1.5 Measurement Uncertainty, Accuracy and Precision

Question 1

Exact number

Answer 1

Number derived by counting or by definition

Question 2

Uncertainty

Answer 2

Estimate amount of by which measurement differs from true value.

Question 3

Significant Figures

Answer 3

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

Question 4

Rounding

Answer 4

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

Question 6

What are the three rules for rounding numbers?

Answer 6

1. When adding or subtracting numbers, round the result to the same number of decimal places as the number with the least number of decimal places (the least certain value in terms of addition and subtraction)
2. When multiplying or dividing numbers, round the result to the same number of digits as the number with the least number of significant figures (the least certain value in terms of multiplication and division)
3. If the digit to be dropped (the one immediately to the right of the digit to be retained) is less than 5, “round down” and leave the retained digit unchanged; if it is more than 5, “round up” and increase the retained digit by 1. If the dropped digit is 5, and it’s either the last digit in the number or it’s followed only by zeros, round up or down, whichever yields an even value for the retained digit. If any nonzero digits follow the dropped 5, round up.

Question 7

Round the following to the indicated number of significant figures:
(a) 31.57 (to two significant figures)
(b) 8.1649 (to three significant figures)
(c) 0.051065 (to four significant figures)
(d) 0.90275 (to four significant figures)

Answer 7

(a) 31.57 rounds “up” to 32 (the dropped digit is 5, and the retained digit is even)
(b) 8.1649 rounds “down” to 8.16 (the dropped digit, 4, is less than 5)
(c) 0.051065 rounds “down” to 0.05106 (the dropped digit is 5, and the retained digit is even)
(d) 0.90275 rounds “up” to 0.9028 (the dropped digit is 5, and the retained digit is even)

Question 8

Rule when adding and subtracting significant figures: When adding or subtracting numbers, round the result to the same number of decimal places as the number with the fewest decimal places (the least certain value in terms of addition and subtraction)

(a) Add 1.0023 g and 4.383 g.
(b) Subtract 421.23 g from 486 g.

Answer 8

(a) 5.385 g (round to the thousandths place; three decimal places)
(b) Answer is 65 g (round to the ones place; no decimal places)

Question 9

Round the following to the indicated number of significant figures:
(a) 0.424 (to two significant figures)
(b) 0.0038661 (to three significant figures)
(c) 421.25 (to four significant figures)
(d) 28,683.5 (to five significant figures)

Answer 9

(a) 0.42
(b) 0.00387
(c) 421.2
(d) 28,684

Question 10

(a) Add 2.334 mL and 0.31 mL
(b) Subtract 55.8752 m from 56.533 m.

Answer 10

(a) 2.64 mL; (b) 0.658 m

Question 11

Multiplication and Division with Significant Figures
Rule: When multiplying or dividing numbers, round the result to the same number of digits as the number with the fewest significant figures (the least certain value in terms of multiplication and division).
(a) Multiply 0.6238 cm by 6.6 cm.
(b) Divide 421.23 g by 486 mL.

Answer 11

(a)
0.6238 cm x 6.6 cm = 4.11708 cm^2 –> result in 4.1 cm^2 (round to two significant figures) four significant figures x two significant figures —> two significant figures answer.
(b) 421.23 g/486 mL = 0.866728… g/mL —> results in 0.867 g/mL (round to three significant figures) five significant figures/three significant figures —> three significant figures answer.

Question 12

(a) Multiply 2.334 cm and 0.320 cm.
(b) Divide 55.8753 m by 56.53 s.

Answer 12

(a) 0.747 cm^2
(b) 0.9884 m/s

Question 13

One common bathtub is 13.44 dm long, 5.920 dm wide, and 2.54 dm deep. Assume that the tub is rectangular and calculate its approximate volume in liters.

Answer 13

V = I x w x d
= 13.44 dm x 5.920 dm x 2.54 dm
= 202.09459… dm^3 (value form calculator)
= 202 dm^3, or 202 L (answer rounded to three significant figures)

Question 14

What is the density of a liquid with a mass of 31.1415 g and a volume of 30.13 cm^3?

Answer 14

1.034 g/mL

Question 15

A piece of rebar is weighed and then submerged in a graduated cylinder partially filled with water, with results as shown.
Rebar mass = 69.658 g
“Final” volume = 22.4 mL
“Initial” volume = 13.5 mL
(a) Use these values to determine the density of this piece of rebar.
(b) Rebar is mostly iron. Does your result in (a) support this statement? How?

Answer 15

The volume of the piece of rebar is equal to the volume of the water displaced:
volume = 22.4 mL - 13.5 mL = 8.9 mL = 8.9 cm^3
(Rounded to the nearest 0.1 mL, per the rule for addition and subtraction)
The density is the mass-to-volume ratio:
density = mass/volume = 69.658 g/8.9 cm^3 = 7.8 g/cm^3
(rounded to two significant figures, per the rule for multiplication and division)

The density of iron is 7.9 g/cm^3, very close to that of rebar, which lends some support to the fact that rebar is mostly iron.

Question 16

An irregularly shaped piece of a shiny yellowish material is weighed and then submerged in a graduated cylinder.
Mass = 51.842 g
“Final” volume = 19.8 mL
“Initial” volume = 17.1 mL
(a) Use these values to determine the density of this material.
(b) Do you have any reasonable guesses as to the identity of this material? Explain your reasoning.

Answer 16

(a) 19 g/cm^3
(b) It is likely gold; the right appearance for gold and very close to the density given for gold.

Question 17

Precision

Answer 17

How closely a measurement matches the same measurement when repeated.

Question 18

Accuracy

Answer 18

How closely a measurement aligns with a correct value.

Question 19

Express each of the following numbers in scientific notation with correct significant figures:
(a) 711.0
(b) 0.239
(c) 90743
(d) 134.2
(e) 0.05499
(f) 10000.0
(g) 0.000000738592

Answer 19

(a) This number has 4 significant figures. In scientific notation, it is written as: 7.110 x 10^2
(b) This number has 3 significant figures. In scientific notation, it is written as: 2.39 x 10^-1
(c) This number has 5 significant figures. In scientific notation, it is written as: 9.0743 x 10^4
(d) This number has 4 significant figures. In scientific notation, it is written as: 1.342 x 10^2
(e) This number has 4 significant figures. In scientific notation, it is written as: 5.499 x 10^-2
(f) This number has 6 significant figures. In scientific notation, it is written as: 1.00000 x 10^4
(g) This number has 6 significant figures. In scientific notation, it is written as: 7.38592 x 10^-7

Question 20

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

Answer 20

(a) 7.04 x 10^2
(b) 3.344 x 10^-2
(c) 5.479 x 10^2
(d) 2.2086 x 10^4
(e) 1.00000 x 10^3
(f) 6.51 x 10^-8
(g) 7.157 x 10^-3

Question 21

Indicate whether each of the following can be determined exactly or must be measured with some degree of uncertainty:
(a) the number of eggs in a basket
(b) the mass of a dozen eggs
(c) the number of gallons of gasoline necessary to fill an automobile gas tank.
(d) the number of cm in 2m
(e) the mass of a textbook
(f) the time required to drive from San Francisco to Kansas City at an average speed of 53 mi/h

Answer 21

(a) The number of eggs in a basket: This can be determined exactly, as you can count the eggs.

(b) The mass of a dozen eggs: This must be measured with some degree of uncertainty, as the mass of each egg can vary slightly.

(c) The number of gallons of gasoline necessary to fill an automobile gas tank: This must be measured with some degree of uncertainty, as it depends on the exact capacity of the tank and the amount of gasoline already in it.

(d) The number of centimeters in 2 meters: This can be determined exactly, as there are 100 centimeters in a meter, so 2 meters is exactly 200 centimeters.

(e) The mass of a textbook: This must be measured with some degree of uncertainty, as the mass can vary slightly depending on the exact contents and materials of the textbook.

(f) The time required to drive from San Francisco to Kansas City at an average speed of 53 mi/h: This must be measured with some degree of uncertainty, as it depends on various factors such as traffic, road conditions, and exact starting and ending points.

Question 22

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

Answer 22

(a) exact
(b) exact
(c) uncertain
(d) exact
(e) uncertain
(f) uncertain

Question 23

How many significant figures are contained in each of the following measurements?
(a) 38.7 g

(b) 2 ×
1018 m

(c) 3,486,002 kg

(d) 9.74150 ×
10−4 J

(e) 0.0613 cm3

(f) 17.0 kg

(g) 0.01400 g/mL

Answer 23

(a) 38.7 g: This has 3 significant figures.

(b) 2 × 10^18 m: This has 1 significant figure.

(c) 3,486,002 kg: This has 7 significant figures.

(d) 9.74150 × 10^-4 J: This has 6 significant figures.

(e) 0.0613 cm³: This has 3 significant figures.

(f) 17.0 kg: This has 3 significant figures.

(g) 0.01400 g/mL: This has 4 significant figures.

Question 24

How many significant figures are contained in each of the following measurements?
(a) 53 cm

(b) 2.05 ×
108 m

(c) 86,002 J

(d) 9.740 ×
104 m/s

(e) 10.0613 m3

(f) 0.17 g/mL

(g) 0.88400 s

Answer 24

(a) two
(b) three
(c) five
(d) four
(e) six
(f) two
(g) five

Question 25

The following quantities were reported on the labels of commercial products. Determine the number of significant figures in each.
(a) 0.0055 g active ingredients

(b) 12 tablets

(c) 3% hydrogen peroxide

(d) 5.5 ounces

(e) 473 mL

(f) 1.75% bismuth

(g) 0.001% phosphoric acid

(h) 99.80% inert ingredients

Answer 25

(a) 0.0055 g active ingredients: This has 2 significant figures.

(b) 12 tablets: This has 2 significant figures (although counting numbers are considered exact and have an infinite number of significant figures, for practical purposes, we consider the digits provided).

(c) 3% hydrogen peroxide: This has 1 significant figure.

(d) 5.5 ounces: This has 2 significant figures.

(e) 473 mL: This has 3 significant figures.

(f) 1.75% bismuth: This has 3 significant figures.

(g) 0.001% phosphoric acid: This has 1 significant figure.

(h) 99.80% inert ingredients: This has 4 significant figures.

Question 26

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 ×
10−3

(f) 0.445

Answer 26

(a) 0.44
(b) 9.0
(c) 27
(d) 140
(e) 1.5 x 10^-3
(f) 0.44

Question 27

Round off each of the following numbers to two significant figures:
(a) 517

(b) 86.3

(c) 6.382 ×
103

(d) 5.0008

(e) 22.497

(f) 0.885

Answer 27

a) 517: Rounded to two significant figures, it is 520.

(b) 86.3: Rounded to two significant figures, it is 86.

(c) 6.382 × 10^3: Rounded to two significant figures, it is 6.4 × 10^3.

(d) 5.0008: Rounded to two significant figures, it is 5.0.

(e) 22.497: Rounded to two significant figures, it is 22.

(f) 0.885: Rounded to two significant figures, it is 0.89.

Question 28

Perform the following calculations and report each answer with the correct number of significant figures.
(a) 628 ×
342

(b) (5.63 ×
102) ×
(7.4 ×
103)

(c) 28.013.483

(d) 8119 ×
0.000023

(e) 14.98 + 27,340 + 84.7593

(f) 42.7 + 0.259

Answer 28

(a) 2.15 x 10^5
(b) 4.2 x 10^6
(c) 2.08
(d) 0.19
(e) 27,440
(f) 43.0

Question 29

Perform the following calculations and report each answer with the correct number of significant figures.
(a) 62.8 ×
34

(b) 0.147 + 0.0066 + 0.012

(c) 38 ×
95 ×
1.792

(d) 15 – 0.15 – 0.6155

(e) 8.78× (0.05000.478)

(f) 140 + 7.68 + 0.014

(g) 28.7 – 0.0483

(h) (88.5−87.57)/45.13

Answer 29

(a) 62.8 × 34:

62.8 has 3 significant figures, and 34 has 2 significant figures.
The result should be rounded to 2 significant figures.
Calculation: 62.8×34=2135.2

Rounded to 2 significant figures: 2100.

(b) 0.147 + 0.0066 + 0.012:

The number with the least decimal places is 0.012 (3 decimal places).
Calculation: 0.147+0.0066+0.012=0.1656

Rounded to 3 decimal places: 0.166.

© 38 × 95 × 1.792:

38 and 95 each have 2 significant figures, and 1.792 has 4 significant figures.
The result should be rounded to 2 significant figures.
Calculation: 38×95×1.792=6465.44

Rounded to 2 significant figures: 6500.

(d) 15 – 0.15 – 0.6155:

The number with the least decimal places is 15 (no decimal places).
Calculation: 15−0.15−0.6155=14.2345

Rounded to no decimal places: 14.
(e) 8.78 × (0.0500 / 0.478):

Calculation: 8.78×(0.4780.0500​)=0.918

Significant figures: 8.78 has 3 significant figures, 0.0500 has 3 significant figures, and 0.478 has 3 significant figures. The result should be rounded to 3 significant figures.
Rounded result: 0.918

(f) 140 + 7.68 + 0.014:

Calculation: 140+7.68+0.014=147.694

Significant figures: The number with the least decimal places is 140 (no decimal places).
Rounded result: 148

(g) 28.7 – 0.0483:

Calculation: 28.7−0.0483=28.6517

Significant figures: The number with the least decimal places is 28.7 (1 decimal place).
Rounded result: 28.7

(h) (88.5 – 87.57) / 45.13:

Calculation: (88.5−87.57)/45.13=0.0205

Significant figures: 88.5 has 3 significant figures, 87.57 has 4 significant figures, and 45.13 has 4 significant figures. The result should be rounded to 3 significant figures.
Rounded result: 0.0205

Question 30

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?

Answer 30

(a) Archer X
(b) Archer W
(c) Archer Y

Question 31

Classify the following sets of measurements as accurate, precise, both, or neither.
(a) Checking for consistency in the weight of chocolate chip cookies: 17.27 g, 13.05 g, 19.46 g, 16.92 g

(b) Testing the volume of a batch of 25-mL pipettes: 27.02 mL, 26.99 mL, 26.97 mL, 27.01 mL

(c) Determining the purity of gold: 99.9999%, 99.9998%, 99.9998%, 99.9999%

Answer 31

(a) Checking for consistency in the weight of chocolate chip cookies: 17.27 g, 13.05 g, 19.46 g, 16.92 g

These measurements are neither accurate nor precise. They vary significantly from each other, indicating a lack of precision, and without a known true value, we can’t determine accuracy.

(b) Testing the volume of a batch of 25-mL pipettes: 27.02 mL, 26.99 mL, 26.97 mL, 27.01 mL

These measurements are precise but not necessarily accurate. They are very close to each other, indicating high precision. However, since the true value should be 25 mL, they are not accurate.

(c) Determining the purity of gold: 99.9999%, 99.9998%, 99.9998%, 99.9999%

These measurements are both accurate and precise. They are very close to each other (precise) and close to the true value of high-purity gold (accurate).