Electricity and Magnetism 203-SN2-RE • CEGEP, Montreal

Revision sheet: lab measurements, uncertainties and graphical analysis (203-SN2-RE)

Q: How do you calculate the uncertainty of a digital multimeter?

You read it in the datasheet, which gives a percentage of the reading plus a number of digits, and you add the two terms. One digit is one unit of the last place displayed on the range used. For a reading of 12.47 volts on the 20.00 volt range with a specification of plus or minus 0.5 percent plus 2 digits: half a percent of 12.47 is 0.062 volt, plus twice 0.01 volt, which is 0.02 volt, for 0.08 volt in total. The display says nothing about accuracy: four displayed digits do not make four correct digits.

Q: Why does repeating a measurement eventually stop helping?

Repeating only reduces the random part of the uncertainty, the one given by the standard deviation divided by the square root of the number of measurements. The datasheet uncertainty is present in every reading and does not move. As soon as the datasheet term exceeds the statistical term, multiplying the measurements gains nothing: on ten readings of a supply set to five volts, thirteen millivolts of statistical scatter are added to forty-five millivolts of datasheet uncertainty, and going to forty measurements would gain less than seven millivolts out of a total of fifty-eight. You then need a better instrument, not more rows in the table.

Q: How do you know whether a measurement is consistent with the expected value?

You compare the intervals, never the central values. The measurement covers the central value plus or minus its uncertainty; if the expected value belongs to that interval, the two are consistent. When the expected value itself carries an uncertainty, like the five percent tolerance of a resistor, you check that the two intervals overlap. Writing that 226 ohms is not 220 ohms and that the measurement is therefore wrong is the costliest reasoning error of the chapter, because it makes you declare a successful experiment failed.

Q: Why does a voltmeter skew the measurement, and how do you know?

Because it is connected in parallel and lets a small current through: the resistance seen by the circuit becomes that of the setup in parallel with that of the voltmeter, therefore smaller, and the voltage read falls. On a divider of two one-hundred-kilohm resistors under twelve volts, a one-megohm voltmeter makes you read 5.71 volts instead of 6.00 volts, almost five percent too low. The error is systematic, so no repetition reveals it: you detect it by comparing the resistance of the voltmeter with that of the circuit, and you correct it by calculation or by choosing resistors a thousand times smaller.

Q: When should you linearize rather than calculate point by point?

As soon as the law is not affine and you have several points. Linearization turns the law into a straight line, for example the logarithm of the voltage against time for a capacitor discharge, and it brings two things that an isolated point will never give: the proof that the model holds, by the alignment of the points, and a diagnosis when it does not, by the shape of the curvature. A single point always supplies a number, even when the phenomenon is not the one you think.

An electricity lab report is graded as much on the uncertainty as on the value. The rules fit in a few lines, but each one has its trap, and they are not the same as in mechanics: here the instrument has no graduation to interpolate, it has a DATASHEET, and it disturbs the circuit it measures.

This sheet gives what is really lost: the rule applied to the wrong shape of formula, the systematic error mistaken for good precision, the consistency conclusion written the wrong way. The corrected exercises of the same chapter practise each of these moves on a real setup.

Mark this sheet as read or add it to your favourites: a free account, no password, keeps your read sheets and favourites from one visit to the next, tells you which chapter to tackle next and lets you ask for the method or exercise you are missing. Create your space, an email is enough.

The thread of the chapter

In electricity, the uncertainty almost never comes from the hand that reads: it comes from the instrument, its datasheet and the circuit. A four-digit display does not have four correct digits, a voltmeter changes the circuit it measures, and a small difference of two nearby voltages destroys the precision. So you work out before the lab what each quantity will contribute, and you improve the contribution that dominates.

This chapter is part of 203-SN2-RE, Electricity and Magnetism

What is on this sheet

1. The essentials
2. The mistakes that cost marks
3. Which method to choose
4. How the answer is expected to be written
5. Check before you hand in
6. The typical problem, taken apart
7. Learn by heart
8. Frequently asked questions

Topics covered

Datasheet uncertainty
Digit and range
Absolute uncertainty
Relative uncertainty
Propagation in a quotient
Extreme-values method
Differential method
Uncertainty of the mean
Systematic error
Loading effect
Component tolerance
Extreme lines

The essentials

Reading an instrument's uncertainty, then propagating it

• Digital multimeter: the uncertainty is in the DATASHEET, in the form $\pm(a\,\%\ \text{of reading}+n\ \text{digits})$ . One digit is one unit of the last displayed place, and it changes with the RANGE: $0.01$ V on the $20.00$ V range, $0.1$ V on the $200.0$ V range.
• Result: $x=(\bar x\pm\Delta x)$ unit. The uncertainty to one significant figure, two if that figure is a $1$ , and the value rounded to the same decimal place.
• Sum or difference: $\Delta(x\pm y)=\Delta x+\Delta y$ , ABSOLUTE uncertainties. Product, quotient, power: $\dfrac{\Delta z}{z}=|a|\dfrac{\Delta x}{x}+|b|\dfrac{\Delta y}{y}$ for $z=kx^{a}y^{b}$ , RELATIVE uncertainties.
• An EXACT factor contributes nothing: a counted number of turns, a conversion $2$ , the constant $\mu_{0}$ . A quantity with no uncertainty does not contribute, even if it multiplies everything else.
• Parallel combination: $R_{p}=\dfrac{R_{1}R_{2}}{R_{1}+R_{2}}$ is NEITHER a sum NOR a product. You propagate with partial derivatives, $\left|\dfrac{\partial R_{p}}{\partial R_{1}}\right|=\left(\dfrac{R_{2}}{R_{1}+R_{2}}\right)^{2}$ .

The slope of

U

against

I

IS the resistance,

219.8\ \Omega

here; the point at

40

mA drops off by

2.9\,\%

because the resistor has heated up.

When the two resistors carry the SAME relative tolerance $t$ , the series combination and the parallel combination both keep it: $\dfrac{\Delta R_{p}}{R_{p}}=t$ . This is what makes the $5\,\%$ tolerance usable directly on a whole network.

Telling random from systematic, then concluding

• RANDOM error: changes sign from one measurement to the next. The mean reduces it, the standard deviation measures it, $\dfrac{s}{\sqrt{N}}$ puts a number on it. This is the TYPE A uncertainty.
• SYSTEMATIC error: always keeps the same sign. The mean keeps it intact, and a tiny standard deviation then gives the ILLUSION of an excellent measurement. It is detected only by analyzing the circuit.
• The datasheet uncertainty is TYPE B: it is present even if you never repeat, and repeating does not reduce it. As soon as it dominates type A, taking more measurements no longer helps.
• Loading effect: a voltmeter of resistance $R_{V}$ connected across a resistance $R$ makes the reading too low. The error is negligible as long as $R_{V}\gg R$ , typically by a factor of a thousand. An ammeter, for its part, must have a resistance much SMALLER than that of the circuit.
• Consistency: two values are consistent when their INTERVALS overlap. Comparing the central values decides nothing.

This divider makes you read

5.71

V instead of

6.00

V, that is

-4.8\,\%

: a systematic error that a hundred repetitions would not reveal.

A component's tolerance and a measurement uncertainty are NEVER added: one describes the scatter of a population of components out of the factory, the other what our measurement does not know about this one. You confront the two intervals, you do not combine them.

The mistakes that cost marks

These are the errors I correct most often in session. Each one costs marks on a paper, even when the reasoning behind it is right.

1. Adding absolute uncertainties in a quotient

2 points, and an uncertainty eighty times too small

What not to write

“ $R=\dfrac{U}{I}$ with $\Delta U=0.05$ V and $\Delta I=0.3$ mA: $\Delta R=0.05+0.0003=0.05$ Ω.”

What to write

“ $\dfrac{\Delta R}{R}=\dfrac{0.05}{6.42}+\dfrac{0.3}{28.4}=1.84\,\%$ , so $\Delta R=226\times 0.0184=4$ Ω.”

Why: Adding volts and amperes does not even make sense. In a product or a quotient, only the RELATIVE uncertainties, which have no unit, add. The variant of the same trap is to write $\Delta R=\dfrac{\Delta U}{\Delta I}$ : you obtain $167\ \Omega$ , an uncertainty of the same order as the resistance, which should have raised an alarm.

2. Forgetting that a square doubles the relative uncertainty

1 to 2 points, and an uncertainty underestimated by a quarter

What not to write

“ $P=\dfrac{U^{2}}{R}$ : $\dfrac{\Delta P}{P}=\dfrac{0.2}{12.0}+\dfrac{2}{47}=5.9\,\%$ .”

What to write

“The voltage is SQUARED: $\dfrac{\Delta P}{P}=2\times\dfrac{0.2}{12.0}+\dfrac{2}{47}=7.6\,\%$ .”

In P = U²/R, enlarging U by ΔU adds TWO strips U·ΔU to the square: the relative uncertainty of U counts twice, 2 × 1.67 % and not 1.67 %.

Why: In $z=kx^{a}y^{b}$ , the EXPONENT multiplies the relative uncertainty. A square doubles, a cube triples, a square root halves. This is the most frequent trap of the chapter, and it also affects the cross-section of a wire, where the squared diameter often makes up $80\,\%$ of the uncertainty budget.

3. Treating a parallel combination as a product

2 points, and an uncertainty exactly twice too large

What not to write

“ $R_{1}$ and $R_{2}$ at $5\,\%$ each, so $R_{p}$ at $10\,\%$ .”

What to write

“ $\Delta R_{p}=\left(\dfrac{220}{320}\right)^{2}\times 5+\left(\dfrac{100}{320}\right)^{2}\times 11=3.44\ \Omega$ , that is $5.0\,\%$ .”

R₁ and R₂ both at their minimum, then both at their maximum: Rp goes from 65.3 to 72.2 Ω, exactly ± 5 %. Adding the relative uncertainties announces ± 10 %, double.

Why: The parallel formula contains a SUM in the denominator: neither the absolute rule nor the relative rule applies to it. You go through partial derivatives. The result is worth remembering: when the two relative tolerances are equal, the combination keeps them, series or parallel.

4. Taking both high bounds in the extreme-values method

2 points: the value obtained is neither the maximum nor the minimum

What not to write

“ $P_{max}=\dfrac{12.2^{2}}{49}$ , with $U$ and $R$ at their maximum value.”

What to write

“ $R$ is in the DENOMINATOR: $P_{max}=\dfrac{12.2^{2}}{45}$ , with $R$ at its MINIMUM.”

P grows with U and decreases with R: the maximum is at the corner (U max, R min). The two high bounds give 3.038 W, neither the maximum nor the minimum.

Why: The extreme-values method asks you to choose each bound according to the DIRECTION OF VARIATION of the result. A quantity in the numerator is taken at its maximum, a quantity in the denominator at its minimum. Check: the central value must fall between $z_{min}$ and $z_{max}$ , roughly in the middle.

5. Believing that a small standard deviation proves a good measurement

3 points, and a wrong conclusion about the whole experiment

What not to write

“Ten readings give $5.71$ V every time: the measurement is excellent.”

What to write

“The standard deviation is small, but the voltmeter loads the divider: the reading is biased by $-4.8\,\%$ , and no repetition shows it.”

The ten readings land in the same place, but next to 6.00 V: the voltmeter loads the divider. A zero standard deviation says nothing about a systematic error.

Why: The standard deviation measures ONLY the random part. A systematic error, loading effect, resistance of the leads, shifted zero, thermal drift, produces exactly the opposite of the expected alarm signal: it makes the measurements very reproducible. Only an analysis of the circuit reveals it.

6. Confusing component tolerance with measurement uncertainty

2 points, and an invalid consistency argument

What not to write

“Resistor marked $220\ \Omega$ at $5\,\%$ , measurement $(226\pm 4)\ \Omega$ : total uncertainty $\pm 15\ \Omega$ .”

What to write

“The manufacturer guarantees $[209, 231]$ , the measurement covers $[222, 230]$ : the two overlap, the component is within specification.”

The measurement falls within the guaranteed interval: the component is within specification. You confront the two intervals; adding tolerance and uncertainty means nothing.

Why: The tolerance describes the scatter of a POPULATION of components; the uncertainty describes what the measurement does not know about THIS ONE. You confront the two intervals, you never add them. The only thing you add to a measurement uncertainty is another measurement uncertainty, for example the $0.1\ \Omega$ of the ohmmeter zero.

7. Announcing a regression slope with no uncertainty

2 points, often on the consistency conclusion that follows

What not to write

“The calculator gives a slope of $219.800$ : $\Delta R$ is negligible.”

What to write

“The extreme lines that pass through all the bars give $216$ and $224$ : $R=(220\pm 4)\ \Omega$ .”

The slope inherits the bars of the points: the extreme lines that pass through all of them frame it, at 216 and 224 Ω with the true bars, hence R = (220 ± 4) Ω.

Why: The number of decimals of a calculator is a precision of CALCULATION, not an accuracy of MEASUREMENT. The slope inherits the uncertainty of the points, and you estimate it by the half-range of the slopes of the extreme lines. Without it, there is no way to say whether the slope is consistent with the expected value.

8. Calculating a small difference of two nearby quantities

3 points, and an uncertainty announced forty times too small

What not to write

“ $r=\dfrac{1.565-1.520}{0.100}=0.450\ \Omega$ , to $0.3\,\%$ like the voltages.”

What to write

“The difference is $0.045$ V with $\Delta=0.010$ V, that is $22\,\%$ ; with the current, $r=(0.45\pm 0.12)\ \Omega$ .”

Each voltage is uncertain to only 0.3 %, but their two ends float: the 45 mV gap carries 10 mV of uncertainty, that is 22 %.

Why: A difference adds the absolute uncertainties, and when its result is SMALL compared with the terms, the relative uncertainty blows up: the uncertainty survives intact while the quantity vanishes. Three remedies: make the difference grow, here by drawing five times more current, measure it directly, or replace the two points by a regression, which measures a slope instead of a gap.

Which method to choose

Which propagation rule, according to the form of the relation

The form of the formula that links the result to the measured quantities

Series: the ABSOLUTE uncertainties of the two resistors are laid end to end. Product: two strips I·ΔU and U·ΔI, the RELATIVE uncertainties add.

If a sum or a difference → add the ABSOLUTE uncertainties

Example: $(12.47\pm 0.08)+(6.42\pm 0.05)=(18.89\pm 0.13)$ V
If a product, a quotient or a power → add the relative uncertainties, each multiplied by its EXPONENT

Example: $P=\dfrac{U^{2}}{R}$ : $2\times 1.67\,\%+4.26\,\%=7.59\,\%$
If a sum in the denominator, like the parallel combination → go through the partial derivatives, differential method

Example: $\left(\dfrac{220}{320}\right)^{2}\times 5+\left(\dfrac{100}{320}\right)^{2}\times 11=3.44\ \Omega$
If a complicated formula that is monotonic in each variable → EXTREME-values method, choosing each bound according to the direction of variation

Example: $\Delta P=\dfrac{P_{max}-P_{min}}{2}=\dfrac{3.308-2.842}{2}=0.23$ W
If the result comes from a regression SLOPE → draw the two extreme lines and take the half-range of their slopes

Example: slopes $-0.42$ and $-0.48$ V/A: $r=(0.45\pm 0.03)\ \Omega$
If the law is not affine, exponential for example → LINEARIZE first, then regress on the transformed quantity

Example: $u=U_{0}e^{-t/\tau}$ gives $\ln u$ affine in $t$ , of slope $-\dfrac{1}{\tau}$

A mixed formula is handled in two steps. For the internal resistance obtained from $\dfrac{U_{1}-U_{2}}{I}$ , you first apply the ABSOLUTE rule to the numerator, then the RELATIVE rule to the quotient. Doing the reverse, or applying a single rule to the whole formula, is the fault that comes back most often.

How the answer is expected to be written

A marker ticks steps. Here they are in order, with the concluding sentence expected word for word.

Presenting the result of an electricity lab

When to use it: The report asks for a quantity calculated from electrical measurements and its comparison with a nominal or tabulated value.

1 Give each measurement with its uncertainty AND say where it comes from: instrument datasheet, component tolerance, observed scatter.
2 Write the relation, name the propagation rule chosen, and detail the contributions one by one, in percent.
3 Round the uncertainty to one significant figure, two if it starts with $1$ , then align the value on the same decimal place.
4 Conclude on CONSISTENCY by comparing the intervals, never the central values, and say which measurement dominates the uncertainty budget.

Concluding sentence

“ $R=(226\pm 4)\ \Omega$ . The marked value, $220\ \Omega$ at $\pm 5\,\%$ , guarantees $[209, 231]\ \Omega$ , which overlaps $[222, 230]\ \Omega$ : the component is within specification. The current supplies $58\,\%$ of the uncertainty, it is what should be measured on a finer range.”

The trap: Concluding “the measurement is good, the gap is only $2.7\,\%$ ” without mentioning the intervals: the marker expects a criterion, not an impression. And declaring an inconsistency while forgetting that the marked value itself carries a tolerance costs as much as the reverse.

Marking: Typically 1 mark for the uncertainties of the measurements, 2 for the propagation, 1 for the rounded write-up, 1 for the justified conclusion.

Check before you hand in

Five minutes of checking recover more marks than one more problem started in a hurry.

The reasonable relative uncertainty

In the electricity lab, a relative uncertainty is most often between $0.5$ and $10\,\%$ . Below $0.2\,\%$ , suspect a forgotten uncertainty, often the datasheet one; above $20\,\%$ , look for a difference of two nearby quantities.

$\Delta R=0.05\ \Omega$ on a multimeter measurement of $226\ \Omega$ : invented uncertainty.
The smallest range that does not overload

The digit term is set by the RANGE, not by the value measured. Measuring $12$ V on the $200$ V range costs three times more uncertainty than on the $20$ V range, without the display giving any sign of it.

$(12.47\pm 0.08)$ V on $20.00$ V against $(12.5\pm 0.3)$ V on $200.0$ V.
The order of magnitude of the result

A parallel equivalent resistance is always SMALLER than the smaller of the two; the terminal voltage of a cell is always below its electromotive force; the power dissipated in a lab circuit rarely exceeds a few watts.

$R_{p}=68.75\ \Omega$ for $100$ and $220\ \Omega$ : consistent, since $68.75<100$ .
The sum of the deviations from the mean

Before calculating a standard deviation, check that the sum of the deviations from the mean is ZERO. A nonzero sum signals an error in the mean or a value copied wrongly, and it is spotted in two seconds.

$+0.007-0.043+0.067-0.063+\ldots=0$ : the table is right.
The check by the intercept

A line $U=RI$ must pass through the origin, a line $U=\varepsilon-rI$ must cut the axis at the electromotive force, a semi-logarithmic line must cut at $\ln U_{0}$ . When this check fails, the setup is carrying something other than what you think.

$b=0.007$ V on the Ohm line: consistent with zero, the setup is sound.

The typical problem, taken apart

The internal resistance of a cell, from five points to the conclusion

A cell supplies a variable resistor. For five settings, we record $I$ (A): $0.10$ , $0.20$ , $0.30$ , $0.40$ , $0.50$ and $U$ (V): $1.52$ , $1.47$ , $1.43$ , $1.38$ , $1.34$ .

Model: $U=\varepsilon-rI$ . The extreme lines through all the uncertainty bars have slopes of $-0.42$ and $-0.48$ V/A.

The intercept gives the electromotive force, the slope gives the opposite of the internal resistance, and the half-range of the extreme slopes gives its uncertainty.

Step 1

Means: $\bar I=0.30$ A and $\bar U=1.428$ V.

Why

The regression line always passes through the mean point: this is what then lets you calculate the intercept without any other formula.

Step 2

Slope: $a=\dfrac{\sum(I_{i}-\bar I)(U_{i}-\bar U)}{\sum(I_{i}-\bar I)^{2}}=\dfrac{-0.045}{0.10}=-0.45$ V/A, so $r=0.45\ \Omega$ .

Why

The internal resistance is the OPPOSITE of the slope. Taking the slope positive would give a cell that gets stronger when you draw on it, which makes no physical sense.

Step 3

Intercept: $\varepsilon=\bar U-a\bar I=1.428+0.135=1.563$ V.

Why

Extrapolating to zero current gives the electromotive force. Measuring it directly with a voltmeter would be wrong: the voltmeter itself draws a small current.

Step 4

Uncertainty: $\Delta r=\dfrac{0.48-0.42}{2}=0.03\ \Omega$ , so $r=(0.45\pm 0.03)\ \Omega$ .

Why

The half-range of the extreme slopes is the standard graphical method. Without it, the slope remains a number with no status, and the next conclusion is impossible.

Step 5

By the quick method, $r=\dfrac{1.565-1.520}{0.100}=0.450\ \Omega$ but $\Delta r=0.12\ \Omega$ .

Why

Same central value, four times worse, because a difference of $45$ mV still carries $10$ mV of uncertainty. The regression is worth the time it costs.

The conclusion, written out

“ $r=(0.45\pm 0.03)\ \Omega$ and $\varepsilon=(1.563\pm 0.020)$ V. The nominal value $1.5$ V does not belong to $[1.54, 1.58]$ V, which is normal for a new cell: a nominal value is not an expected value in the metrological sense.”

The classic mistake on this problem: Taking $r=+0.45$ from the slope without changing the sign, or measuring the electromotive force with the voltmeter at open circuit instead of extrapolating it.

Learn by heart

• Multimeter datasheet: $\pm(a\,\%+n$ digits $)$ , and the digit depends on the RANGE.
• Sum or difference: absolute uncertainties added.
• Product, quotient, power: relative uncertainties added, each multiplied by its exponent.
• Parallel: neither sum nor product, partial derivatives; equal tolerances, tolerance kept.
• An exact factor contributes no uncertainty.
• Systematic: same sign, the mean does not remove it. Random: the mean reduces it.
• $R_{V}\gg R$ for a voltmeter, $R_{A}\ll R$ for an ammeter.
• Tolerance and uncertainty are compared, never added.
• A regression slope with no uncertainty is not a result.
• Small difference of two large quantities: make it grow, measure it directly, or regress.

Frequently asked questions

How do you calculate the uncertainty of a digital multimeter?

You read it in the datasheet, which gives a percentage of the reading plus a number of digits, and you add the two terms. One digit is one unit of the last place displayed on the range used. For a reading of 12.47 volts on the 20.00 volt range with a specification of plus or minus 0.5 percent plus 2 digits: half a percent of 12.47 is 0.062 volt, plus twice 0.01 volt, which is 0.02 volt, for 0.08 volt in total. The display says nothing about accuracy: four displayed digits do not make four correct digits.

Why does repeating a measurement eventually stop helping?

Repeating only reduces the random part of the uncertainty, the one given by the standard deviation divided by the square root of the number of measurements. The datasheet uncertainty is present in every reading and does not move. As soon as the datasheet term exceeds the statistical term, multiplying the measurements gains nothing: on ten readings of a supply set to five volts, thirteen millivolts of statistical scatter are added to forty-five millivolts of datasheet uncertainty, and going to forty measurements would gain less than seven millivolts out of a total of fifty-eight. You then need a better instrument, not more rows in the table.

How do you know whether a measurement is consistent with the expected value?

You compare the intervals, never the central values. The measurement covers the central value plus or minus its uncertainty; if the expected value belongs to that interval, the two are consistent. When the expected value itself carries an uncertainty, like the five percent tolerance of a resistor, you check that the two intervals overlap. Writing that 226 ohms is not 220 ohms and that the measurement is therefore wrong is the costliest reasoning error of the chapter, because it makes you declare a successful experiment failed.

Why does a voltmeter skew the measurement, and how do you know?

Because it is connected in parallel and lets a small current through: the resistance seen by the circuit becomes that of the setup in parallel with that of the voltmeter, therefore smaller, and the voltage read falls. On a divider of two one-hundred-kilohm resistors under twelve volts, a one-megohm voltmeter makes you read 5.71 volts instead of 6.00 volts, almost five percent too low. The error is systematic, so no repetition reveals it: you detect it by comparing the resistance of the voltmeter with that of the circuit, and you correct it by calculation or by choosing resistors a thousand times smaller.

When should you linearize rather than calculate point by point?

As soon as the law is not affine and you have several points. Linearization turns the law into a straight line, for example the logarithm of the voltage against time for a capacitor discharge, and it brings two things that an isolated point will never give: the proof that the model holds, by the alignment of the points, and a diagnosis when it does not, by the shape of the curvature. A single point always supplies a number, even when the phenomenon is not the one you think.

Practise it

Corrected exercises: Lab measurements, uncertainties and graphical analysis, 203-SN2-RE

A method is proved on a paper, not on a sheet. The set for the same chapter takes each of these traps into a problem, with the solution written out step by step.

15 corrected exercises
150 points
225 minutes

Do the exercises

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© Ahmed Squalli Houssaini. Revision sheet published at www.letuteurscientifique.ca/en/fiches/cegep-phys2-lab-uncertainties. Free for personal and classroom use; republishing it elsewhere requires written permission (legal notice).

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Revision sheet: lab measurements, uncertainties and graphical analysis (203-SN2-RE)

The essentials

Reading an instrument's uncertainty, then propagating it

Telling random from systematic, then concluding

The mistakes that cost marks

1. Adding absolute uncertainties in a quotient

2. Forgetting that a square doubles the relative uncertainty

3. Treating a parallel combination as a product

4. Taking both high bounds in the extreme-values method

5. Believing that a small standard deviation proves a good measurement

6. Confusing component tolerance with measurement uncertainty

7. Announcing a regression slope with no uncertainty

8. Calculating a small difference of two nearby quantities

Which method to choose

Which propagation rule, according to the form of the relation

How the answer is expected to be written

Presenting the result of an electricity lab

Check before you hand in

The typical problem, taken apart

The internal resistance of a cell, from five points to the conclusion

Learn by heart

Frequently asked questions

How do you calculate the uncertainty of a digital multimeter?

Why does repeating a measurement eventually stop helping?

How do you know whether a measurement is consistent with the expected value?

Why does a voltmeter skew the measurement, and how do you know?

When should you linearize rather than calculate point by point?

Corrected exercises: Lab measurements, uncertainties and graphical analysis, 203-SN2-RE

See also

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