Topic 11 – Measurement and Data Processing

Big ideas:

• All measured values must be expressed with appropriate units in order to have meaning.
• Quantitative data are obtained from measurements, and are always associated with random errors/uncertainties, determined by the apparatus and by human limitations such as reaction time.
• Propagation of random errors in data processing shows the impact of the uncertainties on the final result.
• Experimental design, procedure and equipment can all lead to systematic errors in measurement, which cause a deviation in a particular direction.
• Random errors affect the precision of a measurement while systematic errors affect its accuracy.
• Repeat trials and measurements will reduce random errors but not systematic errors.

Students will be able to…

• Convert within both metric and non-metric unit systems.
• Record uncertainties in all measurements as a range (±) to an appropriate precision.
• Distinguish between random and systematic errors.
• Distinguish between accuracy and precision when evaluating results.
• Propagate uncertainties in processed data, including the use of percentage uncertainties.

Theory of Knowledge questions/connections:

• Scientists report uncertainties on a measurement to quantify its precision, but some non-scientists interpret these uncertainties as meaning “but we don’t really know”.**
• Does the inclusion of an uncertainty make a measurement more or less clear?**
• How do measurement uncertainties impact how scientific results are received by the public?*
• How can we improve communication and understanding with respect to scientific results and issues?

Quote

“As we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don’t know we don’t know. And if one looks throughout the history of our country and other free countries, it is the latter category that tend to be the difficult ones.”
– Donald Rumsfeld, U.S. Secretary of Defense, Feb. 12, 2002

Precision and accuracy

The quality of any given measurement can be described in terms of accuracy or precision.
Accuracy: The closeness of a measured value to a standard or known value.

Example: The density of water (at 4ºC) is known to be 1.00 g/mL

	Single measurement	Set of three measurements
“Accurate”	-	➢ ➢
“Inaccurate”	➢ ➢	➢ ➢

Precision: The reproducibility of multiple measurements, often described by an uncertainty (±)
The number of significant figures (digits) reported also reflects a measurement’s precision.
In a data set that includes multiple measurements, the spread or variability in the individual measurements (the reproducibility) is a reflection of the precision.*

	Single measurement	Set of three measurements
“Precise” or “Reproducible”	➢ ➢ ➢	➢
“Imprecise” or “Not reproducible”	➢ ➢ ➢	- ➢

Note that the terms accuracy and precision are neither interchangeable nor mutually exclusive. They describe different properties of a measurement so it is possible to be both, neither, or one but not the other.

Let’s illustrate this another way using a dartboard as an analogy, where the “bullseye” or centre of the dartboard corresponds to the known value. Draw four Xs (darts) on each dartboard below such that they correspond to the stated levels of precision and accuracy:

Measurement-1

high precision high accuracy


high precision low accuracy



low precision high accuracy



low precision low accuracy

Measurement-1

Further Practice: Try the questions on the reverse side of this page!

Measurement-2

Practice Problems on precision and accuracy:

1. An object of mass 2.000 kg is placed on four different balances and for each balance the reading, in units of kg, is recorded five times:

Balance	1	2	3	4	5
A	2.000	2.004	1.998	2.002	2.006
B	2.009	1.999	2.001	1.988	2.004
C	2.013	2.015	2.016	2.017	2.014
D	1.998	1.999	1.999	2.000	1.999

Rank each balance from most precise to least precise and most accurate to least accurate**

2. The mass of a particular marble is known to be exactly 3.987 grams.

Craft an example of a single measurement of the marble’s mass that is:

a) precise but not accurate: _____________ b) accurate but not precise: _____________

3. Repeat #2, but craft examples of measurement sets (at least three measurements in each set, please):
   1. precise but not accurate: ____________________________________________________
   2. accurate but not precise: ____________________________________________________
4. Repeat #2 and #3, but this time for measurements of Avogadro’s constant, 6.02×1023.

a) precise but not accurate: _____________ b) accurate but not precise: _____________

100. precise but not accurate: ____________________________________________________
101. accurate but not precise: ____________________________________________________

5. What are some experimental factors that might limit the precision of a particular measurement?
6. What are some experimental factors that might limit the accuracy of a particular measurement?
7. To what extent are the terms precise and accurate subject to the judgement of the experimenter? Can you give an example of a situation where two experimenters might look at the same measurement(s) and disagree as to their precision or accuracy?
8. What are the limitations of the “dartboard analogy” as a tool for better understanding the terms precision and accuracy? Can you think of other analogies you have learned in your science education (i.e. “Concept X is like a _________…”) that may also have limited applications?
9. Obviously,** we use the terms precise and accurate in other contexts besides measurement.** Is there a difference between using precise language and using accurate language? Example, please.

Measurement-3

Significant figures in measured values

One of the central concepts of our first unit of study is that every experimental measurement has some uncertainty or error associated with it. In other words, all measurements are imprecise to some degree.

Looking at the number of significant figures in a particular measured value is one way to understand just how uncertain or imprecise that measurement is. A significant figure (“sig fig”) can be defined as:

• Digits in a measured value that carry meaning.
• Digits in a measured value that were actually measured.
• Digits in a measured value that reflect its precision rather than its magnitude (= place value).
• Digits in a measured value that appear in the coefficient when expressed in scientific notation.

These definitions can be a bit confusing, so let’s break things down to a few simple rules and try to understand this concept by exploring a few examples:

Rule #1: All non-zero digits are significant. Example: Mr. Furugori is 142 cm tall (3 s.f.)

• All non-zero digits have been measured and thus carry meaning.

Rule #2: Trailing zeros to the LEFT of a decimal place are not significant.

• These digits describe magnitude (place value), but are not measured digits.
• These digits do not contribute to the precision of a measurement.

Example: “This rock is 500,000,000 years old” is the same as saying “this rock is 5×109 years old”

The zeros were not measured; the meaning is clearly “approximately 500,000,000 years old”, not “exactly 500,000,000 years old to the nearest year”.

As such, there is only one significant figure (measured digit) in this measurement: the “5”

Rule #3: Zeros between significant figures are significant.

Example: “This rock is 500,000,038 years old.” Now the measurement has 9 s.f., not just 1 s.f. Why?

Rule #4: Trailing zeros to the RIGHT of a decimal place are significant.

Example: Consider the following measurements: 2 kg 2.0 kg 2.00 kg 2.000 kg Rule #5: Zeros before a non-zero digit are not significant.

Example: Sound travels from my mouth to your ears in 0.009 seconds (= 9×10-3 seconds) has only 1 s.f. Rule #6: In scientific notation, all digits are significant (not including “×10n”).

Compare these three representations of “five hundred metres”: 500m 5.0×102m 5.00×102 m

(1 s.f) (2 s.f) (3 s.f)

Rule #7: “Counting” or “exact” numbers are treated like they are perfectly or infinitely precise.

Example: “12 eggs” means exactly twelve, and is treated as though it was written “12.0000000….”

Exact values such as p, ½, and many unit conversion factors (e.g.1kg=1000g, 1inch = 2.54cm) are perfectly precise and are considered as having an “infinite” number of s.f.

Measurement-4

Calculations with Significant Figures

When performing calculations with measured quantities, the precision of the final result is limited to the precision of the least precise measurement.

As an example, consider someone trying to measure the area of a rectangular room. She measures the length very precisely: 6.9255 m (5 s.f.), but only makes a rough estimate for the width: 4 m (1 s.f.).

She then calculates: Area = length × width = 6.9255 m × 4 m = 27.702 m2

However, since the width was only known to a precision of 1 s.f., the precision of the calculated area can only be known to 1 s.f. as well.

The calculated area is thus most fairly expressed like so: ________

RULE #1: When multiplying or dividing, round the final answer to the same number of

significant figures as the least precise measurement used in the calculation.

5.00 × 7.000 = 1280000 = (3 s.f.) (4.s.f) 640

1200 × 0.0950 = 0.01511335… 754.3

RULE #2: When adding or subtracting, round the final answer to the same place value as

the least precise measurement used in the calculation.

950.1 + 23.75 9 5 0 . 1 13475 – 1200 1 3 4 7 5

+ 2 3. 7 5 – 1 2 0 0

Mixed Operations:

Perform calculations involving addition or subtraction AND multiplication or division by following BEDMAS rules and treating the significant figures in a step-by-step manner:

105.3 × 0.25 + 540.00 × 1.221 = 26.325 + 659.34 = 2 6 . 3 2 5 (4 s.f.) (2 s.f.) (5 s.f.) (4 s.f.) + 6 5 9 . 3 4

6 8 5 . 6 6 5 9.3125×10

250.0 × (3.72 5×10 -6) = =~~ - 4 = 0.216569…

0.1732 – 0.1689 0.0043

Measurement-5

Further Practice

1. Determine the number of significant figures in each of the following numbers:

Measurement-5

1. 967
2. 967.000
3. 96.7
4. 9.67
5. 30.4
6. 2.700
7. 5.10
8. 0.023
9. 7.0200
10. 0.04010

____ 11. 3.0 x 10-4 ____ 12. 7.08 x 102 ____ 13. 0.009 ____ 14. 0.90 ____ 15. 0.9

____ 16. 909

____ 17. 0.00881 ____ 18. 0.006007 ____ 19. 0.500 ____ 20. 0.050

____ 21. 0.1110 ____ 22. 0.005670 ____ 23. 670004 ____ 24. 670000 ____ 25. 670.000 ____ 26. 708 ____ 27. 780 ____ 28. 7800 ____ 29. 78000 ____ 30. 16.0

____ 31. 45.908 ____ ____ 32. 24091800 ____ ____ 33. 0.800008 ____ ____ 34. 0.00872 ____ ____ 35. 14000000 ____ ____ 36. 6.00 x 10-3 ____ ____ 37. 54000 ____ ____ 38. 222 ____ ____ 39. 39.01020 ____ ____ 40. 10203040 ____

Measurement-5

2. Change from standard notation to scientific notation (or vice versa).

In all cases, make sure the value retains the same level of precision (i.e. same number of sig figs)

Measurement-5

1. 31200
2. 0.005070
3. 68009000
4. 0.00000007400
5. 0.000010000

________________ ________________ ________________ ________________ ________________

6. 4.88000×104
7. 1.150×10-3
8. 7.752×105
9. 120.0×10-6
10. 900×10-7

________________ ________________ ________________

________________ ________________

Measurement-5

3. Report each number in scientific notation to the given order of magnitude (exponent or “power of 10”):
4. 3750000000 ___________ ×106 5. 9.06×10-6 ___________ ×10-4
5. 30700 ___________ ×105 6. 3.027×10-3 ___________ ×10-4
6. 0.005320 ___________ ×10-5 7. 1.9×10-2 ___________ ×10-4
7. 0.0000440 ___________ ×10-2 8. 5.44×10-5 ___________ ×10-4
8. Perform the following operations and give the answer to the correct number of significant figures:

(a) 15.1 + 75.32 (b) 178.90456 – 125.8055 (c) 0.0000481 – 0.000817

(d) 375.59 × 1.5 (e) 1.99 ¸ 31 (f) 1200.0 ¸ 3.0

(g) 7.95 + 0.583 (h) 97300 + 8513.2 (i) 7.819×105 – 8.166×104

(j) 299 × 5 (k) 0.02400 ¸ 6.000 (l) 5.31×10-4 ¸ (3.187×10-8)

(m) 4.56×10-5 + 3.1×10-5 (n) 1252.7 – 9.4×102 (o) 0.0589×10-6 + 7.785×10-8

5. Perform the following mixed operations and give the answer to the correct number of significant figures:

(a) 25.00 × 0.1000 – 15.87 × 0.1036 (b) 35.0 × 1.525 + 50.0 × 0.975

(c) (0.865 – 0.800) × (1.593 + 9.04) (d) (0.3812 + 0.4176) ¸ (0.0159 – 0.0146)

(e) (3.65 ¸ 0.3354) – (6.14 ¸ 0.1766) (f) 5.3 × 0.1056 ¸ (0.1036 – 0.0978)

(g) (0.341 × 18.64 – 6.00) × 3.176 (h) 9.34 × 0.07146 – 6.88 × 0.08115

Answers: B. 1. 3.12×104, 2. 5.070×10-3, 3. 6.8009×107, 4. 7.400×10-8, 5. 1.0000×10-5

6. 48800.0, 7. 0.001150, 8. 775200, 9. 0.0001200, 10. 0.00009

3. 1. 3750×106, 2. 0.307×105, 3. 532.0×10-5, 4. 0.00440×10-2, 5. 0.0906×10-4, 6. 30.27×10-4, 7. 190×10-4, 8. 0.544×10-4
4. (a) 90.4, (b) 53.0991, (c) –0.000769, (d) 5.6×102 or 560, (e) 0.064, (f) 4.0×102, (g) 8.53, (h) 105800,
   1. 7.002×105, (j) 1000, (k) 0.004000, (l) 1.67×104, (m) 7.7×10-5, (n) 3.1×102 or 310, (o) 1.368×10-7
5. (a) 0.856, (b) 102.1, (c) 0.69, (d) 610, (e) –23.9, (f) 96, (g) 1.1, (h) 0.109

Measurement-6

The SI Unit system and Unit Conversions

Over time, people have used countless different unit systems to measure volume, mass, length, and other quantities. In 1960, the international community agreed on a common system of units for scientific communication, which they called the International System of Units (abbreviated the “SI system” from the French Système Internationale d’Unités). Six of the base units in the SI unit system are:

Quantity	Written Unit	Unit Symbol
Length	Metre	m
Mass	Kilogram	kg
Time	Second	s
Amount of Substance	Mole	mol
Temperature	Kelvin	K
Current	Ampere	A

Base units can be scaled up or down by orders of magnitude (powers of 10) using metric prefixes:

Name	deca-	hecto-	kilo-	mega-	giga-	tera-
Symbol	da	h	k	M	G	T
Factor	1 10	102	103	106	109	12 10
Name	deci-	centi-	milli-	micro-	nano-	pico-
Symbol	d	c	m	µ	n	p
Factor	-1 10	-2 10	-3 10	-6 10	-9 10	-12 10

For example, a very small sample of a substance weighing 0.00005 grams could have its mass more conveniently expressed in milligrams (mg) or micrograms (µg):

0.00005 g = 0.05×10-3 g = 0.05 mg 0.00005 g = 50×10-6 g = 50 μg

Perhaps more intuitively, one could recognize that there are 1000 milligrams or 1000000 micrograms in one gram and simply multiply:

0.00005 g × 1000 = 0.05 mg 0.00005 g × 1000000 = 50 μg

Somewhat less intuitively, you could use the fact that the ratios (1000 mg : 1 g) and (1000000 μg : 1 g) are both equal to one. We call these ratios conversion factors and since they are equal to one, multiplying or dividing by these ratios does not change the actual magnitude of a value, only its units:

0.00005 g ×~~ 10~~ 0~~ 0~~ mg~~ = 0.05 mg 0.00005 g ×~~ 100~~ 0~~ 0~~ 0~~ 0~~~~ μg~~~~ = 50 μg

1 g 1 g

For conversions this simple, the last method (called the factor-label method or dimensional analysis) will seem like overkill, but it is very useful when handling more complex unit conversions (see next page).

Measurement-7

Conversions involving units raised to a power:

Example 1: Convert 320 cm2 to m2: 320 cm2 × 1 m = 0.032 m2

100 cm

Example 2: Convert 0.27 cubic feet (ft3) to cubic inches (in3):

Conversions involving fractional units:

Example 3: Convert 175 metres per minute (m/min) to kilometres per hour (km/h)

175~~ m~~ ×~~ 1~~ k~~ m~~~~ ×~~ 60 mi~~ n~~~~ = 10.5 km/h

min 1000 m 1 h

Example 4: Convert 0.0250 moles per kilogram (mol/kg) to micromoles per gram (μmol/g).

Example 5: The pressure in a can of pop is 12 g/in2. Express this pressure in kg/ft2. Further Practice:

1. Re-write these numbers using the appropriate base SI unit.
   1. 5.7 cm b) 10 ms c) 0.5 MA d) 0.2 mmol e) 237 ng f) 32 ds g) 12.5 μm
2. Convert 54 kilograms to units of: a) g b) hg c) cg
3. Express 6.8 megaseconds in units of: a) minutes b) hours c) days
4. Convert 225 micrometres to units of: a) mm b) cm c) nm
5. Express 25 centiLitres in units of: a) mL b) dL c) cm3 d) dm3
6. A common way of reporting the concentration of a solution is in units of mass per unit volume.

Express 10.73 g/mL in units of kg/L.

7. Another common way of reporting concentration is in units of moles per unit volume. Express 66 mmol/mL in units of cmol/L (centimoles per litre).
8. Sound travels at 343 m/s in air at 20ºC. Convert this speed to units of feet per minute.
9. Sound travels at 1484 m/s in water at 20ºC. Express this speed in units of kilometres per hour (km/h).
10. The density of ice is 920.1 kg m-3 (kg/m3). Express this in units of g cm-3 (g/cm3).
11. In Canada, land area is often measured in hectares. A hectare is equal to exactly 10000 square metres. How large is a hectare, in units of square kilometres?
12. A dyne is a unit of force required to accelerate a mass of one gram at a rate of one centimetre per second squared (1 g•cm/s2). A newton is a unit of force required to accelerate a mass of one kilogram at a rate of one metre per second squared (1 kg•m/s2). How many dynes are equivalent to one newton?
13. Air pressure in a car tire is typically expressed in pounds per square inch (psi). The recommended pressure for some tires is 34 psi. Express this pressure in kilograms per square centimetre.

Hint: One pound (lb) is equivalent to 453.592 grams

1. a) 0.057 m, b) 0.01 s, c) 5 ×105 A, d) 2 ×10-4 mol, e) 2.37 ×10-7 g, f) 3.2 s; g) 0.0000125 or 1.25×10-5 m
2. a) 5.4 ×104 g, b) 540 hg, c) 5.4 ×106 cg; 3. 1.1×105 min, b) 1900 h, c) 31 days; 4. a) 0.225 mm, b) 0.0225 cm, c) 225000 nm

5. a) 250 mL, b) 2.5 dL, c) 250 cm3, d) 0.25 dm3; 6. 10.73 kg/L; 7. 6.6 cmol/L; 8. 67500 ft/min; 9. 5342 km/h; 10. 0.9201 g cm-3;

11. 0.01 km2; 12. 100000 (105) dynes; 13. 2.4 kg/cm2;

Measurement-8

Uncertainty

No measurement is exact.** When a quantity is measured, the outcome depends on the measuring equipment, the measurement procedure, the skill of the operator, the environment, and other factors.

Even if a quantity were to be measured several times, in the same way and under the same circumstances, a different measured value would often be obtained each time. We call this random error.

A measurement uncertainty is a number that characterizes the random error in a measured value.

No measurement is complete without an appropriate uncertainty.

An uncertainty is written after the measured value, but before the unit. Examples: 27 ± 1ºC

0.0108 ± 0.0002 g

1800 ± 100 mL

The last (significant) digit of any measured value is always uncertain.

Hence, an uncertainty is always reported to the same precision (place value) as its measurement. Examples of incorrectly reported measurements: 12 ± 0.01 g 22.50 ± 0.5 mL

When reading glassware or other pieces of analog equipment, the experimenter must make a reasonable judgment as to what the final digit and its uncertainty should be.

Common practice is to set the uncertainty as one-half, one-fifth or one-tenth of the smallest marked division on the scale, depending on the spacing between the divisions. If divisions are well-spaced, often one-tenth is most appropriate. When markings are very close together, one-half may be more appropriate.

Measurement-9

Now try some practice on your own. Remember, the magnitude of the uncertainty is a judgement call based on the size of the spacing between divisions or, in some cases, environmental factors too.

The most common uncertainties are one-half, one-fifth, or one-tenth of the smallest marked division. Tips: Determine magnitude of the SMD and the uncertainty first, then determine the measured value.

Don’t forget to ensure that the precision of the measured value and the uncertainty agree.

3.6 3.7 3.8 3.9 ![ref1]![ref1]![ref1]

24ºC

22ºC 120 130 140 150 160 ![ref2]![ref2]![ref3]![ref3]![ref3]

20ºC

Use a ruler to measure the length of each line below. Include units and an appropriate uncertainty! Use a ruler to measure the diameter of each circle below. Include units and an appropriate uncertainty!

Calculate the area of the circles above using your diameter measurement. Don’t forget to include an appropriate uncertainty in the area!

Measurement-10

Random and systematic error

All experimental error is due to either random error or systematic error.

• Random errors are caused by factors that vary unpredictably from one measurement to the next.
  • Statistical fluctuations in both directions (larger and smaller) in the measured data
  • Primarily due to the limitations of the measuring equipment; expressed as an uncertainty.
• Random errors limit the precision of a measurement. How can we minimize random errors?
• Systematic errors are reproducible inaccuracies that are consistently in the same direction (that is to say, they bias the measurement either up or down). They may occur because:
  • An instrument is flawed or has not been calibrated correctly before use
  • An instrument is being consistently incorrectly used by the experimenter
  • There is a flaw in the procedure leading to a measurement being incorrect
• Systematic errors limit the accuracy of a measurement, but are usually difficult to detect.

Consider each statement and determine whether it is describing a random or a systematic error. If random, assess the magnitude of the error. If systematic, assess the magnitude and direction of the error.

1. One student uses a 50 mL graduated cylinder to measure 50.0±0.2 mL of solution, while another student uses a 50.00 mL pipette to measure 50.00±0.05 mL.**
2. A tape measure used to measure the length of a piece of aluminum foil has been stretched out from years of use by abnormally strong IB students.
3. A student uses an electronic balance to measure the mass of 175 g of sugar into a beaker, but forgets to “tare” (zero) the balance after placing the beaker on top.
4. A student using a stopwatch to time how long it takes a steel ball to fall from a fixed height repeats the experiment ten times but obtains a slightly different measurement each time.
5. A biologist studying the reproduction rate of a particular strain of bacteria makes multiple measurements but obtains very different results each time.
6. As a joke, somebody drew a decimal place on the readout of your voltmeter between the “tens” place and the “ones” place (so that, for example, 57 V would actually be read as 5.7 V. Hilarious.)
7. Vibrations from a nearby centrifuge are affecting the stability of a buret making it difficult for a student to read the wobbly meniscus, so she assigns an uncertainty of ±0.1 mL instead of ±0.05 mL.
8. A student is studying the rate of a reaction but accidentally spills one of the two reactants on the floor instead of adding it to the flask containing the other reactant. They record the reaction rate as “zero”.
9. A gas pump is calibrated to measure volume at 15ºC, but is being operated in the Arctic where temperatures are often much lower and liquids tend to be denser.
10. A student conducts three trials of the same experiment to determine the boiling point of water and obtains the following data: 87.2ºC, 87.6ºC, 87.4ºC.

Measurement-11

Propagating uncertainties in calculated results

When you perform calculations using measured values and their uncertainties, you must carry the uncertainties over (propagate them) to the calculated value. This ensures that the random errors inherent in the original measured values are appropriately expressed in the new, calculated value.

Definitions:

Absolute uncertainty:* Simply the magnitude (value) of the measurement uncertainty

Example: 10.00 ± 0.05 mL

Relative uncertainty:* The percentage (or decimal) uncertainty

Uncertainty propagation for addition and subtraction: Example 1: (12 ± 2 cm) + (25 ± 5 cm) =

**When adding and subtracting, add the absolute uncertainties. **Example 2: (45 ± 1) mL – (20.0 ± 0.7) mL =

RECALL: ✓ Uncertainties are only reported to only 1 s.f.

✓ The precision of the measured value and uncertainty must agree to the same place value.

Uncertainty propagation for multiplication and division:

**When multiplying and dividing, add the relative (percentage) uncertainties. **Example 3: 0.70 ± 0.05 m × 22 ± 1 m Step 1: Always get the correct answer first!

0.70 m × 22 m = 15.4 m2

Step 2: Add the percentage uncertainties:

0.05/0.70 = 0.071, or 7.1% 1 / 22 = 0.045, or 4.5%

Example 4: 4750 ± 50 mm3 Step 1: Always get the correct answer first!

37 ± 1 mm 4750 mm3 / 37 mm = 128.378 mm2 Step 2: Add the percentage uncertainties:

50/4750 = 0.011, or 1.1% 1 / 37 = 0.027, or 2.7%

Measurement-12

When scaling a measurement up or down (i.e. multiplying or dividing) by a constant, the relative uncertainty does not change, but the absolute uncertainty does:

Example 5: If the mass of one marble, m, is 2.50 ± 0.05 cm…

Then the mass of twelve marbles, 12m = 12(2.50 ± 0.05 cm) = 30.0 ± 0.6 cm

Let’s try some mixed examples now. Follow the same principles we just discussed, only apply the rules for order of operations (“BEDMAS” rules) for the measurements and their uncertainties.

0. ± 0.05 g + 5.0 ± 0.1 g 7 ± 0.15 g

1. d = =

3.25 ± 0.05 mL 3.25 ± 0.05 mL

7 ± 2.14%

• 

3.25 ± 1.54%

• 2.1538… ± 3.68% g/mL
• 2.1538… ± 0.07926… g/mL =

2. l = (0.84 ± 0.01m2 ÷ 1.7 ± 0.1m) – 15.0 ± 0.5 cm

Further Practice:

1. Let x = (10.0 ± 0.2) m, y = (25.1 ± 0.9) m, and z = (1.05 ± 0.01) m.

a) A = y – x b) B = 5xyz c) C = 2y – 4x – 7z

4. D = x2 + xy e) E = x/z + y/x f) F = (x + y)(y – 10z)

2. A student measures the initial volume of alcohol in a petri dish to be 17.5 ± 0.5 mL. After 64± 2min passes during which time some of the alcohol evaporates; the final volume is measured to be 8.4 ± 0.2 mL. Determine how much alcohol evaporated, with an appropriate uncertainty. Determine the rate that the alcohol evaporated, in mL/s, with an appropriate uncertainty.
3. What is the volume, in mL, of a can with a diameter of 83 ± 1 mm and a height of 150 ± 2 mm?
4. How many litres of water can a cubic container with a side length of 8.00 ± 0.02 inches hold?
5. A particularly bored student, instead of listening during class, watched an ant crawl the length of her ruler (30.0 ± 0.1 cm) in 12.0 ± 0.5 seconds. How fast was the ant crawling, in km/h?
6. If there are 235 ± 5 students in the IB program this year (the way you people scurry around the halls so fast it makes you hard to count, hence the uncertainty) and each one will do 120 ± 20 minutes of homework every weeknight, how much collective homework will be done this school year by IB students?

Answers: 1a) 15 ± 1 m b) 1320 ± 90 m3 c) 3 ± 3 m d) 350 ± 20 m2

5. 12.0 ± 0.4 f) 510 ± 50 m2 2) 9.1 ± 0.7 mL, 2.4 ± 0.3 mL/s
6. 810 ± 30 mL 4) 8.39 ± 0.06 L 5) 0.090 ± 0.004 km/h

Measurement-13

Measurement Review Problem Set

1. How many sig figs? 0.320, 4.550 x 10-5, 0.0700, 7400, 2130.0, 0.012005, 90000, 50.00 x 107
2. Perform the following calculations and report the answer to the correct number of significant figures:

a) 7 + 1.3 + 4.31 b) 382 x (1.04×10-2 + 2.2×10-3) c) 45.0 × 2.075 + 60.0 × 0.725

d) (173 + 11.6) / (930 – 2.6) e) (7.21×105 + 2.4×104) / (5.00×102) f) (2.56 x 107) / (6.40×105)

3. Perform the following unit conversions; report the answer to the correct number of significant figures:

a) 3520.0 mm to km b) 135.2 nm to µm c) 7.94 x 10-6 days to milliseconds

d) 20.0±0.4 g/ft2 to mg/in2 e) 63.5±0.5 g/mL to kg/m3 f) 7.20±0.05 m2/hr to cm2/s

4. Read the scales below, then report the measurement with an appropriate uncertainty and unit:

1. c) d) e)

5. Craft an example of a single measurement of the speed of light (3.00×108 m/s) that includes an uncertainty and is:
   1. Precise, but not accurate b) Accurate, but not precise
6. Craft an example of a set of five repeat measurements for the density of iron (7.874 g/cm3) that is:
   1. Precise, but not accurate b) Accurate, but not precise c) Both precise and accurate
7. Perform the following calculations and report the answer with an uncertainty:

Use: w = (1.8 ± 0.1) cm, x = (35 ± 5) cm, y = (120 ± 10) cm, z = (10.0 ± 0.5) cm

a) wxy b) w + x – z c) 15w + 5x + y d) z2/x + w e) y/x + z/w f) y / (3w + z) g) (z+w) / (x+y)

8. A runner’s maximal aerobic capacity can be expressed using a parameter called VO2 max, the maximum rate of oxygen consumption measured during exercise. VO2 max is reported in units of cubic centimetres of oxygen per kilogram of body mass per minute: cm3 kg-1 min-1. An 68 ± 1 kg runner used his GPS watch to determine that his VO max was 66 cm3 kg-1 min-1 (the watch manufacturer’s manual reports a percentage uncertainty of 3% in this

2

value). What is the maximum volume of oxygen this runner will consume on an intense run lasting 29±1 minutes?

9. The density of aluminum is 2.70±0.01 g/mL. A particular roll of aluminum foil available at the grocery store is 15±1 μm thick, and measures 18.0±0.1 inches wide by 25.0± 0.5 feet long. If the price of aluminum is $1.23/pound, what is the value of the aluminum in this roll? (with an uncertainty!)

Time (hr)	Diameter (km)
5.0±0.1	2.7 ± 0.1
10.0±0.1	7.5 ± 0.5
15.0±0.1	23 ± 1
25.0±0.1	75 ± 5

10. The diameter of an oil spill was monitored using satellite imagery over time. Calculate the average rate of growth of the oil spill, in units of square km per hour:

a) from t = 5h to t = 10h b) from t = 10h to t = 15h c) from t = 5h to t = 25h

11. A student used a buret to measure 25.00 mL of a mystery liquid into a beaker.

Give one example of a random error and one example of a systematic error that would affect this measurement. For the systematic error, state how (i.e. in which direction) it would affect the measurement.

12. A particular chemical reaction undergoes two dramatic colour changes, one immediately after the reactants are mixed and another the instant they finish reacting. A student plans to study the rate of this reaction using a stopwatch to measure the time, in seconds, between the two colour changes. Outline strategies the student could use to minimize systematic and random errors in this experiment (clearly stating which type of error each strategy will minimize).

Measurement Review – Selected Answers

1. 3, 4, 3, 2, 5, 5, 1, 4

2. a) 13 b) 4.81 c) 136.9 d) 0.20 e) 1490 f) 40.0

3. a) 3.5200 x 10-3 km b) 0.1352 mm c) 686 ms

   4. 139 ± 3 mg/in2 e) 63500 ± 500 kg/m3 f) 20.0±0.1 cm2/s

4. a) BEST: 236 ± 2 cm, JUST OK: 235 ± 5 cm b) BEST: 2.198 ± 0.002 mm, OK: 2.200 ± 0.005 mm

c) 13.92 ± 0.02 mL or 13.90 ± 0.05 mL d) SMD/5: 8.6 or 8.7 ± 0.1 mL, or SMD/2: 8.8 ± 0.3 mL

5. 270 ± 10 mL is a good measurement here; reading more precisely could yield ±5, ±4, or ±2 mL instead.

5. a) Your measurement should have a small relative uncertainty (and hence many significant figures), but not be close to the accepted value. For example, something like 0.07654±0.00002 m/s.

b) Your measurement should have a large relative uncertainty (and hence few significant figures), but be very close to the accepted value. For example, something like (3±2)×108 m/s or (3.0±0.9)×108 m/s.

6. a) Your measurements should have small relative uncertainties and be closely clustered together (i.e. highly reproducible), but not be close to the accepted value. For example: 0.99±0.01 g/cm3, 1.00±0.01 g/cm3, 1.01±0.01 g/cm3 (maybe this person confused water for iron ☺)

7. Your measurements should have large relative uncertainties and vary significantly from one to the other (i.e. not be reproducible), but the average of the measurements should be close to the accepted value. This one is a little more difficult to craft a realistic example for, but consider something like: 1.0±0.5 g/cm3, 2.5±0.5 g/cm3, 20±10 g/cm3 (compute this average to convince yourself that this set is accurate, in a sense)
8. Pretty straight-forward here. Just like (a) but close to the accepted value. Something like 7.873±0.001 g/cm3, 7.874±0.001 g/cm3, 7.875±0.001 g/cm3 works nicely.

9. a) 8000 ± 2000 cm3 b) 27 ± 6 cm c) 320 ± 40 cm d) 4.7 ± 0.8 cm 8 ± 1

   5. 9 ± 1 f) 8 ± 1 g) 0.08 ± 0.01 Question: Why are the answers to e, f and g unitless?

10. 130 ± 10 L

11. $0.38 ± 0.04… but you can bet they charge a lot more than that in the store! ☺

12. a) 1.0±0.2 km2/h (rounded from 0.96±0.16 km2/h) b) 3.1 ± 0.4 km2/h c) 3.6 ± 0.3 km2/h

13. The best example of random error is the built-in limitation in the glassware itself, i.e. the ±0.05 mL uncertainty of the buret. Note that this is actually a very small amount of random error; burets are among our most precise pieces of glassware and have very small relative uncertainties. A beaker, for example, introduces a lot more random error into measurements (as we clearly saw during Part A of the Coke lab). You could have also discuss things like temperature fluctuations or shaky tables, but they are weaker examples.
    A great example of systematic error is someone who doesn’t eject the air bubble from the buret tip before dispensing the 25.00 mL. This would introduce a negative bias into their measurement since they will be dispensing less than 25.00 mL of liquid each time (the first mL or so will be air!). Another example is dispensing from too great a height so that some liquid splashes out of the beaker (also a negative bias). Note that something like consistently reading the buret from below eye-level is not a systematic error. Why not?

14. What a great question! So great, in fact, that I think I’ll add something very similar to it on your test!

In summary, remember that there are two primary ways that a measurement can be precise. Precise single measurements have small uncertainties and hence many significant figures. Precise data sets (i.e. repeated measurements/trials) are highly reproducible (there is little variability from one trial to the next).

Being accurate is less nuanced: simply make a measurement that is “correct” (close to the accepted/known value).

Final piece of advice: On our Unit Test…make sure your calculations are accurate, and your language is precise. ☺