theoretically using approximations along the way, like assuming an ideal gas, while
other times many accepted values are dependent on environmental conditions, like temperature
and pressure, and your experimental conditions might not actually match those. In other words,
you might actually be more accurate than the accepted value!