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Error Analysis LectureCHM 335
Error Analysis LectureCHM 335 – p. 1/39
Reading Assignment
Text, 4-7, 18-23, 27-42 (Pages 29-32, 43-48and 52-66)
Syllabus, Pages 22-32
Error Analysis LectureCHM 335 – p. 2/39
ExampleWhat is incomplete in the statement: “The distance between
Kingston and New York City is 150 miles?”
150 miles ± 1 mile
150 miles ± 100 miles
All future reports must provide results with errors. The error
is as important as the measured result.
Error Analysis LectureCHM 335 – p. 3/39
Types of ErrorsSystematic errors
1. Definite algebraic sign
2. Accuracy
Random errors
1. No definite algebraic sign
2. Precision
Can control the precision by increasing the number of
independent measurements.
Error Analysis LectureCHM 335 – p. 4/39
The Central LimitTheorem
The results of independent measurements fall ona Gaussian distribution (a bell curve)
p(x) =1√2πσ2
e−(x−µ)2/2σ2
x = what we are measuringµ = the exact valueσ = standard deviation, a measure of the width
p(σ + µ)
p(µ)=
1√e
Error Analysis LectureCHM 335 – p. 5/39
GaussianDistribution
-2 0 2 40
0.1
0.2
0.3
0.4
µµ−σ µ+σ
Error Analysis LectureCHM 335 – p. 6/39
Properties∫
∞
−∞
p(x) dx = 1
∫ µ+σ
µ−σp(x) dx ≈ 0.67
∫ µ+2σ
µ−2σp(x) dx ≈ 0.95
∫
∞
−∞
xp(x) dx = µ
Error Analysis LectureCHM 335 – p. 7/39
Properties (cont.)∫
∞
−∞
x2p(x) dx− µ2 = σ2
Interpretation - There is a 95% chance that a particular
measurement of x will fall in the range µ± 2σ.
Error Analysis LectureCHM 335 – p. 8/39
Finite Set of NMeasurements
Assume
µ = 〈x〉 = 1
N
N∑
i=1
xi
〈f〉 = 1
N
N∑
i=1
f(xi)
σ2 = 〈x2〉 − 〈x〉2
=1
N
N∑
i=1
x2i −(
1
N
N∑
i=1
xi
)2
Error Analysis LectureCHM 335 – p. 9/39
Standard Deviation ofthe Mean
The deviation of the ith measurement from the exact result
is given by
εi = xi − µ
Then εi can be shown to be normally distributed with
standard deviation
σN =σ√
N − 1.
σN is called the standard deviation of the mean.
Error Analysis LectureCHM 335 – p. 10/39
InterpretationThere is a ∼67% chance the correct result lies in the range
〈x〉 ± σN .
There is a ∼95% chance the correct result lies in the range
〈x〉 ± 2σN .
We report 〈x〉 ± 2σN .
Error Analysis LectureCHM 335 – p. 11/39
ExampleMeasure the length of a book 5 times obtaining 9.30 in, 9.20
in, 9.30 in, 9.40 in and 9.30 in.
〈x〉 = 1
5(9.3 + 9.2 + 9.3 + 9.4 + 9.3) in = 9.3 in
〈x2〉 = 1
5(9.32 + 9.22 + 9.32 + 9.42 + 9.32) in2 = 86.494 in2
σ = (86.494− 9.32)1/2 in = 0.06 in
Error Analysis LectureCHM 335 – p. 12/39
Report?
σN =σ√
N − 1=
0.06 in√5− 1
= 0.03 in
〈x〉 ± 2σN = (9.30± 0.06) in
In today’s experiment you will measure the flow times of
water and a polyvinyl alcohol solution using a viscosimeter.
For the flow times you make 10 measurements, and report
the flow times as 〈t〉 ± 2σN .
Error Analysis LectureCHM 335 – p. 13/39
Error EstimationIn real laboratory research we always calculate statistical
errors by measuring the standard deviation of the mean. In
CHM 335 we do not have time, so we estimate random
errors using simple rules.
Rule of thumb: We assume an error of 2σN = ±2 in the last
digit of a measuring device.
Example: If we use a balance and determine a mass of
2.4354 grams, we assume the mass is 2.4354± 0.0002
grams.
Exceptions: Pipettes and volumetric flasks have estimated
statistical errors etched on the glassware. Alternatively, for
pipettes, see page 26 of the syllabus for a table.Error Analysis LectureCHM 335 – p. 14/39
Roundoff
> 5 increase
< 5 no change
= 5 increase if the result is even
Error Analysis LectureCHM 335 – p. 15/39
Roundoff Examples24.43
24.43 = 24.4
24.46
24.46 = 24.5
24.45
24.45=24.4
24.55
24.55=24.6
Error Analysis LectureCHM 335 – p. 16/39
Error Propagation
Suppose you measure variables x, y, z, . . . withassociated errors ǫ(x), ǫ(y), ǫ(z), . . .. Supposef = f(x, y, z, . . .). What is the error in f?
Error Analysis LectureCHM 335 – p. 17/39
The DifferentialLet f(x) be a function of x. The differential of f is defined by
df =df
dxdx
The differential of f gives the infinitesimal variation of f
given infinitesimal variations dx.
Example: The radius of the earth at the equator is 6.356
×106 meters. Imagine a wire strung around the equator that
is exactly 2π × 6.356× 106 meters in length. The wire is then
raised on polls around the earth such that the radius of the
wire ring is increased by exactly 1 meter. What is the
increase in the circumference of the wire?
Error Analysis LectureCHM 335 – p. 18/39
Solution
C(r) = 2πr
dC = 2πdr
∆C = 2π × 1 m ≈ 6.28318 meters.
Error Analysis LectureCHM 335 – p. 19/39
The Total DifferentialLet f(x, y, z, . . .) be a function of x, y, z, . . .. The total
differential or equivalently the exact differential of f is
defined by
df =
(
∂f
∂x
)
y,z,...
dx+
(
∂f
∂y
)
x,z,...
dy +
(
∂f
∂z
)
x,y,...
dz + . . .
The total differential of f gives the infinitesimal variation of f
given infinitesimal variations of the arguments of f ; i.e.
dx, dy, dz, . . ..
Error Analysis LectureCHM 335 – p. 20/39
Example
f(x, y) = x2y + yex
df = (2xy + yex)dx+ (x2 + ex)dy
Error Analysis LectureCHM 335 – p. 21/39
Differentials andErrors
We can replace the differentials of the arguments
dx, dy, dz, . . . by the small errors in the arguments
ǫ(x), ǫ(y), ǫ(z), . . ..
Error Analysis LectureCHM 335 – p. 22/39
Propagation ofSystematic Errors
df =
(
∂f
∂x
)
y,z,...
dx+
(
∂f
∂y
)
x,z,...
dy
+
(
∂f
∂z
)
x,y,...
dz + . . .
ǫ(f) ≈(
∂f
∂x
)
y,z,...
ǫ(x) +
(
∂f
∂y
)
x,z,...
ǫ(y)
+
(
∂f
∂z
)
x,y,...
ǫ(z) + . . .
Error Analysis LectureCHM 335 – p. 23/39
ExampleThe expression for the gram molecular mass (molecularweight) G of an ideal gas at temperature T , pressure P ,volume V and mass m is given by
G =mRT
PV.
Suppose measurements are made with results
m = 10.0 g T = 269. K P = 1.00 bar
V = 0.50 L,
and suppose the volume has a systematic error ofǫ(V ) = +0.02 L. Find G.
Error Analysis LectureCHM 335 – p. 24/39
Solution
G =mRT
PV=
(10.0 g)(0.08314 L bar/mol K)(269 K)(1. bar)(0.50 L)
= 447. g/mol
dG = −mRT
PV 2dV
dG
G= −mRT/PV 2
mRT/PVdV = −dV
V
ǫ(G) = −Gǫ(V )
V= −447. g/mol
(
0.02
0.50
)
= −18 g/mol
Error Analysis LectureCHM 335 – p. 25/39
Report?
G = Gcalc + ǫ(G) = 447. g/mol − 18. g/mol = 429. g/mol
ǫ(G) = −Gǫ(V )
V
Both the errors in G and V have a definite sign.
The value of G decreases, because the volume V appears
in the denominator.
Error Analysis LectureCHM 335 – p. 26/39
Random ErrorsRandom errors have no definite sign, so we square and
average.
ǫ2(f) ≈[
(
∂f
∂x
)
y,z,...
ǫ(x) +
(
∂f
∂y
)
x,z,...
ǫ(y)
+
(
∂f
∂z
)
x,y,...
ǫ(z) + . . .
]2
≈(
∂f
∂x
)2
ǫ2(x) +
(
∂f
∂y
)2
ǫ2(y) +
(
∂f
∂z
)2
ǫ2(z) + . . .
[Assume 〈ǫ(x)ǫ(y)〉 = 0]
Error Analysis LectureCHM 335 – p. 27/39
Same Example
G =mRT
PV
m = (10.0± 0.2) g T = (269.± 1.) K
P = (1.0± 0.1) bar V = (0.50± 0.02) L
Assume no systematic error and R is exact (nottrue).
Error Analysis LectureCHM 335 – p. 28/39
Example Propagation
G =mRT
PV
dG =
(
∂G
∂m
)
T,V,P
dm+
(
∂G
∂T
)
m,V,P
dT
+
(
∂G
∂V
)
T,m,P
dV +
(
∂G
∂P
)
T,V,m
dP
=
(
RT
PV
)
dm+
(
Rm
PV
)
dT
−(
mRT
PV 2
)
dV −(
mRT
V P 2
)
dP
Error Analysis LectureCHM 335 – p. 29/39
Example (cont.)Divide both sides by G
dG
G=
RT/PV
mRT/PVdm+
Rm/PV
mRT/PVdT
−mRT/PV 2
mRT/PVdV − mRT/V P 2
mRT/PVdP
=dm
m+
dT
T− dV
V− dP
P
Replace differentials by errors, square and average
(
ǫ(G)
G
)2
=
(
ǫ(m)
m
)2
+
(
ǫ(T )
T
)2
+
(
ǫ(V )
V
)2
+
(
ǫ(P )
P
)2
Error Analysis LectureCHM 335 – p. 30/39
Example (cont.)
G =mRT
PV
m = (10.0± 0.2) g T = (269.± 1.) K
P = (1.0± 0.1) bar V = (0.50± 0.02) L
G = 447. g mol−1
ǫ(G) = G
[
(
ǫ(m)
m
)2
+
(
ǫ(T )
T
)2
+
(
ǫ(V )
V
)2
+
(
ǫ(P )
P
)2]1/2
Error Analysis LectureCHM 335 – p. 31/39
Example (cont.)
= 447. g mol−1
[
(
0.2
10.
)2
+
(
1.
269.
)2
+
(
0.02
0.50
)2
+
(
0.1
1.
)2]1/2
= 49 g mol−1
Report?
G = 450± 50 g mol−1
The first digit in the error defines the number of digits to be
reported in the final result.
Error Analysis LectureCHM 335 – p. 32/39
Another ExampleThe concentration of hydrogen ions in a particular solution is
measured with the result [H+]± ǫ([H+]). What is the error in
the pH of the solution?
pH = − log10([H+]) = − 1
2.3ln([H+])
dpH = − d[H+]
2.3[H+]
ǫ(pH) =ǫ([H+])
2.3[H+]
Error Analysis LectureCHM 335 – p. 33/39
Sums andDifferences Only
f = a+ b− c
df = da+ db− dc
ǫ(f) = [ǫ2(a) + ǫ2(b) + ǫ2(c)]1/2
Error Analysis LectureCHM 335 – p. 34/39
Products andQuotients Only
f =ab
c
df =b
cda+
a
cdb− ab
c2dc
df
f=
b/c
ab/cda+
a/c
ab/cdb− ab/c2
ab/cdc
=da
a+
db
b− dc
c
ǫ(f) = f
[
(
ǫ(a)
a
)2
+
(
ǫ(b)
b
)2
+
(
ǫ(c)
c
)2]1/2
Error Analysis LectureCHM 335 – p. 35/39
GraphsSome graphs are to be drawn by hand. When hand-drawn
graphs are required, do not use computer generated graphs.
Instructions associated with graphs will be given for each
experiment by the TA.
Give all graphs a title, and label the x and y axes.
Graphs should fill the entire page.
Include all points with associated error bars or error boxes.
Error Analysis LectureCHM 335 – p. 36/39
Example Graph
200 250 300 350 400 450 500 550T(K)
20
25
30
35
40
45
50
PV
(L
atm
)
PV vs. T
Error Analysis LectureCHM 335 – p. 37/39
Errors in GraphicalData
Error in slope m
ǫ(m) ≈ |m+ −m−|2
Error in intercept b
ǫ(b) ≈ |b+ − b−|2
Error Analysis LectureCHM 335 – p. 38/39
Laboratory ReportsInclude all calculations of your final results, and include all
calculations of errors.
Find errors using a combination of error propagation and
graphical analysis (when required).
In intermediate calculations of results and errors, keep as
many figures as you wish.
Report the final results and errors with the proper number of
significant figures as dictated by the calculated errors.
Incorrect numbers of significant figures in the final results
are subject to significant penalties.
The error analysis represents between 30% and 40% of
your report grades.Error Analysis LectureCHM 335 – p. 39/39