how closely a measurement aligns with a correct value

number derived by counting or by definition

how closely a measurement matches the same measurement when repeated

procedure used to ensure that calculated results properly reflect the uncertainty in the measurements used in the calculation

(also, significant figures) all of the measured digits in a determination, including the uncertain last digit

(also, significant digits) all of the measured digits in a determination, including the uncertain last digit

estimate of amount by which measurement differs from true value

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Chemistry 2e · 1.5 Measurement Uncertainty, Accuracy, and Precision

Significant figures: practice problems

6 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

How to approach these problems

To count significant figures, start at the first nonzero digit. Zeros between nonzero digits count, trailing zeros after a decimal point count, and leading zeros never do; a number like 1000 is ambiguous, so write it in scientific notation to show what was measured. When you multiply or divide, keep as many significant figures as the measurement with the fewest. When you add or subtract, keep as many decimal places as the measurement with the fewest. Exact numbers, like 60 seconds in a minute, limit nothing. When the digit you drop is exactly 5, this book rounds to the even neighbour: 0.445 becomes 0.44. The usual mistakes: using the multiplication rule for a sum, and rounding at every step instead of once at the end.

Exercises 45
Express each of the following numbers in exponential notation with correct significant figures:
(a) 704
(b) 0.03344
(c) 547.9
(d) 22086
(e) 1000.00
(f) 0.0000000651
(g) 0.007157
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Exercises 47
Indicate whether each of the following can be determined exactly or must be measured with some degree of uncertainty:
(a) the number of seconds in an hour
(b) the number of pages in this book
(c) the number of grams in your weight
(d) the number of grams in 3 kilograms
(e) the volume of water you drink in one day
(f) the distance from San Francisco to Kansas City
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Exercises 49
How many significant figures are contained in each of the following measurements?
(a) 53 cm
(b) 2.05 $\times$ 10⁸ m
(c) 86,002 J
(d) 9.740 $\times$ 10⁴ m/s
(e) 10.0613 m³
(f) 0.17 g/mL
(g) 0.88400 s
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Exercises 51
Round off each of the following numbers to two significant figures:
(a) 0.436
(b) 9.000
(c) 27.2
(d) 135
(e) 1.497 $\times$ 10⁻³
(f) 0.445
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Exercises 53
Perform the following calculations and report each answer with the correct number of significant figures.
(a) 628 $\times$ 342
(b) (5.63 $\times$ 10²) $\times$ (7.4 $\times$ 10³)
(c) $\frac{28.0}{13.483}$
(d) 8119 $\times$ 0.000023
(e) 14.98 + 27,340 + 84.7593
(f) 42.7 + 0.259
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Exercises 55
Consider the results of the archery contest shown in this figure.
(a) Which archer is most precise?
(b) Which archer is most accurate?
(c) Who is both least precise and least accurate?
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Key terms · 7

accuracy: how closely a measurement aligns with a correct value
exact number: number derived by counting or by definition
precision: how closely a measurement matches the same measurement when repeated
rounding: procedure used to ensure that calculated results properly reflect the uncertainty in the measurements used in the calculation
significant digits: (also, significant figures) all of the measured digits in a determination, including the uncertain last digit
significant figures: (also, significant digits) all of the measured digits in a determination, including the uncertain last digit
uncertainty: estimate of amount by which measurement differs from true value

Learn 1.5 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 1.5 in the textbook · This section on openstax.org

Significant figures: practice problems

6 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

How to approach these problems

Exercises 45

Express each of the following numbers in exponential notation with correct significant figures:

(a) 704

(b) 0.03344

(d) 22086

(e) 1000.00

(f) 0.0000000651

(g) 0.007157

Exercises 47

Indicate whether each of the following can be determined exactly or must be measured with some degree of uncertainty:

(a) the number of seconds in an hour

(b) the number of pages in this book

(d) the number of grams in 3 kilograms

(e) the volume of water you drink in one day

(f) the distance from San Francisco to Kansas City

Exercises 49

How many significant figures are contained in each of the following measurements?

(a) 53 cm

(b) 2.05 $\times$ 10⁸ m

(d) 9.740 $\times$ 10⁴ m/s

(e) 10.0613 m³

(f) 0.17 g/mL

(g) 0.88400 s

Exercises 51

Round off each of the following numbers to two significant figures:

(a) 0.436

(b) 9.000

(d) 135

(e) 1.497 $\times$ 10⁻³

(f) 0.445

Exercises 53

Perform the following calculations and report each answer with the correct number of significant figures.

(a) 628 $\times$ 342

(b) (5.63 $\times$ 10²) $\times$ (7.4 $\times$ 10³)

(d) 8119 $\times$ 0.000023

(e) 14.98 + 27,340 + 84.7593

(f) 42.7 + 0.259

Exercises 55

Consider the results of the archery contest shown in this figure.

(a) Which archer is most precise?

(b) Which archer is most accurate?

4 targets are shown each with 4 holes indicating where the arrows hit the targets. Archer W put all 4 arrows closely around the center of the target. Archer X put all 4 arrows in a tight cluster but far to the lower right of the target. Archer Y put all 4 arrows at different corners of the target. All 4 arrows are very far from the center of the target. Archer Z put 2 arrows close to the target and 2 other arrows far outside of the target.

Key terms · 7

accuracy: how closely a measurement aligns with a correct value
exact number: number derived by counting or by definition
precision: how closely a measurement matches the same measurement when repeated
rounding: procedure used to ensure that calculated results properly reflect the uncertainty in the measurements used in the calculation
significant digits: (also, significant figures) all of the measured digits in a determination, including the uncertain last digit
significant figures: (also, significant digits) all of the measured digits in a determination, including the uncertain last digit
uncertainty: estimate of amount by which measurement differs from true value

Learn 1.5 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 1.5 in the textbook · This section on openstax.org