1.5 Measurement Uncertainty, Accuracy and Precision Flashcards
(31 cards)
Exact number
Number derived by counting or by definition
Uncertainty
Estimate amount of by which measurement differs from true value.
Significant Figures
(also, significant digits) all of the measured digits in a determination, including the uncertain last digit.
Rounding
Procedure used to ensure that calculated results properly reflect the uncertainty in the measurements used in the calculation.
What are the three rules for rounding numbers?
- 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)
- 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)
- 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.
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)
(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)
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.
(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)
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)
(a) 0.42
(b) 0.00387
(c) 421.2
(d) 28,684
(a) Add 2.334 mL and 0.31 mL
(b) Subtract 55.8752 m from 56.533 m.
(a) 2.64 mL; (b) 0.658 m
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.
(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.
(a) Multiply 2.334 cm and 0.320 cm.
(b) Divide 55.8753 m by 56.53 s.
(a) 0.747 cm^2
(b) 0.9884 m/s
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.
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)
What is the density of a liquid with a mass of 31.1415 g and a volume of 30.13 cm^3?
1.034 g/mL
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?
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.
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.
(a) 19 g/cm^3
(b) It is likely gold; the right appearance for gold and very close to the density given for gold.
Precision
How closely a measurement matches the same measurement when repeated.
Accuracy
How closely a measurement aligns with a correct value.
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
(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
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
(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
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
(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.
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
(a) exact
(b) exact
(c) uncertain
(d) exact
(e) uncertain
(f) uncertain
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
(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.
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
(a) two
(b) three
(c) five
(d) four
(e) six
(f) two
(g) five