Uncertainties in Measurements
Pooran Appadu
1
Review: Uncertainty
“Every measurement has a degree of uncertainty associated with it. The uncertainty derives from the
measuring device and from the skill of the person doing the measuring.” 1
Let's use volume measurement as an example. Say you are in a chemistry lab and need 7 mL of water: 2

You could take an unmarked coffee cup and add water until you think you have about 7 milliliters.
In this case, the majority of the measurement error is associated with the skill of the person doing
the measuring.3

You could use a beaker, marked in 5 mL increments. With the beaker, you could easily obtain a
volume between 5 and 10 mL, probably close to 7 mL, give or take 1 mL. If you used a pipette
marked to with 0.1 mL, you could get a volume between 6.99 and 7.01 mL pretty reliably. 4
Absolute Uncertainties 5
1.1




Uncertainty of the last digit of a measurement
Example, 45.47 mL is a measurement of volume and the last digit is uncertain
The absolute uncertainty is + 0.01 mL
The measurement lies between 45.46 and 45.48 mL

When adding or subtracting, add absolute uncertainties!
Relative Uncertainties 6
1.2


Absolute uncertainty divided by the number itself
Example,
0.01 𝑚𝐿
= 2.00 × 10−4
45.47 𝑚𝐿

1.2.1


When multiplying or dividing, add relative uncertainties !
Percent Relative Uncertainty
Relative uncertainty represented as a percent;
Example,
0.01 𝑚𝐿
× 100 = 0.02 %
45.47 𝑚𝐿
1
http://chemistry.about.com/od/mathsciencefundamentals/a/sigfigures.htm
Ibid.
3 Ibid
4 Ibid.
5 Jespersen, N.D. 2010. Barron’s AP Chemistry. Barron's Educational Series: USA. Pg. 590
6 Ibid.
2
Compiled By: Pooran Appadu
2
Review: Significant Figures
“All measurements are approximations—no measuring device can give perfect measurements without
experimental uncertainty. By convention, a mass measured to 13.2 g is said to have an absolute
uncertainty of plus or minus 0.1 g and is said to have been measured to the nearest 0.1 g.” 7
The number of significant figures in a result is simply the number of figures that are known with some
degree of reliability. The number 13.2 is said to have 3 significant figures. The number 13.20 is said to
have 4 significant figures. 8
2.1
Rules for deciding the number of significant figures in a measured quantity:
(1) All nonzero digits are significant:


1.234 g has 4 significant figures
1.2 g has 2 significant figures.
(2) Zeroes between nonzero digits are significant:


1002 kg has 4 significant figures,
3.07 mL has 3 significant figures
(3) Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate
the position of the decimal point:


oC
has only 1 significant figure,
0.012 g has 2 significant figures.
(4) Trailing zeroes that are also to the right of a decimal point in a number are significant:


0.0230 mL has 3 significant figures,
0.20 g has 2 significant figures.
(5) When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not
necessarily significant:


190 miles may be 2 or 3 significant figures,
50,600 calories may be 3, 4, or 5 significant figures.
The potential ambiguity in the last rule can be avoided by the use of standard exponential, or "scientific,"
notation. For example, depending on whether the number of significant figures is 3, 4, or 5, we would
write 50,600 calories as:



5.06 × 104 calories (3 significant figures)
5.060 × 104 calories (4 significant figures), or
5.0600 × 104 calories (5 significant figures).
By writing a number in scientific notation, the number of significant figures is clearly indicated by the
number of numerical figures in the 'digit' term as shown by these examples.
7
8
http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/index.html
Ibid.
Compiled By: Pooran Appadu
Rules for Mathematical Operations 9
2.2
Addition & Subtraction
Another way to state this rule is as follows: in addition and subtraction, the result is rounded off so that it
has the same number of digits as the measurement having the fewest decimal places (counting from left
to right). For example,


100 (assume 3 significant figures) + 23.643 (5 significant figures) = 123.643,
should be rounded to 124 (3 significant figures).
Note: however, that it is possible two numbers have no common digits (significant figures in the same
digit column). ???
Multiplication & Division
The result should be rounded off so as to have the same number of significant figures as in the
component with the least number of significant figures. For example,


3
3.1
3.0 (2 significant figures ) × 12.60 (4 significant figures) = 37.8000
Which should be rounded to 38 (2 significant figures).
More Important Terms and Concepts
Mean
“is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of
the collection” 10
𝑛
𝑥̅ =
1
∑ 𝑎𝑖
𝑛
𝑖=1
Question
1. Richards & Willard (1910) determined the atomic mass of Lithium and collected the following
data: 11
Exp’t
1
2
3
4
5
6
7
Molar Mass (g/mol)
6.9391
6.9407
6.9409
6.9399
6.9407
6.9391
6.9406
(a) Find the mean atomic mass determined.
2. Find the mean of the following sets of data:
12
9
Ibid.
http://en.wikipedia.org/wiki/Mean
11 Skoog, D., et al., (2004). Fundamentals of Analytical Chemistry 8th Edition. USA: Thomson Publishers
12 Ibid.
10
Compiled By: Pooran Appadu
(a) 0.0110
(b) 4.52 x 10-3
3.2
0.0104
4.47 x 10-3
0.0105
4.63 x 10-3
4.48 x 10-3
4.58 x 10-3
4.53 x 10-3
Standard Deviation
“The mean does a nice job in telling where the center of the data set is. However, consider that a
pharmaceutical engineer develops a new drug that regulates iron in the blood. She finds out that the
average sugar content after taking the medication is the optimal level. This does not mean that the drug
is effective. There is a possibility that half of the patients have dangerously low sugar content while the
other half have dangerously high content. Instead of the drug being an effective regulator, it is a deadly
poison. What the pharmacist needs is a measure of how far the data is spread apart. For this, the
standard deviation comes in handy.”13
The standard deviation is defined as:
n
1
s=√
∑(x − x̅)2
n−1
i=1
Example
“The owner of the Ches Tahoe restaurant is interested in how much people spend at the restaurant. He
examines 10 randomly selected receipts for parties of four and writes down the following data.” 14
44, 50, 38, 96, 42, 47, 40, 39, 46, 50
(a) Calculate the mean
𝐱̅ =
44 + 50 + 38 + 96 + 42 + 47 + 40 + 39 + 46 + 50
= 49
10
(b) Calculate the standard deviation
𝐱
44
50
38
96
42
47
40
39
46
50
𝐱 − 𝐱̅
-5.2
0.8
11.2
46.8
-7.2
-2.2
-9.2
-10.2
-3.2
0.8
Total
(𝐱 − 𝐱̅)𝟐
27.04
0.64
125.44
2190.24
51.84
4.84
84.64
104.04
10.24
0.64
2600.40
2600.40
s = √(
) = 17.00
9
13
14
http://www.ltcconline.net/greenl/courses/201/descstat/mean.htm
Ibid.
Compiled By: Pooran Appadu
The standard deviation can be thought of measuring how far the data values lie from the mean, we take
the mean and move one standard deviation in either direction.
(c) Calculate x̅ ± s
Upper: 49.20 + 17.00 =
Lower: 49.20 − 17.00 =
Question 15
A laboratory technician analyses water samples for degree of hardness, which is expressed in parts per
million (ppm) of calcium carbonate. The results are given below:
Sample #
1
2
3
4
5
6
7
8
9
10
CaCO3, ppm
250
245
265
300
225
230
248
295
235
220
Table 1 – Degree of Hardness of Water Samples16
(a) Calculate mean and standard deviation of the values (4 marks)
(b) What does the standard deviation value indicate about the variability in the degree of hardness?
(1 mark)
(c) The ten samples of water are taken from ten different sites. How can the reliability of the data
obtained from each sample be improved? (1 mark)
3.3
Precision & Accuracy
Precision: “Describes the agreement among several results obtained in the same way. Precision
can be determined by measuring replicate samples.” 17
Accuracy: “Indicates the closeness of the measurement to the true or accepted value. Accuracy
is more often difficult to determine because the true value is usually unknown. An accepted value
must be used instead.”18
15
CXC. 2005. CAPE Chemistry Unit 1 Module 3.
CXC. 2005. CAPE Chemistry Unit 1 Module 3.
17 Skoog, D., et al., 2004. Fundamentals of Analytical Chemistry 8th Edition. USA: Thomson Publishers.
Pg. 93
18 Ibid, Pg. 93
16
Compiled By: Pooran Appadu
Figure 1 – Illustration of Accuracy and Precision using the patter of darts on a Dartboard
3.4
19
Calibration Curves
“A calibration curve is a general method for determining the concentration of a substance in an unknown
sample by comparing the unknown to a set of standard samples of known concentration” 20
3.4.1
Characteristics of a Calibration Curve
Figure 2 – A calibration curve plot showing limit of detection (LOD), limit of quantification (LOQ),
dynamic range, and limit of linearity (LOL) 21
LOD: 22
19
Skoog, D., et al., 2004. Fundamentals of Analytical Chemistry 8th Edition. USA: Thomson Publishers.
http://en.wikipedia.org/wiki/Calibration_curve
21 Ibid
22 http://en.wikipedia.org/wiki/Detection_limit
20
Compiled By: Pooran Appadu

lowest quantity of a substance that can be distinguished from the absence of that substance (a
blank value) within a stated confidence limit
LOD defined as 3 x standard deviation of the blank

LOQ 23


The LOQ is the limit at which we can reasonably tell the difference between two different values.
10 * standard deviation of the blank;
Dynamic Range

LOQ to LOL
Sensitivity 24


Smallest change in amount we can see with a known level of confidence.
Often based on smallest change that an instrument can display
LOL

3.5
Point where the calibration curve departs from linearity
Standards
“Standards are materials containing a known concentration of an analyte. They provide a reference to
determine unknown concentrations or to calibrate analytical instruments. The accuracy of an analytical
measurement is how close a result comes to the true value. Determining the accuracy of a measurement
usually requires calibration of the analytical method with a known standard. This is often done with
standards of several concentrations to make a calibration or working curve.” 25
3.5.1
Primary Standards
“A primary standard is typically a reagent which can be weighed easily, and which is so pure that its
weight is truly representative of the number of moles of substance contained. Features of a primary
standard include:” 26
1.
2.
3.
4.
5.
6.
7.
High purity
Stability (low reactivity)
Low hygroscopicity and efflorescence
High solubility (if used in titration)
High equivalent weight
Non-toxicity
Ready and cheap availability
Some primary standards for titration of acids:


sodium carbonate: Na2CO3, mol wt. = 105.99 g/mol
tris-(hydroxymethyl)aminomethane (TRIS or THAM): (CH2OH)3CNH2, mol wt. = 121.14 g/mol
23
http://en.wikipedia.org/wiki/LOQ
http://ull.chemistry.uakron.edu/chemometrics/07_Calibration.pdf
25 http://www.files.chem.vt.edu/chem-ed/data/standard.html
26 http://en.wikipedia.org/wiki/Primary_standard
24
Compiled By: Pooran Appadu
Some primary standards for titration of bases:


potassium hydrogen phthalate (KHP): KHC8H4O4, mol wt. = 204.23 g/mol
potassium hydrogen iodate: KH(IO3)2, mol wt. = 389.92 g/mol
Some primary standards for redox titrations:

3.5.2
potassium dichromate: K2Cr2O7, mol wt. = 294.19 g/m
Secondary Standards
A secondary standard is a standard that is prepared in the laboratory for a specific analysis. It is usually
standardized against a primary standard.
4
Degree of Uncertainty
4.1
Pipettes
A pipette (also called a pipet, pipettor, or chemical dropper) is a laboratory tool used to transport a
measured volume of liquid.
Pipettes are used to measure small volumes of fluid:






1
2
3



5
10
20
25
50
100
Both the maximum capacity and the size of the increments are printed on the end of the pipette. 27
Question:
Precision and Accuracy of a 25 mL Pipette
Three chemistry majors at the University of Guyana performed the following experiment in their Analytical
Chemistry II laboratory session:




A pipette was used to measure 25 mL of water;
A beaker was placed on an analytical balance and tared;
The water in the pipette was poured into the beaker and the mass (4 dp) was recorded.
Each student did 4 replicates and the results are as follows:
Mass, g Recorded
by Student #1
24.8043
Mass, g Recorded
by Student #2
24.8067
Mass, g Recorded
by Student #3
24.7123
24.8086
24.8162
24.7396
24.8102
24.8194
24.7403
24.8160
24.8228
24.7512
Note: the original experiment was done by eight students and twelve replicates were recorded,
however, statistical analyses were performed using software packages.
27
Lab Math: A Handbook of Measurements, Calculations, and Other Quantitative Skills for Use at the
Bench
Compiled By: Pooran Appadu
(a) Assuming that the density of water at 30 oC = 0.9957 g/cm3 AND assume that the balance has
been calibrated, convert the mass (g) to volume (mL)
(b) Calculate the mean and standard deviation of the results for each student.
(c) Comment on the accuracy (Vo = 25 mL) and precision of the pipette (Hint: consider standard
deviation);
Question28 - A typical CAPE Exam question
The table below shows the readings obtained during the calibration of a 25 cm 3 pipette. During the
calibration exercise the masses of 25 cm3 samples of water were recorded. The volumes of the
equilibratied water used were then determined using the density of water with a correction for buoyancy
error.
Volume, cm3
25.008
25.006
25.017
25.010
25.009
25.015
25.007
25.008
25.015
25.006
Sample
1
2
3
4
5
6
7
8
9
10
(a) Calculate the mean, standard deviation of the volumes delivered by the pipette. (4 marks)
(b) State the capacity of the pipette (1 mark)
(c) What additional measurements would have been necessary for the conversion of mass to
volume? Explain your answer. (2 marks)
4.1.1
Types of Pipettes
Volumetric
Figure 3 29
28http://fmcapechemistry.weebly.com/uploads/5/6/3/3/5633072/unit_2_chem_lessons_mod_2_titrimetric_
analyses.doc
29 http://www.hoskin.ca/catalog/images/Volumetric%20Pipette.jpg
Compiled By: Pooran Appadu
Mohr
Figure 4 30
Serological
Figure 5 31
Burettes 32
4.2





Used to dispense controlled small amounts of a liquid with great precision;
Used to perform titrations for quantitative analyses;
Used to determine accurate concentrations of stock solutions;
Available in glass and plastic
o Glass are easy to clean;
o Unaffected by organic solvents;
Capacity : 50 mL, 100 mL are common;
30
http://www.uwplatt.edu/chemep/chem/chemscape/labdocs/glossary/pic/anmohr.gif
http://www.thesciencefair.com/Merchant2/graphics/00000001/SerologicalPipets_3877_M.jpg
32 Thompson, RB & Thompson, BF. 2012. Illustrated Guide to Home Chemistry Experiments: All Lab, No
Lecture. : USA
31
Compiled By: Pooran Appadu
o
o

Flexibile
Accurate
Values are reported as :
o 25.00 + 0.0
Volumetric Flasks 33
4.3








Used to make up a precise volume of solution
One graduation line that indicates the ‘nominal’ volume
Typical procedure
o Partially fill with water
o Add chemicals
o Swirl until dissolved
o Fill the flask to the graduation line
Designed to be accurate at 20 oC
Small variations do not affect accuracy
If making up solutions that are endothermic or exothermic, wait until they cool to room
temperature, then make up to mark
MOST volumetric flaks are made of ‘flint glass’. Others are made of Pyrex or Borosilicate.
o Initially dissolve exothermic solutes in a beaker or conical flask
o After the solution is cool, transfer to the volumetric flask
o Rinse the beaker or conical flask into the volumetric flask;
o Make up to mark
o Also known as a Quantitative transfer
Labs typically have: 25, 50, 100, 250, 500, 1 L, 2 L flasks
Flint Glass: is a type of glass known for its high transparency and refractive value
Pyrex:
Borosilicate Glass
34
Top Loading Balances35
4.4


Used to weigh solid material when a precision of 0.1 g is adequate;
For more accurate mass measurements or small amounts, use an analytical balance
33
Thompson, RB & Thompson, BF. 2012. Illustrated Guide to Home Chemistry Experiments: All Lab, No
Lecture.
34 www.wisegeek.com/what-is-flint-glass.htm
35 http://www.dartmouth.edu/~chemlab/techniques/top_balance.html
Compiled By: Pooran Appadu
Figure 6 36
Analytical Balances37
4.5


Measures masses to within 0.0001 g.
Use these balances when you need this high degree of precision.
Figure 7 38
36
http://mettler-toledo.centralcarolinascale.com/PL-Balance.htm
http://www.dartmouth.edu/~chemlab/techniques/a_balance.html
38 http://www.sysmatec.ch/Images/KERNABT.jpg
37
Compiled By: Pooran Appadu