# 1.5 Measurement uncertainty, accuracy, and precision  (Page 5/11)

 Page 5 / 11

Suppose a quality control chemist at a pharmaceutical company is tasked with checking the accuracy and precision of three different machines that are meant to dispense 10 ounces (296 mL) of cough syrup into storage bottles. She proceeds to use each machine to fill five bottles and then carefully determines the actual volume dispensed, obtaining the results tabulated in [link] .

Volume (mL) of Cough Medicine Delivered by 10-oz (296 mL) Dispensers
Dispenser #1 Dispenser #2 Dispenser #3
283.3 298.3 296.1
284.1 294.2 295.9
283.9 296.0 296.1
284.0 297.8 296.0
284.1 293.9 296.1

Considering these results, she will report that dispenser #1 is precise (values all close to one another, within a few tenths of a milliliter) but not accurate (none of the values are close to the target value of 296 mL, each being more than 10 mL too low). Results for dispenser #2 represent improved accuracy (each volume is less than 3 mL away from 296 mL) but worse precision (volumes vary by more than 4 mL). Finally, she can report that dispenser #3 is working well, dispensing cough syrup both accurately (all volumes within 0.1 mL of the target volume) and precisely (volumes differing from each other by no more than 0.2 mL).

## Key concepts and summary

Quantities can be exact or measured. Measured quantities have an associated uncertainty that is represented by the number of significant figures in the measurement. The uncertainty of a calculated value depends on the uncertainties in the values used in the calculation and is reflected in how the value is rounded. Measured values can be accurate (close to the true value) and/or precise (showing little variation when measured repeatedly).

## Chemistry end of chapter exercises

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

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 $×$ 10 2 ; (b) 3.344 $×$ 10 −2 ; (c) 5.479 $×$ 10 2 ; (d) 2.2086 $×$ 10 4 ; (e) 1.00000 $×$ 10 3 ; (f) 6.51 $×$ 10 −8 ; (g) 7.157 $×$ 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 2 m

(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

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 $×$ 10 18 m

(c) 3,486,002 kg

(d) 9.74150 $×$ 10 −4 J

(e) 0.0613 cm 3

(f) 17.0 kg

(g) 0.01400 g/mL

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

(a) 53 cm

(b) 2.05 $×$ 10 8 m

(c) 86,002 J

(d) 9.740 $×$ 10 4 m/s

(e) 10.0613 m 3

(f) 0.17 g/mL

(g) 0.88400 s

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

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

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

(a) 0.44; (b) 9.0; (c) 27; (d) 140; (e) 1.5 $×$ 10 −3 ; (f) 0.44

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

(a) 517

(b) 86.3

(c) 6.382 $×$ 10 3

(d) 5.0008

(e) 22.497

(f) 0.885

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

(a) 628 $×$ 342

(b) (5.63 $×$ 10 2 ) $×$ (7.4 $×$ 10 3 )

(c) $\frac{28.0}{13.483}$

(d) 8119 $×$ 0.000023

(e) 14.98 + 27,340 + 84.7593

(f) 42.7 + 0.259

(a) 2.15 $×$ 10 5 ; (b) 4.2 $×$ 10 6 ; (c) 2.08; (d) 0.19; (e) 27,440; (f) 43.0

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) $8.78\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(\frac{0.0500}{0.478}\right)$

(f) 140 + 7.68 + 0.014

(g) 28.7 – 0.0483

(h) $\frac{\left(88.5-87.57\right)}{45.13}$

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?

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

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%

following processes: Solid phosphorus pentachloride decomposes to liquid phosphorus trichloride and chlorine gas b. Deep blue solid copper(II) sulfate pentahydrate is heated to drive off water vapor to form white solid copper(II) sulfate
How to know periodic table oftend
how to get atomic number of an element
how do you solve the examples in a much more explanatory way
Ogunleye
The reaction of aceto nitrile with propane in the presence of the acid
Explain this paragraph in short
What is solid state?
What is chemical reaction
Manish
transforming reactants to product(s)
Andre
process
Andre
Example of Lewis acid
Example of Lewis acid
Chidera
Chlorine
Anything with an empty orbital... the hydrogen ion is the most common example. BH3 is the typical example, but any metal in a coordination complex can be considered a Lewis acid.
Eszter
okay thanks
Jovial
aluminium and sulphur react to give aluminium sulfide.How many grams of Al are required to produce 100g of aluminium sulphide
aluminium and sulphur react to give aluminium sulphide how many grams of Al are required to produce 100g of aluminium sulphide?
Soni
aluminium and sulphur react to give aluminium sulphide how many grams of Al are required to produce 100g of aluminium sulphide?
Soni
2Al+3S=Al2S3
galina
m(Al)=100×27×2/150=36g
galina
150 comes from?
Soni
thank you very much
Soni
molar mass of Al2S3
galina
150.158
thiru
Why can't atom be created or destroyed
matter simply converts to pure energy
that's nice
Meshach
explain how to distinguish ethanol from a sample of ethanoic acid by chemical test
explain how ethanol can be distinguished from ethanoic acid by chemical test
Alice
Using a suitable experiment, describe how diffusion occurs in gases.
when the excited energy which are in gaseous state collides with another to liberate from one place to another
Meshach
what is electrolytes?
substance which splits into ions during melting or dissolving
galina
on passing electric current though electrode
Kv
State that use law of partial pressure in a gas jar containing a gas and water what is the total pressure composed of 272cm^3 of carbon (iv) oxide were collected over water at15°c and 782mmHg pressure. calculate the volume of the dry gas at stp(SVP of water at 15°c is 12mmHg)
was Dalton's second postulate"atoms of the same kind have have similar/same mass and size" Or " the one mentioned in B here?