Chapter 6: Random Errors in Chemical Analysis

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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


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IF: Different Mean, Different SD


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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


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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


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Cont’d


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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


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


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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

Chapter 6: Random Errors in Chemical Analysis

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