UMass Boston Physics 181

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.

Introduction

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 7/10th chance that a subsequent determination of the same

quantity will fall within this range. Equivalently, this predicts a roughly 7/10th 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.

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

UMass Boston Physics 181

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

reports.

UMass Boston Physics 181

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.

UMass Boston Physics 181

How to minimize experimental error: some examples

Type of Error |
Example | How to minimize it |

Random errors 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. |
||

Systematic errors 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 strip. 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 direction. • How would one can compensate for the incorrect results when the tape is stretched out? • How would you take care of improperly tared scale? |

Statistics

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

is Microsoft Excel =AVERAGE( ). |
𝑋1 = ∑ 34 5 5 67 8 |

A single determination of X’s standard deviation in one-dimensional statistics is

𝑆3 = :;6758 (𝑋𝑁6 –𝑋11)?

UMass Boston Physics 181

Microsoft Excel =STDEVA( ).

The standard deviation is usually symbolized with the Greek letter sigma: 𝜎3.

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 ∑ 𝑥6𝑦6 – ∑ 𝑥6 ∑ 𝑦6

𝐷

𝑏 =

∑ 𝑥6? ∑ 𝑦6 – ∑ 𝑥6 ∑ 𝑥6𝑦6

𝐷

𝑤𝑖𝑡ℎ 𝐷 = 𝑁 ; 𝑥6? – 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

𝑆O

= :;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 =

𝑆3

√𝑁

Microsoft Excel =STDEVA(A:A)/SQRT(COUNT(A:A))

UMass Boston Physics 181

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.

R2 tending toward 1 implies linear dependence.

R2 tending toward 0 implies independence.

R2 tending toward some number in-between 0 and 1 implies dependence, but not necessarily

linear.

How reliably we know where R2 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

R2 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

goes. See supporting appendix for additional examples and walkthroughs.

UMass Boston Physics 181

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:

If 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

𝑆Z

𝐵

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

If

𝐴 = 𝐵 ± 𝐶 ± 𝐷 ± ⋯

Then

(𝑆Y)? = (𝑆Z)? + (𝑆h)? + (𝑆i)? + ⋯

𝑆Y = j(𝑆Z)? + (𝑆h)? + (𝑆i)? + ⋯

If

𝐴 = 𝐵k ∗ 𝐶d ∗ 𝐷l ∗ …

Then

[𝑆𝐴Y? = [𝑙 𝑆𝐵Z? + [𝑚 𝑆𝐶h? + [𝑛 𝑆𝐷i? + ⋯

𝑆Y = 𝐴m[𝑙 𝑆𝐵Z? + [𝑚 𝑆𝐶h? + [𝑛 𝑆𝐷i? + ⋯

UMass Boston Physics 181

For a more general example, suppose

𝑧 =

(𝑥 + 𝑦)o

j𝑥 – 𝑦

[𝑆𝑧p? = [3 𝑥𝑆r+sO𝑦? + [- 1 2 𝑥𝑆r–tO𝑦?

[𝑆𝑧p? = [3 𝑥𝑆r+sO𝑦? + [1 2 𝑥𝑆r–tO𝑦?

𝑆r

sO = 𝑆rtO = u(𝑆r)? + v𝑆Ow?

[𝑆𝑧p? =

⎛⎝

3

u(𝑆r)? + v𝑆Ow?

𝑥 + 𝑦 ⎠ ⎞

?

+ ⎝ ⎛1 2 u(𝑆r𝑥)?-+𝑦v𝑆Ow?⎠ ⎞?

[𝑆𝑧p? = 9 (𝑆r()𝑥?++𝑦v)𝑆?Ow? + 1 4 (𝑆r()𝑥?-+𝑦v)𝑆?Ow?

𝑆p

= 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

regression—the standard error of slope, 𝑚, and the standard error of intercept, 𝑏, are

𝑆d

= 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.

UMass Boston Physics 181

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.

If 𝑋1 and 𝑆3 are the best estimates of their theoretical true values, 𝜇 and 𝜎3, then

𝑋1 5 ÄÅ→ÅÇ 𝜇 and 𝑆3 5 ÄÅ→ÅÇ 𝜎3

𝑆31

5→

ÄÅÅÇ 0

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.

If 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 σy, then

𝑚

5→

ÄÅÅÇ 𝛼 and 𝑆

d

5→

ÄÅÅÇ 0

𝑏

5→

ÄÅÅÇ 𝛽 and 𝑆~

5→

ÄÅÅÇ 0

𝑆O

5→

ÄÅÅÇ 𝜎

O

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 =

𝑆r 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 =

𝑥

árà – 𝑥âää

𝑥

âää

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

âää

é ∗ 100

UMass Boston Physics 181

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

scale.

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

result.

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.

UMass Boston Physics 181

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

𝑔

𝑐𝑚o , 𝑆c = 442.2365

𝑔

𝑐𝑚o , % 𝑑𝑖𝑓𝑓 = 23.94185 %

You would formally report this as

𝑄 = 6 ± 2 𝑚𝑜𝑙

𝜌 = 3 (? ) ± 400

𝑔

𝑐𝑚3

% 𝑑𝑖𝑓𝑓 = 24 %

Examples

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

𝑚

á = 9.109784 𝑥 10to8 𝑘𝑔, 𝑆d

õ

= 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 %

𝑚

á = 9.10978𝑥10to8 ± 1.4𝑥10to• 𝑘𝑔

% 𝑑𝑖𝑓𝑓 = 0.0044 %

𝑚

á = (9.10978 ± 0.00014 )𝑥10-31 𝑘𝑔

% 𝑑𝑖𝑓𝑓 = 0.0044 % OR

OR

UMass Boston Physics 181

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 x.

𝑺

𝒙

𝒆𝒙𝒑

? ™𝒙𝒆𝒙𝒑 – 𝒙𝒂𝒄𝒄™

Case one: 𝑺

𝒙

𝒆𝒙𝒑

> ™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

𝒙

𝒆𝒙𝒑

< ™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 systematic

errors. This is when you want to look inside the absolute value to see if 𝑥árà > 𝑥âää or 𝑥árà <

𝑥

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!