1. Chapter 3
Experimental Error

2. - Significant figures
- Precision (Reproducibility)
- Accuracy (Error)
- Uncertainty

3. 3-1 Significant Figures
Significant figures: minimum number of digits required to express a value in scientific notation without loss of
accuracy (with the appropriate accuracy).
Guidelines for Sig. Figs., when looking at a number
1) All nonzero digits are significant figures
142.7 = x 102
= x10-6
4 significant figures (sf)
Zeros are simple place holders
9.25 x 104
9.250 x 104
x 104
(3 sf)
92500
(4 sf)
(5 sf)
2) Zeros are counted as significant figures only if:
i) occur between other digits in the number
ii) occur at the end of a number on the right-hand side of the
decimal point
0.1060
(3 sf)
(4 sf)

4. 9.25 × 104  3 significant figures 9.250 × 104  4 significant figures
The number of significant figures is the minimum number of digits needed to write a given value in scientific notation without loss of precision.
92, 500 
9.25 ×  3 significant figures
9.250 ×  4 significant figures
×  5 significant figures
Zeros are significant 1) in the middle of a number, 2) at the end of a number on the right-hand side of a decimal point.

5. 3-2 Significant Figures in Arithmetic
How many digits to retain in the answer after you have performed arithmetic operations with your data?
Addition and Subtraction:the same decimal place in the final answer
1.362 × 10 -4
× 10 -4
4.473 × 10 -4
5.345
12.073
7.26 × 10 14
-6.69 × 10 14
0.57 × 10 14
(F)
(F)
(Kr)
1.632 × 105
× 103
× 106 × × × ×

6. - Multiplication and Division
In multiplication and division, we are normally limited to the number of digits contained in the number with the fewest significant figures.
3.26 × 10 -5
×1.78
5.80 × 10-5
× 1012
× ×
× 10-6
34.60 ÷
14.05
- Logarithms and antilogarithms
IF n = 10a, then we say that a is the base 10 logarithm of n :

7. A logarithm is composed of a characteristic and a mantissa.
The characteristic is the integer part and the mantissa is the decimal part
log 339= log 3.39×10-5=
Characteristic Mantissa Characteristic Mantissa
= = = =0.470
=340(339.6)
=339(338.8)
=338(338.1)
antilog (-3.42) = = 3.8 × 10 -4
log = antilog4.37=2.3 × 10 4
Log1 237= =2.3 × 10 4
Log3.2= =2.51 × 10 -3

8. Significant Figures and Graphs

9. (and maybe uncontrollable)
3-3 Types of Error
Experimental error from uncertainties of measurements
- SYSTEMATIC ERROR:
Systematic error , also called determinate error arises from a flaw in equipment or the design of experiment.
- Random error:
called indeterminate error arises from uncontrolled
(and maybe uncontrollable)
(Gross errors) ⇒ mainly originated by person
⇒ statistical calibration

10. Box 3-1. Case study : systematic error in Ozone Measurement
Systematic errors: determinate error
⇒ definite value, assignable cause.
⇒ bias -> 모든 결과에 같은 크기의 영향을 미치며, 부호를
가짐.
Box 3-1. Case study : systematic error in Ozone Measurement

11. Sources of systematic errors
Three types of  systematic errors
    ) instrumental errors
         ) method errors
        ) personal errors  

12. ■ Instrumental errors ⇒ measuring device error
Ex) ① Pipet, burette, mass flask, etc.
Reasons : The calibration temp. is nonequal to the measuring temp.
Dry or heating cause the distortion of glass
Contamination inside surface of vassel
       ② Electronic instrument error
Reasons: potential change by dry cell life time
Wrong calibration
electric devise error by temperature change
Noise from AC power source
③ Instrument operation in error conditions
이유: pH meter in strong acid solutions -> acid error
      ⇒ vibration → detectable, correctable        

13. ■ Method errors ⇒ From nonideal behaviors of reactions and reagents.
⇒ Slow reaction, incomplete reaction, using unstable chemical,
nonselectiveity of reagents, side reactions, etc..
⇒ The most difficult to remove it.
Ex) using excess reagents
- Due to chemical properties of nicotinic acid – degradation using hot and
conc. H2SO4 - pyridine ring bearing nicotinic acid ->
incomplete degradation. 
- Addition of potassium sulfate and elevate the boilng temp. -> complete
degradation.  

14. ■ Personal errors ⇒ personal decision needs in many measurements.
⇒ error, toward one direction
예) • Reading in position of pointer between two points.
• Color at the end point
• liquid level of burret
• Color sensitivity
⇒ personal bias
예) • 정밀도를 증가시키는 방향으로 눈금을 읽을 때
• 측정의 참값을 미리 마음 속으로 정해 놓을 때
• 숫자에 대한 편견이 있을 때
- 눈금 위의 바늘의 위치를 읽을 때 숫자 0과 5를 선호, - 큰 수보다
작은 수, - 홀수보다 짝수 선호

15. The effect of the systematic error on analytical results
⇒ Systematic errors is constant or proportional
⇒ constant errors size
• 측정되는 양의 크기에 따라 달라지지 않음
• 절대오차는 시료크기에 대하여 일정
• 상대오차는 시료의 크기에 따라 변함
⇒ proportional errors size
• 분석에 사용된 시료의 크기에 따라 증가 또는 감소
• 절대오차는 시료크기에 따라 변함
• 상대오차는 시료의 크기에 대하여 일정

16. Detection of systematic instrumental & personal errors
⇒ Instrumental error
• can be founded and corrected by calibration
• periodical calibration
• Instrumental error by interferences in samples
→ 단순한 검정으로 영향제거 불가능
⇒ Personal error
• It can be minimize by precaution, excise, etc.
• Check instrument reading, notebook entries & calculations
• chose the adequate method -- error minimizing

17. Analysis of standard samples
Detection of systematic method errors
■ Method errors
Analysis of standard samples
      ⇒ The best way of estimating the bias of analytical method is by the analysis of standard reference materials(SRMs)
⇒ SRMs : 정확하게 잘 알려진 농도를 가지고 있는 analytes를 하나
또는 그 이상 포함한 물질
⇒ 합성하여 사용
• 순수한 성분들을 혼합하여 조성을 알 수 있는 균일시료 제조
• 합성 표준물질의 조성은 분석시료의 조성과 거의 같아야 함
• 표준시료의 합성이 불가능하거나, 쉽지 않고 시간이 많이 걸리는 경우 가 있음 → 실제적이지 못할 수 있음
⇒ • 미국 정부기관인 NIST(National Institute od Standards & Technology)에서 종 이상의 SRMs 공급
• 몇몇 시판 공급회사에서도 공급

21. 105.4 Toxic Substances in Urine (powder form)
SRMs 2670a, 2671a and 2672a are for determining toxic substances in human urine.
They consist of freeze-dried urine and are provided in sets of four 30 mL bottles -- two each at low and elevated levels.
NOTE:The values listed for these SRMs apply only to reconstituted urine.

22. ■ Independent analysis
⇒ 표준시료를 사용할 수 없을 경우
⇒ 같은 시료를 또 다른 독립적이고 신뢰성 있는 분석법으로 분석 (parallel analysis)
⇒ 두 방법은 가능한 한 많이 달라야 함 → 두 방법에 모두 영향을 줄 수 있는 공통요인을 최소화 하기 위함
⇒ 두 방법간의 차이가 random error 또는 방법에서 오는 bias
때문인지를 평가하기 위해 반드시 통계적 test를 실시 (7B-2)

23. ■ Blank determinations
• reagents and solvents without analyte.
• Similar condition of analyte environment (sample
matrix):
- >addition of sample constituents.
⇒ blank determination
• every step in analysis : blank material analysis
needs.

24. Precision and Accuracy
Precision describes the reproducibility of a result, if you measure
A quantity several time and the values agree closely with on another
your measurement is precise.
Accrue describes how close a measured value is to the “true” value
If a known stands is available , accuracy hoe close your value is to the known value.

25. Precision ⇒ 측정의 재현성(reproducibility of measurements)
⇒ 정확히 똑같은 방법을 이용하여 얻은 측정값들 사이의 일치성
⇒ 반복시료를 사용하여 반복적인 측정을 함
• Three terms to describe the precision
Standard deviation
Variance
Coefficient of variance (분산계수)
⇒ deviation from the mean (di)의 함수.

26. Absolute and Relative uncertainty
Absolute uncertainty expresses the margin of uncertainty associated with a measurement.
Relative uncertainty compares the size of the absolute uncertainty with the size of its associated measurement.

27. 3-4 Propagation of Uncertainty from Random Error
We can usually estimate or measure the random error associated with measurement,
such as the length of an object or the temperature of a solution.
- Addition and Subtraction
1.76 (± 0.03)
(±0.02)
-0.59(±0.02)
3.06(±e4) e1 e2 e3
Percent relative uncertainty=0.041 / 3.06 × 100= 1.3%
3.06 (± 0.04): absolute uncertainty,
3.06 (± 1 %): relative uncertainty

28. (EXAMPLE) Uncertainty in a Buret Reading

29. -Multiplication and Division

30. - Mixed Operations

34. Uncertainty for powers and roots: Uncertainty for logarithm:
Exponents and logarithms
Uncertainty for powers and roots:
(3-7)
Uncertainty for logarithm:
(3-8)
(3-9)
Uncertainty for
Natural logarihm
(3-10)
Uncertainty For 10x:
Uncertainty for ex
(3-11)

36. 3-5 Propagation of Uncertainty from Systematic Error
The standard deviation ( defined in section 4-1 ) for this distribution,
Also called the standard uncertainty , is ±a√3= ±0.0003/ √3= ±

37. Uncertainty in molecular mass

38. - Uncertainty in Molecular Mass

39. Multiple Deliveries From One Pipet: The Triangular Distribution
The standard uncertainty ( standard deviation) in the triangular
Distribution is ± a √6 = ±0.03 √6= ±0.012ml
The standard uncertainty is ±4 × 0.012= ±0.048 ml , not
± √ = ±0.024ml
Calibration improves certainty by removing the systematic error