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Chapter 6: Random Errors in Chemical Analysis
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Contents in Chapter 061. The Nature of Random Errors
1) Random Errors and Their Mathematical Equations 2) Central Limit Theorem
2. Gausian (Normal) Distribution1) Define Gausian Distribution2) Z Value Transformation3) Z value Applications4) Constructing Gaussian Curve From Experimental Data
3. Standard Deviation of Calculated Results4. Significant Figures
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1. The Nature of Random Errors1) Random Errors and Their Mathematical Equations
Random (Indeterminate, Statistic) error:Errors affecting the precision (uncertainty), caused by uncontrollable variables. Positive and negative () fluctuation occur with approximate equal frequency.
Population:The set of infinite objects in the system being investigated.
Sample:The finite members of a population that we actually collect and analyze.
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Mean (Average)
Standard deviation
- For population:
- For sample:
samplefor x ,populationfor :symbol n
x
mean ii
1
)(1
2
N
xx
s
N
ii
( )x
N
ii
N2
1
μ x σ, s 20, n If ●
Spread (range, w): The difference between extreme values in a set of d
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t
N
kk
N
jj
N
ii
pooled NNNN
xxxxxx
S
...
...)()()(
321
1
23
1
22
1
21
321
Pooled standard deviation:For several subsets of data, estimating standard deviation by pooling (combining) the data.
Variance = s2 *variance are additive
Relative standard deviation (RSD) by 100%(also called coefficient of variation, CV):
100%s
RSD in percentx
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Degree of freedom (dof)In statistics, the number of independent observations on which a result is based.
- For standard deviation:dof = n-1,n is munber of measurement.
- For pooled standard deviation:dof = N – M,N: Total number of measurementsM: Number of the subset.
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2) Central Limit Theoremi) Definition For the probability distribution plot for the frequency
of individual values. The measurements subject to indeterminate errors arise a normal (Gaussian) distribution.
The sources of individual error must be independent. The individual error must have similar magnitude (no
one source of error dominates the final distribution).
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Combination of uncertainty Magnitude of
Combination
Frequency of
combinations
Relative
frequency
+U1+U2+U3+U4 +4U 1 1/16=0.0625
-U1+U2+U3+U4
+U1-U2+U3+U4
+U1+U2-U3+U4
+U1+U2+U3-U4
+2U 4 4/16=0.250
-U1-U2+U3+U4
+U1+U2-U3-U4
+U1-U2+U3-U4
-U1+U2-U3+U4
-U1+U2+U3-U4
+U1-U2-U3+U4
0 6 6/16=0.375
+U1-U2-U3-U4
-U1+U2-U3-U4
-U1-U2+U3-U4
-U1-U2-U3+U4
-2U 4 4/16=0.250
-U1-U2 -U3 -U4 -4U 1 1/16=0.0625
(Frequency of Combinations of four equal-sized uncertainty, u)ii) Simulated Central Limit Theorem
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4 random uncertainties
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10 random uncertainties
Infinite number of random uncertainties
- A Gaussian (normal) distribution curve.
- Symmetrical about the mean.
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2. Gausian (Normal) Distribution1) Define Gausian Distribution
A “bell-shaped” probability distribution curve for measurements showing the effect of random error, which encountered continuous distribution.The equation for Gaussian distribution:
e2
1 f(x)y 2
2
2)(x
deviation standard:
value true:
1)~(0frequency relative:y
valueindividual:x
12
1 22 2/)(
dxe x
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IF: Same Mean, Different SD 2
2
2
)(
2
1y
x
e
982_Ch04_Treat_RandDistr_Data.xls
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IF: Different Mean, Same SD
982_Ch04_Treat_RandDistr_Data.xls
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IF: Different Mean, Different SD
982_Ch04_Treat_RandDistr_Data.xls
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2) Z value Transformation For x-axis: transform x to z by:
x
z
2
2
2
1)(y
z
ezf
For y-axis: transform to f(z) by :
12
1 2/2
dze z
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Now, the Gaussian Curve Always Consistent
x
z
2
2
2
1y
z
e
982_Ch04_Treat_RandDistr_Data.xls
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教戰守則
1.經過 z transform, Gaussian curve 長得一模一樣,此時 x-axis 的單位為 σ。
2. 看到標示為 x ,單位為 σ 時,代表已經經過 z transform 。
3. Relative frequency (y) at mean (x), 約為 0.4 。4. The entire area of the Gaussian curve is 1.
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3) Z value ApplicationsArea (probability, percentage) in defined z interval
1dze2π
1 within Area
0.9973dze2π
1 3σ within Area
0.9546dze2π
1 2σ within Area
0.6826dze2π
1 1σ within Area
/2z
33
/2z
22
/2z
11
/2z
2
2
2
2
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Step 1: Raw data collectionStep 2: Arrange the data in order from lowest to
highestStep 3: Condense the data by grouping them into cellsStep 4: Pictorial representation of the frequency
distributionsStep 5: Estimating the σ from sStep 6: Plotting relative frequency versus x or z
4) Constructing Gaussian Curve From Experimental Data
i) General procedure
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Example: Replicate Data for the Calibration of a 10 mL Pipet
982_Ch04_Treat_RandDistr_Data.xls
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Cont’d
982_Ch04_Treat_RandDistr_Data.xls
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982_Ch04_Treat_RandDistr_Data.xls
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Number in range Percentage in range
Relative frequency versus x
2
2
2
2
2(0.006)
)982.9(x
2σ
μ)(x
e2π(0.006)
)(50)(0.003
e2πs
bar)per umepipet)(vol (totaly
2
2
2
2
2(0.006)
)982.9(x
2σ
μ)(x
e2π(0.006)
(0.003)
e2πs
bar)per (volumey
962_Ch04_Treat_RandDistr_Data.xls
X-axis 刻度為體積之間隔
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Matching Histogram with Gaussian curve
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Z Transformed Gaussian Curve
982_Ch04_Treat_RandDistr_Data.xls
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4. Standard Deviation of Calculated Results(for Random Errors)
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1) Sum or Difference
2 2y 48(0.006) (0.006) 0.008s
Example 1: The calibration result of class A 10 mL pipet showed that the marker reading is 9.9920.006 mL. When it is used to deliver two successive volumes. What is the absolute and percent relative uncertainties for the total delivered volume.Solution:Total volume = 9.992 mL + 9.992 mL = 19.984 mL
Ans: 19.984(0.008) mL
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2 2y 8(0.02) (0.02) 0.02s
Example 2: For a titration experiment, the initial reading is 0.05(0.02) mL and final reading is 17.88 (0.02) mL. What is the volume delivered?Solution:Delivered volume = 17.88 mL – 0.05 mL = 17.83 mL
Ans: 17.83(0.03) mL
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y 2 27
y 7 2
0.01 1( ) ( ) 0.06
0.15 120
18 0.06 1.
s
y
s
Example 1: The quantity of charge Q = I x t. When a current of 0.150.01 A passes through the circuit for 1201 s, what is the total charge?Solution:Total charge 0.15 A x 120 s = 18 C
Ans: 18(1) C
2) Product or Quotient
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3) Mixed operations
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Ans: 1.9 (0.1)x10–4 M
4) Exponents and logarithms
Example: The pH of a solution is 3.720.03, what is the [H3O+] of this solution?
Solution: [H3O+] = 10–pHy10 2.3026 sy
xx
sy
y y04
1
4 50 1 0 3
2.3026 0.03 0.07y 1.9 10
(0.07 ) (1.9 10 )(0.07 ) 1. 10y
s s
s y
y = 10–3.72 =1.91x10–4
y(sy) = 1.91 (0.13)x10–4
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4. Significant Figures1) Statement of significant figures The digits in a measured quantity, including all digits
known exactly and one digit (the last figure) whose quantity is uncertain.
The more significant digits obtained, the better the precision of a measurement.
The concept of significant figures applies only to measurements.
The Exact values (e.g., 1 km = 1000 m) have an unlimited number of significant figures.
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0 1 2 3
Recording: 1.51 or 1.52 or 1.53
0 1 2 3
Recording: 1.4 or 1.5 or 1.6
2) Recording with significant figures
2 significant figure,1 certain, 1 uncertain
3 significant figure,2 certain, 1 uncertain
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3) Rules for Zeros in Significant Figures
Zeros between two other significant digits are significante.g. 10023 no. of sig. fig.: 5
A zero preceding a decimal point is not significante.g., 0.10023 no. of sig. fig.: 5
Zeros preceding the first nonzero digit are not significante.g. 0.0010023 no. of sig. fig.: 5
Zeros at the end of a number are significant if they are to the right of the decimal pointe.g. 0.1002300 no. of sig. fig.: 7
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Zeros at the end of a number may or may not be significant if the number is written without a decimal pointe.g. 92500 no. of sig. fig.: N/A
Scientific notation is required: e.g. 9.25x104 no. of sig. fig.: 3 e.g. 9.250x104 no. of sig. fig.: 4 e.g. 9.2500x104 no. of sig. fig.: 5
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2. Significant Figures in Arithmetic1) Rules for rounding off numbersi) If the digit immediately to the right of the last sig. fig.
is more than 5, round up.ii) If the digit immediately to the right of the last sig. fig.
is less than 5, round down.iii) If the digit immediately to the right of the last sig. fig.
is 5 followed by nonzero digits, round up.iv) If the digit immediately to the right of the last sig. fig.
is 5 round up if the last sig. fig. is odd. round down if the last sig. fig. is even.v) If the resulting number has ambiguous zeroes, it should
be recorded in scientific notation to avoid ambiguity.
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Examples:
35.76 in 3 sig. fig. is 35.8
35.74 in 3 sig. fig. is 35.7
24.258 in 3 sig. fig. is 24.3
24.35 in 3 sig. fig. is 24.4 (rounding to even digit)
24.25 in 3 sig. fig. is 24.2 (rounding to even digit)
13,052 in 3 sig. fig. is 1.31 x 104
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2) Addition and Subtraction: The reported results should have the same number of
decimal places as the number with the fewest decimal places
Example 2: MW of KrF2
18.9984032 (F)+ 18.9984032 (F)+ 83.798 (Kr) 121.7948064
Ans: 121.795 g/molAns: 112.2 m
49.146 m+ 72.13 m– 9.1 m 112.176 m
Example 1:
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Note:To avoid accumulating “round-off ” errors in calculations:1) Go through all calculation by calculator, then
rounding on the final answerOR2) Retaining one extra insignificant figure (a subscribe
digit) for intermediate results, then rounding on the final answer
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Example 3:Write the answer with correct number of digits:2.432x106 + 6.512x104 – 1.227x105 = ?
Ans: 2.374x106
2.432 x 106
+ 0.06512 x 106
– 0.1227 x 106
2.37442 x 106
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3) Multiplication and Division: The reported results should have no more significant
figures than the factor with the fewest significant figures
Example 1: 1.827 m × 0.762 m = ?1.827 m x 0.762 m = 1.392174 m2 = 1.39 m2
Ans: 1.39 m2
Example 2:(4.3179 x 1012)(3.6x10–19) = 1.554444x10–6 = 1.6x10–6
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Stoichiometric coefficients in a chemical formula, and unit conversion factors etc, have an infinite number of significant figures.
Example:Results of four measurements: 36.4 g, 36.8 g, 36.0 g, 37.1 g. What is the average?Solution:(36.4+36.8+36.0+37.1)/4 = 36.6Ans: 36.6 g
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4) Logarithms and Antilogarithmsi) Mathematic terms For exponential expression, e.g. 4.2 x 10–2:
“4.2” is called the coefficient,“–2” is called the exponent.
For logarithm operation, e.g. log (4.2 x 10–2) = –1.38“–1” is called the characteristic, the integer part“38” is called the mantissa, the decimal part
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ii) Logarithms operation
Example 1: log(56.7 x 106) = ?Solution:log(5.67 x 107) = 7 + log(5.67) = 7.754 3 sig. fig.
Example 2: log(0.002735) = ?Solution:log(0.002735) = – 2.5630 4 sig. fig.Orlog(0.002735) = 2.735x10–3 = (– 3) + log(2.735)= (–3) + (0.4370) = – 2.5630
Mantissa remain equal number of sig. fig. as the original digit in coefficient
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Example 3: Percent transmittance (%T) is related to the absorbance (A) by the equation: A = –log (%T/100). What is correct digits of A if %T is 72.9. Solution:A = –log (%T/100) = –log (0.729) = – (– 0.137) = 0.137Or log (0.729) = – log(7.29x10–1) = – [(–1) + log (7.29)]= – [(–1) + (0.863)] = 0.137
Example 4: The pH is defined as pH = –log [H3O+]. If the [H3O+] is 3.8 x 10–2 M, what is the pH of the solution?Solution:pH = –log [H3O+] = –log (3.8 x 10–2) = –[(–2) + log(3.8)] =– (–1.42) = 1.42
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iii) Antilogarithms operation
Example 1: antilog(2.671) = ?Solution:antilog(2.671) = 102.671 = 100.671x102 = 4.69 x 102
Example 2: What is the correct digits of %T if the absorbance is 0.931? Solution:Antilog (–0.931) = 10–0.931 = 0.117%T = 0.117 x 100% = 11.7%
Final digit has same number of sig. fig. as number of digits in mantissa
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Example 3: If the pH is 10.3, what is the [H3O+]?Solution:[H3O+] = 10–pH = 10–10.3 = 5 x 10–11
Or 10–10.3 = 100.7 x 10–11 = 5 x 10–11
Example 4: If the pH is 2.52, what is the [H3O+]?Solution:[H3O+] = 10–pH = 10–2.52 = 3.0 x 10–3
Or 10–2.52 = 100.48 x 10–3 = 3.0 x 10–3
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Homework (Due 2014/3/6)
Skoog 9th edition, Chapter 06 Questions and Problems6-16-46-56-9 (a) (c) (e)6-11 (a) (c)6-15
End of Chapter 06