Mechanics 203-SN1-RE • CEGEP, Montreal

Revision sheet: measurement, uncertainty and graph analysis in the lab (203-SN1-RE)

A 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: an absolute rule applied to a product, a forgotten exponent, a rounding copied from the calculator, a conclusion based on an impression.

This sheet serves the three physics courses of the revised Natural Science program, 203-SN1-RE, SN2-RE and SN3-RE. It gathers the traps, the tree that picks the rule from the formula, the write-up of a result and a complete pendulum example.

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

A measurement without an uncertainty is not a result, it is a number: the uncertainty says how many digits to write, whether two values can be declared equal, and which measurement to improve. Each trap applies the wrong rule to the shape of the formula, or judges without the interval.

This chapter is part of 203-SN1-RE, Mechanics

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

Absolute uncertainty
Relative uncertainty
Significant figures
Propagation of uncertainty
Extreme-values method
Differential method
Experimental standard deviation
Uncertainty of the mean
Linearization
Linear regression
Extreme lines
Consistency of a measurement

The essentials

Reading an instrument: half a division

• Instrument with divisions, ruler or thermometer: the reading uncertainty is HALF of the smallest division, $0.05$ cm for a ruler graduated in millimetres.
• Stopwatch started by hand: $0.2$ s, the reaction time, whatever the number of digits displayed.
• Measuring $N$ objects or $N$ periods at once, then dividing by an exact $N$ , divides the uncertainty by $N$ : ten oscillations in $(15.3\pm 0.2)$ s give $T=(1.53\pm 0.02)$ s.

The end falls between 23.4 and 23.5 cm: you estimate 23.47, to within half a division, that is ± 0.05 cm.

Writing and propagating an uncertainty

• Result: $x=(\bar x\pm\Delta x)$ unit. The uncertainty to one significant figure (two if that figure is a $1$ ), the value rounded to the same decimal place.
• Sum or difference: you ADD the absolute uncertainties, $\Delta(x\pm y)=\Delta x+\Delta y$ .
• Product, quotient, power, for $z=kx^{a}y^{b}$ : you add the RELATIVE uncertainties weighted by the exponents, $\dfrac{\Delta z}{z}=|a|\dfrac{\Delta x}{x}+|b|\dfrac{\Delta y}{y}$ .
• Extreme-values method: $\Delta z=\dfrac{z_{max}-z_{min}}{2}$ . Differential method: $\Delta z=\sum\left|\dfrac{\partial z}{\partial x_{i}}\right|\Delta x_{i}$ , angles in radians.

The error bars of each point frame all the plausible lines: half the gap between the steepest and the shallowest gives the uncertainty on the slope.

The three methods give the same order of magnitude for small uncertainties. The choice is made on the shape of the formula: products and powers, the relative rule; a twisted formula, extreme values or derivatives.

Analyzing a data set

• Repeated measurements: mean $\bar x$ , standard deviation $s=\sqrt{\dfrac{\sum(x_{i}-\bar x)^{2}}{N-1}}$ , uncertainty of the mean $\dfrac{s}{\sqrt{N}}$ .
• Linearizing: rewrite the law as $Y=aX+b$ . Pendulum: $T^{2}=\dfrac{4\pi^{2}}{g}L$ ; free fall: $h=\dfrac{g}{2}t^{2}$ .
• The slope of the regression gives the quantity sought; a clearly nonzero intercept signals a systematic error.
• Consistency: the expected value belongs to $[\bar x-\Delta x, \bar x+\Delta x]$ ; two measurements are consistent if their intervals overlap.

The eight fall times: s = 4.4 ms measures the scatter of ONE measurement, while s/√8 = 1.6 ms, almost three times narrower, is the uncertainty of the mean.

Repeating a measurement reduces the effect of random errors by one over the square root of N; no amount of repeating corrects a systematic error.

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 marks, and an uncertainty more than ten times too large

What not to write

“ $\rho=\dfrac{m}{V}$ with $\Delta m=0.1$ g and $\Delta V=0.5$ cm³: $\Delta\rho=0.1+0.5=0.6$ g/cm³.”

What to write

“ $\dfrac{\Delta\rho}{\rho}=\dfrac{0.1}{85.2}+\dfrac{0.5}{31.4}=1.71$ %, so $\Delta\rho=0.0171\times 2.71=0.05$ g/cm³.”

Why: Adding grams and cubic centimetres does not even make sense: for a product or a quotient, only the RELATIVE uncertainties, which have no unit, add up.

2. Forgetting the exponent of a squared quantity

1 to 2 marks

What not to write

“ $\rho=\dfrac{4m}{\pi d^{2}h}$ : $\dfrac{\Delta\rho}{\rho}=0.12\,\%+0.40\,\%+0.78\,\%=1.30$ %.”

What to write

“The diameter is squared, its relative uncertainty counts twice: $0.12\,\%+2\times 0.40\,\%+0.78\,\%=1.70$ %.”

Enlarging the side d by Δd adds TWO strips d·Δd to the square: the relative uncertainty of d² is 2 Δd/d. Counting only one means forgetting the exponent.

Why: A power multiplies the relative uncertainty by its exponent: $x^{2}=x\times x$ accumulates the error on $x$ twice. It is often the squared quantity that dominates the result.

3. Copying every digit from the calculator

0.5 to 1 mark in every result of the report

What not to write

“ $g=(9.81234\pm 0.0467)$ m/s².”

What to write

“ $g=(9.81\pm 0.05)$ m/s²: uncertainty to one significant figure, value to the hundredth.”

Why: If the hundredths are already uncertain, writing hundred-thousandths claims a precision the measurement does not have. The marker reads it as a misunderstanding of what an uncertainty is.

4. Pairing the bounds the wrong way in the extreme-values method

2 marks, and an uncertainty underestimated by a factor of 2.6

What not to write

“ $g_{max}=\dfrac{4\pi^{2}\times 0.802}{1.800^{2}}=9.772$ and $g_{min}=\dfrac{4\pi^{2}\times 0.798}{1.790^{2}}=9.832$ : $\Delta g=0.03$ m/s².”

What to write

“ $T$ is in the denominator: $g_{max}$ takes $L_{max}$ with $T_{min}$ , that is $9.882$ , and $g_{min}$ takes $L_{min}$ with $T_{max}$ , that is $9.723$ ; $\Delta g=0.08$ m/s².”

g grows with L and falls with T: its maximum is at the corner (L max, T min). The two upper bounds give 9.772, less than the two lower bounds, 9.832.

Why: Pairing the two upper bounds makes the effects cancel, since L is in the numerator and T in the denominator. The “maximum” obtained is even smaller than the “minimum”.

5. Subtracting the uncertainties when taking a difference

2 marks

What not to write

“ $\Delta T=23.1-22.4=0.7$ °C, with an uncertainty of $0.1-0.1=0$ °C.”

What to write

“The uncertainties add: $\Delta T=(0.7\pm 0.2)$ °C, that is $29$ % relative uncertainty.”

Each reading can lie anywhere inside its bracket: the real gap runs from 0.5 to 0.9 °C, that is 0.7 plus or minus 0.2.

Why: Nothing guarantees that the two errors cancel: if the first reading is low and the second high, the difference is overestimated by both. A small difference of large values is always very uncertain.

6. Judging consistency by eye

the conclusion of the report, often 2 to 4 marks

What not to write

“ $g=(9.72\pm 0.05)$ m/s² is close to $9.81$ : the measurement is consistent.”

What to write

“ $[9.67, 9.77]$ does not contain $9.81$ : the measurement is not consistent; the gap is almost two uncertainties.”

The vertical line at 9.81 crosses the lower bracket but misses the upper one: only the lower measurement is consistent.

Why: The word “close” means nothing without the uncertainty: it is the uncertainty that sets the scale. A measurement that is not consistent is not a mistake, it calls for a search for a systematic error.

7. Leaving the uncertainty of an angle in degrees inside a derivative

2 marks, and an uncertainty fifty times too large

What not to write

“ $\Delta R=\left|\dfrac{2v_{0}^{2}\cos 2\theta}{g}\right|\times 1=5.1$ m for $\Delta\theta=1°$ .”

What to write

“ $\Delta\theta=1°=0.01745$ rad: $\Delta R=5.10\times 0.01745=0.089$ m.”

Leaving Δθ = 1 in the derivative means counting one radian, an angle of 57.3°: the uncertainty comes out 57 times too large. One degree is only 0.01745 rad.

Why: The derivative of $\sin\theta$ is $\cos\theta$ only if $\theta$ is in radians. In degrees, the factor $\dfrac{\pi}{180}$ is missing.

Which method to choose

Which propagation rule for which formula

The shape of the relation between the result and the measurements

Sum: the two uncertainties are laid end to end, so you add the ABSOLUTE ones. Product: two strips y·Δx and x·Δy, so you add the RELATIVE ones Δx/x + Δy/y.

If a sum or a difference → add the ABSOLUTE uncertainties

Example: $(47.8\pm 0.1)-(12.3\pm 0.1)=(35.5\pm 0.2)$ cm
If a product, a quotient, powers → add the RELATIVE uncertainties multiplied by the exponents

Example: $g=\dfrac{4\pi^{2}L}{T^{2}}$ : $0.25\,\%+2\times 0.28\,\%=0.81$ %
If a sine, a logarithm, a mixed formula → differential method, angles in radians

Example: $R=\dfrac{v^{2}\sin 2\theta}{g}$ : $0.24+0.09=0.33$ m
If a complicated formula with no convenient derivative → extreme-values method: $\dfrac{z_{max}-z_{min}}{2}$ , bounds correctly paired

Example: $\dfrac{9.882-9.723}{2}=0.08$ m/s²
If a series of repeated measurements → mean and $\dfrac{s}{\sqrt{N}}$

Example: $s=4.4$ ms, $N=8$ : $1.6$ ms
If the slope of a graph → half the difference between the slopes of the extreme lines

Example: $\dfrac{4.97-4.84}{2}=0.065$ m/s²

A mixed formula, such as a work obtained as the difference of two energies, is handled in two steps: the relative rules for each energy, then the absolute rule for the difference.

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 a lab result

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

1 Give each measurement with its uncertainty and say where that uncertainty comes from (reading, resolution, scatter).
2 Calculate the value, then the uncertainty by the appropriate rule, writing the propagation formula before the numbers.
3 Round the uncertainty to one significant figure (two if it starts with 1), then the value to the same decimal place.
4 Conclude on consistency with the reference value by quoting the interval, and name a plausible source of error if it is not consistent.

Concluding sentence

“ $g=(9.78\pm 0.15)$ m/s². The reference value $9.81$ m/s² belongs to the interval $[9.63, 9.93]$ : the measurement is consistent. The uncertainty is dominated by the timing, which counts twice.”

The trap: Concluding “the result is good, the gap is only 0.3 %” without mentioning the interval: the marker expects a criterion, not an impression.

Marking: Typically 1 mark for the uncertainties of the measurements, 2 for the propagation, 1 for the rounded result, 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.

A reasonable relative uncertainty

In a CEGEP lab, a relative uncertainty is most often between 0.5 and 10 %. Below 0.1 %, suspect a forgotten uncertainty; above 30 %, a difference of nearby quantities.

$\Delta g=0.001$ m/s² on a hand-timed pendulum: made-up uncertainty.
The same decimal place

After rounding, the last decimal of the value and that of the uncertainty must coincide.

$(9.8\pm 0.05)$ or $(9.812\pm 0.05)$ : badly rounded; $(9.81\pm 0.05)$ : correct.
The weight of each term

In a sum of relative uncertainties, spot the dominant term: it is the measurement to improve, and it is what the conclusion must name.

$0.31\,\%+1.23\,\%$ : the timing dominates.
The slope against the single value

A quantity taken from a slope should be close to the one taken from each point: a large gap signals an outlier or a forgotten intercept.

$g=9.76$ from the slope, $9.75$ to $9.81$ point by point: consistent.

The typical problem, taken apart

Measuring g with a pendulum, from the raw measurement to the conclusion

You measure the length of a pendulum, $L=(0.650\pm 0.002)$ m, and the time for $20$ oscillations, $(32.4\pm 0.2)$ s. Theory: $T=2\pi\sqrt{\dfrac{L}{g}}$ .

Calculate g with its uncertainty and conclude on consistency with 9.81 m/s².

The length is measured to the centre of the mass; you time twenty oscillations to divide the starting error by twenty.

Step 1

Period: $T=\dfrac{32.4}{20}=1.620$ s and $\Delta T=\dfrac{0.2}{20}=0.010$ s.

Why

Dividing by an exact number divides the uncertainty too: timing twenty oscillations makes the starting error twenty times smaller per period.

Step 2

$g=\dfrac{4\pi^{2}L}{T^{2}}=\dfrac{4\pi^{2}\times 0.650}{1.620^{2}}=9.778$ m/s².

Why

You keep all the digits during the calculation; you round only at the end, once the uncertainty is known.

Step 3

$\dfrac{\Delta g}{g}=\dfrac{0.002}{0.650}+2\times\dfrac{0.010}{1.620}=0.31\,\%+1.23\,\%=1.54$ %.

Why

The formula is a product of powers: relative uncertainties, and the exponent 2 of T doubles its contribution.

Step 4

$\Delta g=0.0154\times 9.778=0.15$ m/s², so $g=(9.78\pm 0.15)$ m/s².

Why

The uncertainty starts with a 1: you keep two figures, and the value is rounded to the hundredth.

Step 5

Interval $[9.63, 9.93]$ : it contains $9.81$ .

Why

The conclusion rests on the interval, not on the relative gap of 0.3 %, which means nothing without the uncertainty.

The conclusion, written out

“ $g=(9.78\pm 0.15)$ m/s², consistent with $9.81$ m/s². The timing supplies $80$ % of the uncertainty: it is what should be improved, for example by timing fifty oscillations.”

The classic mistake on this problem: Forgetting the factor 2 in front of the relative uncertainty of T: you announce 0.9 % instead of 1.5 %, and an uncertainty that is too optimistic.

Learn by heart

• Sum, difference: absolute uncertainties added.
• Product, quotient: relative uncertainties added; power $n$ : multiply by $n$ .
• Extremes: $\dfrac{z_{max}-z_{min}}{2}$ , pairing the bounds according to numerator and denominator.
• Derivatives: $\sum\left|\dfrac{\partial z}{\partial x_{i}}\right|\Delta x_{i}$ , angles in radians.
• Repeated measurements: $\dfrac{s}{\sqrt{N}}$ ; four times as many measurements to divide by two.
• Rounding: one significant figure for the uncertainty, two if it starts with 1, and the same decimal place for the value.
• Consistent: the reference is inside the interval, or the intervals overlap.

Frequently asked questions

How do you calculate the uncertainty of a product or a quotient?

You add the relative uncertainties of each measurement, that is each uncertainty divided by its value, multiplying each one by the exponent of the quantity in the formula. This gives the relative uncertainty of the result, which you multiply by the calculated value to recover the absolute uncertainty.

How many significant figures should an uncertainty have?

You round the uncertainty to a single significant figure, or to two when that first figure is a 1, then you round the measured value to the same decimal position as the uncertainty. Writing more digits would claim a precision that the measurement does not have.

When is a measurement consistent with the theoretical value?

When the theoretical value lies inside the interval formed by the measured value plus or minus its uncertainty. Two measurements of the same quantity are consistent when their uncertainty intervals overlap. The apparent closeness of the numbers is not enough: the uncertainty sets the scale.

Why linearize the data in a lab report?

Because a straight line is recognized by eye and handled by regression: its slope and intercept give the quantities sought directly, using all the measurements at once. For a pendulum, you plot the square of the period against the length, whose slope is four pi squared divided by g.

Practise it

Corrected exercises: Measurement, uncertainty and graph analysis in the lab, 203-SN1-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

Something missing from this sheet? A method, a trap, an exercise to practise it, a mistake you spotted: tell me, I read every request.

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Next sheet Kinematics in one and two dimensions

© Ahmed Squalli Houssaini. Revision sheet published at www.letuteurscientifique.ca/en/fiches/cegep-phys1-measurement-uncertainty. Free for personal and classroom use; republishing it elsewhere requires written permission (legal notice).

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Revision sheet: measurement, uncertainty and graph analysis in the lab (203-SN1-RE)

The essentials

Reading an instrument: half a division

Writing and propagating an uncertainty

Analyzing a data set

The mistakes that cost marks

1. Adding absolute uncertainties in a quotient

2. Forgetting the exponent of a squared quantity

3. Copying every digit from the calculator

4. Pairing the bounds the wrong way in the extreme-values method

5. Subtracting the uncertainties when taking a difference

6. Judging consistency by eye

7. Leaving the uncertainty of an angle in degrees inside a derivative

Which method to choose

Which propagation rule for which formula

How the answer is expected to be written

Presenting a lab result

Check before you hand in

The typical problem, taken apart

Measuring g with a pendulum, from the raw measurement to the conclusion

Learn by heart

Frequently asked questions

How do you calculate the uncertainty of a product or a quotient?

How many significant figures should an uncertainty have?

When is a measurement consistent with the theoretical value?

Why linearize the data in a lab report?

Corrected exercises: Measurement, uncertainty and graph analysis in the lab, 203-SN1-RE

See also

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