Revision sheet: measurement in the laboratory, significant figures and uncertainty (202-SN1-RE)
This revision sheet covers measurement in the laboratory in CEGEP general chemistry: significant figures, rounding rules, glassware and tolerances, propagation of uncertainties, accuracy and precision, the calibration curve. It is the first chapter of the course and the first lab of the semester, and its rules then apply to every report of the year. It is written for CEGEP students in Science (Sciences de la nature), under the revised 202-SN1-RE (formerly 202-NYA) outline.
The thread: the last digit of a chemistry result is not given by the calculator, it is given by the least precise instrument in the procedure. Everything else follows, including the least intuitive conclusion of the chapter, namely that improving an instrument that is already excellent changes nothing at all. Once the sheet is read, the corrected exercise set on the same chapter puts every reflex to the test.
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The thread of the chapter
The last digit of a result is not given by the calculator: it is given by the least precise instrument in the procedure. The whole chapter consists of finding which one it is, and lining up on it.
•Every nonzero digit counts. A zero SANDWICHED between digits counts. A TRAILING zero counts if there is a decimal point.
•Zeros placed BEFORE the first nonzero digit never count: they only position the decimal point. 0.00450 g carries three significant figures, not five.
•In scientific notation only the coefficient counts: 2.50×103 carries three, the power of ten contributes none. It is the only way to write 2500 without ambiguity.
•EXACT numbers, of unlimited precision, never limit a result: coefficients of a balanced equation, subscripts of a formula, definitions of units such as 1 L =1000 mL, counts.
•A tabulated constant is NOT exact: NA=6.022×1023 written like this carries four significant figures and can therefore limit a calculation.
The meniscus falls between two marks: 36 is certain, the 5 is estimated. A measurement is written with all its certain digits plus ONE estimated digit, and that is all.
The quick check: rewrite the number in scientific notation. The leading zeros disappear by themselves, and what remains in the coefficient is exactly what counts.
The two rounding rules, which do not count the same thing
•MULTIPLICATION and DIVISION: the result keeps the smallest number of SIGNIFICANT FIGURES among the factors.
•ADDITION and SUBTRACTION: the result keeps the smallest number of DECIMAL PLACES among the terms.
•Confusing the two is the most common error of the chapter, and it never produces a visible absurdity: only a digit too many or too few.
•Round ONLY ONCE, at the end. Rounding at every step piles up errors that do not cancel: keep the unrounded value in the calculator and round only for display.
•An uncertainty is rounded to ONE significant figure, and the result lines up on it: C=(0.0948±0.0006) mol/L, not 0.09479.
On the left we count significant figures, on the right decimal places. The two rules apply to different things, and that is where marks are lost.
From uncertainties to the precision of a result
•Relative uncertainty =measured valueabsolute uncertainty, as a percentage. It is the ONLY quantity that allows two instruments of different capacities to be compared.
•Sum or difference: the ABSOLUTE uncertainties add. A difference of two burette readings therefore carries the reading uncertainty twice.
•Product or quotient: the RELATIVE uncertainties add. A concentration, which is a quotient, therefore adds the relative uncertainties of the mass and of the volume.
•Accuracy: agreement with the true value. Precision: agreement of the measurements with one another. The two are INDEPENDENT, and a set of measurements can be very precise and entirely wrong.
•A constant, reproducible offset signals a SYSTEMATIC error: only a calibration corrects it, never repeating the measurement.
The relative uncertainties add, and the burette contributes two thirds of the total: it is the one to improve, while the pipette and the titrant hardly weigh anything.
Tolerances to know: 100 mL beaker ±5 mL, 50 mL graduated cylinder ±0.5 mL, burette ±0.05 mL PER READING, 25.00 mL volumetric pipette ±0.03 mL, 250.0 mL volumetric flask ±0.15 mL. A flask CONTAINS a volume, a pipette DELIVERS one.
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.Counting the leading zeros of a small number
the number of figures in everything else in the problem
What not to write
“0.00450 g: there are five digits after the decimal point, so five significant figures.”
What to write
“Zeros before the first nonzero digit do not count: what remains is 4, 5 and the final zero, that is THREE significant figures. In scientific notation, 4.50×10−3 g.”
The three grey zeros measure nothing, they place the decimal point: they disappear in scientific notation. What remains is 4, 5 and the final zero, three significant figures.
Why: A leading zero measures nothing, it positions the decimal point. The proof is immediate: the same mass is written 4.50 mg, and nobody sees five significant figures there. Changing units does not change the precision of a measurement.
2.Applying the significant-figures rule to a subtraction
1 to 2 marks, and everything that follows from this mass
What not to write
“52.39−48.12: both numbers have four significant figures, so the result keeps four, 4.270 g.”
What to write
“In a subtraction we count DECIMAL PLACES: two on each side, so two in the result, 4.27 g. That is now only three significant figures.”
Two weighings with four significant figures, a difference with three. The lost figure never comes back.
Why: The two rules do not apply to the same thing. Here the subtraction of two CLOSE numbers destroyed a significant figure without any rule being broken: this is loss by cancellation, and it rules out weighing by difference on a coarse balance.
3.Believing that a stoichiometric coefficient limits the precision
a result given to one figure instead of four
What not to write
“2H2+O2→2H2O: the coefficient 2 has only one significant figure, so my result will have only one.”
What to write
“The coefficient 2 comes from balancing: it is an EXACT number, of unlimited precision. It limits nothing, and the measured data are what decides.”
Why: A counting number is not a measurement. The same goes for the subscripts of a formula, for 1 L =1000 mL and for a calculated dilution factor. The opposite error exists too, and it costs more: believing that a TABULATED constant is exact, when 6.022×1023 carries only four figures.
4.Rounding at every step of a chained calculation
the last significant figure, systematically
What not to write
“n=58.442.4530=0.0420 mol, then C=0.25000.0420=0.168 mol/L.”
What to write
“We keep 0.0419747 in the calculator, we divide, and we round only at the final display: C=0.1679 mol/L.”
Why: Rounding errors do not cancel, they pile up. Over a chain of three or four operations, rounding at every step regularly shifts the last digit, and that digit is precisely the one the marker looks at.
5.Comparing two instruments by their raw tolerance
an inverted conclusion about a whole procedure
What not to write
“The balance is ±0.0001 g and the flask ±0.15 mL: clearly the flask is the coarse one, but the balance is so fine that the result will be excellent.”
What to write
“We compare the RELATIVE values: 2.45300.0001=0.0041% for the balance against 250.00.15=0.060% for the flask. The flask dominates by a factor of 15, and it is the one to improve.”
Expressed relative to the quantity measured, the two tolerances become comparable: the flask weighs fifteen times more than the balance, contrary to what the raw numbers suggested.
Why: Two tolerances expressed in different units cannot be compared. Expressed relative to the quantity measured, they become comparable, and the ranking is often reversed. This is the move that decides what you buy and what you change in a procedure.
6.Forgetting the initial reading of a burette
a factor of two on the dominant step of the titration
What not to write
“The burette reads 23.17 mL at the endpoint and is readable to ±0.05 mL, so V=(23.17±0.05) mL.”
What to write
“Two READINGS were made, 0.45 then 23.17. The volume delivered is 22.72 mL and the absolute uncertainties of a difference ADD: ΔV=0.05+0.05=0.10 mL.”
The volume delivered is the DIFFERENCE of two readings, each good to ±0.05 mL: the two uncertainties add, ±0.10 mL.
Why: A volume delivered is never read, it is CALCULATED as a difference of two readings. The propagation rule for a difference therefore applies, and it doubles the uncertainty. This is also why a titration is aimed at an endpoint near three quarters of the burette: the same 0.10 mL weighs half as much on 45 mL as on 22 mL.
7.Concluding that a precise set of measurements is an accurate one
the whole experiment, and the confidence placed in it
What not to write
“My four measurements give 0.801, 0.800, 0.802 and 0.801 g/mL: they are tightly grouped, so my procedure is good.”
What to write
“They are precise, not accurate: the true value is 0.789 g/mL, and all four are offset by +0.012. A constant offset signals a SYSTEMATIC error, a pycnometer not properly dried or a balance not tared.”
The four points almost touch, so the set is PRECISE; but all of them lie +0.012 from the true value, so it is not ACCURATE. Repeating does not reduce this bias.
Why: Scatter says nothing about accuracy. It is even the most dangerous case, since reproducibility inspires confidence and no repetition of the measurement will ever reveal the bias: only a calibration, or a comparison with a reference value, will bring it out.
8.Extrapolating a calibration curve outside its range
an entirely invented result, presented with four figures
What not to write
“My line gives ρ=0.007480C+0.9982 over [0,20]%, so a sample at 1.25 g/mL is 33.7%.”
What to write
“1.25 g/mL lies outside the calibrated range: the line is worthless there. And the chemistry settles it on its own, since sodium chloride saturates near 26% by mass at 20 °C: such a solution does not exist.”
The line is valid only where the points are. Extended in red, it announces 33.7%, beyond saturation: a solution that does not exist.
Why: A calibration line is a model FITTED over an interval, and nothing guarantees that linearity continues beyond it. The cheapest check is the y-intercept: if it does not land on the value of the blank, the calibration has to be redone before it is even used.
Which method to choose
How many figures to keep in a result
The operation that produced the result, and nothing else
If a multiplication or a division → the smallest number of SIGNIFICANT FIGURES among the factors
Example: 4.2638÷100.09, five and five, gives 0.042600
If an addition or a subtraction → the smallest number of DECIMAL PLACES among the terms
Example: 25.00+50.0, two and one, gives 75.0
If a mix of both in the same expression → apply the rule of each operation IN ORDER, without rounding in between
Example: (52.3874−48.1236)÷100.09: first four decimal places, then five figures
If an uncertainty has been calculated → it rules everything: round it to one figure and line the result up on it
Example: ±0.0006 imposes C=0.0948 mol/L, three figures and not four
If the factor is an exact number → it does not count in the comparison, ignore it
Example: the coefficient 2 of an equation, or the 1000 of 1 L =1000 mL
The order of priority is always the same: if a calculated uncertainty exists, it overrides the two rounding rules, which are only a convenient substitute when no propagation has been done.
Which glassware to use
The role this volume plays in the final calculation
The pipette withdraws a fixed volume, the flask holds one, the burette measures it by two readings; the beaker serves only for volumes that go into no calculation.
If it goes directly into the calculation and must be withdrawn → volumetric pipette, which DELIVERS a fixed volume
Example: the 25.00 mL aliquot of a titration
If it goes into the calculation and must be contained, after dissolving → volumetric flask, which CONTAINS a fixed volume
Example: making up to the mark of a 250.0 mL flask
If it is delivered gradually up to an observed mark → burette, and two readings, hence twice the uncertainty
Example: the 22.72 mL delivered up to the endpoint
If it goes into no calculation → beaker or graduated cylinder, and certainly nothing better
Example: the 30 mL of distilled water for a rinse
A flask does not withdraw and a pipette does not prepare: confusing them costs the whole experiment. And measuring precisely a volume that goes into no calculation makes the result neither more accurate nor more precise, it only ties up the slowest glassware to rinse.
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 result with its uncertainty
When to use it: As soon as a laboratory report asks for a quantity calculated from several measurements.
1Calculate the quantity without rounding, keeping every digit of the calculator.
2List each source of uncertainty and calculate its RELATIVE uncertainty as a percentage.
3Add the relative uncertainties if the quantity is a product or a quotient, the absolute ones if it is a sum or a difference.
4Convert the total relative uncertainty into an absolute uncertainty, then round it to ONE significant figure.
5Round the result to the same place as the uncertainty, and write the two together with the unit.
Concluding sentence
“The relative uncertainties are 0.44% for the burette, 0.12% for the pipette and 0.096% for the titrant, that is 0.66% in total. The absolute uncertainty is therefore 0.0006 mol/L and the result is written C=(0.0948±0.0006) mol/L.”
The trap: Reporting every digit of the calculator, 0.09479, when the uncertainty falls on the fourth decimal place. The fifth decimal place is out of reach of the procedure, and the marker sees it before reading the calculation.
Marking: Typically 1 mark for each relative uncertainty correctly calculated, 1 mark for the right propagation rule, 1 mark for rounding the uncertainty to one figure, 1 mark for the lined-up final statement. The numerical value alone never earns full marks.
Using a calibration curve
When to use it: The laboratory provides a series of points of known concentration and asks for the concentration of an unknown.
1Check that the relation is linear over the interval, by seeing whether the intermediate points stray from the line by less than the scatter of the measurements.
2Calculate the slope from the two extreme points, with its unit.
3Read the y-intercept and CHECK that it lands on the value of the blank.
4Write the equation of the line, then invert it for the unknown.
5Check that the value found lies INSIDE the calibrated interval before concluding.
Concluding sentence
“The slope is 7.480×10−3 (g/mL) per percent and the y-intercept 0.9982 g/mL, which is indeed the density of pure water. The sample at 1.0525 g/mL is therefore 7.26%, a value lying within the calibrated interval.”
The trap: Giving the result with as many figures as the starting measurement. The subtraction 1.0525−0.9982=0.0543 leaves only three significant figures, and that is the fundamental limit of the method, not a slip in the calculation.
Marking: 1 mark for the slope with its unit, 1 for the interpreted y-intercept, 1 for the equation, 2 for the inverse reading, 1 for checking the range. The sentence that places the result inside the interval is almost always counted.
Check before you hand in
Five minutes of checking recover more marks than one more problem started in a hurry.
Rewrite the number in scientific notation
The leading zeros disappear, the significant zeros stay in the coefficient. What remains written is exactly what counts.
0.00450 becomes 4.50×10−3: three figures, and the question is settled without discussion.
Compare the uncertainty with the last digit announced
The uncertainty must fall on the LAST digit of the result, neither before nor after. If it falls two places higher, too many figures have been kept.
±0.0006 mol/L on 0.09479: the uncertainty is at the fourth decimal place, so we publish 0.0948.
Check the y-intercept of a calibration
It must land on the value of the blank, measured or tabulated. It is the cheapest check of the session, and it is done before the line is used.
0.9982 g/mL, the density of pure water at 20 °C: the calibration is usable.
Check that no result lies outside the calibrated range
A value outside the interval of the calibration points is not a measurement, it is an extrapolation. Say so, or redo the calibration further out.
33.7% sodium chloride is beyond saturation, which levels off near 26% at 20 °C.
Redo the uncertainty calculation changing a single step
Halve the tolerance of the step you believe dominates. If the total barely moves, you picked the wrong weak link.
Improving the balance from 0.0001 to 0.00005 g takes the total from 0.064 to 0.062%: it was not the problem.
The typical problem, taken apart
A titration carried from the burette reading to the publishable concentration
25.00 mL of hydrochloric acid, taken with a volumetric pipette to ±0.03 mL, is titrated with a sodium hydroxide solution of concentration 0.1043 mol/L, known to ±0.0001 mol/L.
The burette, readable to ±0.05 mL, reads 0.45 mL before the titration and 23.17 mL at the endpoint.
The reaction is HCl+NaOH→NaCl+H2O, in a one-to-one ratio.
The pipette delivers 25.00 mL of acid; the burette, read twice, gives the volume of sodium hydroxide delivered up to the endpoint.
Step 1
V=23.17−0.45=22.72 mL. Two readings at ±0.05 mL, and the absolute uncertainties of a difference add: ΔV=0.10 mL.
Why
The volume delivered is not read, it is calculated. Forgetting the initial reading halves the uncertainty and spoils everything that follows.
Step 2
n(NaOH)=0.1043×0.02272=2.3697×10−3 mol.
Why
Four significant figures on each side, so four in the product. The unrounded value is nevertheless kept for the next step.
Step 3
One-to-one ratio, so n(HCl)=2.3697×10−3 mol and C=0.025002.3697×10−3=0.094788 mol/L.
Why
The coefficient 1 of the equation is an exact number: it does not limit the number of figures, contrary to what is often believed.
Three tolerances in three different units cannot be compared. Expressed relative to their quantity, they become comparable and additive.
Step 5
The concentration is a product of quotients, so the relative uncertainties add: 0.44+0.12+0.096=0.66%.
Why
This is the propagation rule for products and quotients. Adding them in absolute terms would make no sense, since the units do not match.
Step 6
Absolute uncertainty: 0.0066×0.094788=6.2×10−4 mol/L, rounded to 0.0006 mol/L.
Why
An uncertainty is rounded to one significant figure: announcing 0.000625 would claim to know the uncertainty to three figures, which makes no sense.
Step 7
The uncertainty falls on the fourth decimal place, so the result is written C=(0.0948±0.0006) mol/L.
Why
The uncertainty decides the last digit, never the calculator. The 0.09479 displayed announced a fifth decimal place that the procedure cannot support.
Step 8
The burette contributes 0.44 of the 0.66 percentage points, that is two thirds.
Why
Identifying the dominant step is what makes it possible to improve a procedure. The other two steps are not worth touching.
Step 9
By titrating 50.00 mL of acid, the volume delivered would rise to about 45.4 mL and the relative uncertainty of the burette to 45.440.10=0.22%.
Why
The remedy costs nothing: the instrument is not changed, the aliquot is. This is why an endpoint near three quarters of the burette is preferred.
The conclusion, written out
“The concentration of the hydrochloric acid is C=(0.0948±0.0006) mol/L, three significant figures. The uncertainty is dominated two thirds by the burette, and titrating a 50.00 mL aliquot instead of 25.00 mL would cut it by a third.”
The classic mistake on this problem: Publishing C=0.09479 mol/L because the calculator displays it. The uncertainty is 0.0006 mol/L: the fifth decimal place is out of reach of the procedure, and announcing it is a methodological fault, not a touch of extra precision.
Learn by heart
•Leading zeros: never. Sandwiched zeros: always. Trailing zeros after a decimal point: always.
•Multiply or divide, count significant figures. Add or subtract, count decimal places.
•Equation coefficient, formula subscript, definition of a unit: exact numbers, they limit nothing.
•Round only once, at the end.
•Sum or difference: absolutes add. Product or quotient: relatives add.
•An uncertainty is rounded to one figure, and the result lines up on it.
•Two burette readings, so twice the reading uncertainty.
•Precise does not mean accurate: a constant offset is a systematic error, which only calibration corrects.
•A flask contains, a pipette delivers.
•A calibration line is valid only within its range.
Frequently asked questions
How do I know whether a zero counts as a significant figure?
A zero counts if it is sandwiched between nonzero digits, or if it comes after the decimal point at the end of the number. It never counts if it is placed before the first nonzero digit, because it then only positions the decimal point. The surest way to decide is to rewrite the number in scientific notation: the leading zeros disappear by themselves and only the coefficient remains, with exactly the right number of figures.
Why is the rule not the same for an addition as for a multiplication?
Because the two operations do not propagate the error in the same way. In a multiplication, it is the relative errors that add, so the factor with the fewest significant figures decides. In an addition, it is the absolute errors, so the term whose last digit is furthest to the left decides, that is, the one with the fewest decimal places. Confusing the two rules is the most common error of the chapter.
Does a very precise balance guarantee a precise result?
No. What matters is not the raw tolerance of an instrument but its relative uncertainty, that is, its tolerance expressed relative to the quantity measured. A balance reading to a tenth of a milligram often contributes far less uncertainty than a volumetric flask, and improving the balance then changes nothing at all. In a chain of measurements, you calculate the relative uncertainty of each step, find the one that dominates, and improve only that one.
What is the difference between accuracy and precision?
Accuracy is agreement with the true value, precision is agreement of the measurements with one another. The two are independent. A set of tightly grouped measurements that are all offset by the same amount is precise and wrong: it is the signature of a systematic error, due for example to a balance that was not tared or a pycnometer that was not properly dried. Repeating the measurement will never reveal it; only a calibration or a comparison with a reference value will.
Can a calibration curve be used beyond the measured points?
No. A calibration curve is a model fitted over a given interval, and nothing guarantees that the relation stays linear beyond it. A result obtained by extrapolation can be numerically impeccable and chemically impossible, for example a concentration above the saturation of the solute. Either redo the calibration with a wider range of points, or dilute the sample to bring it back into the known range.
Should I round at every step of a multi-step calculation?
No, never. Rounding errors do not cancel, they accumulate, and over three or four operations they regularly shift the last digit of the result. Keep the full calculator value throughout the chain and round only once, when writing the final result. Only then do you apply the significant-figures rule, or better, line the result up on the calculated uncertainty.
Practise it
Corrected exercises: Measurement, significant figures and uncertainty, 202-SN1-RE
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