1. Experimental Error
Every Physical Measurement has Error
• Preparation of a Solution
• Measurement of pH
No way to measure the “TRUE” value of anything

2. Experimental Error
Every Physical Measurement has Error
• Preparation of a Solution
• Measurement of pH
No way to measure the “TRUE” value of anything
Repetition: evaluate the reproducibility (Precision)
Different Methods: Confidence near the “truth” (Accuracy)

3. Experimental Error
Every Physical Measurement has Error
• Preparation of a Solution
• Measurement of pH
No way to measure the “TRUE” value of anything
Repetition: evaluate the reproducibility (Precision)
Different Methods: Confidence near the “truth” (Accuracy)
• Precision
– describes the reproducibility of results
– involves a comparison btwn measurements
• Accuracy
– nearness of a measurement to its accepted value

4. Types of Error
Types of Error
1. Determinate (systematic)
2. Indeterminate (random)

5. Types of Error
1. Systematic (Determinate) Error
Definite value; Assignable Cause
In principle measured and accounted
Unidirectional  low or high

6. Types of Error
1. Systematic (Determinate) Error
2. Random (Indeterminate) Error

7. Types of Error
1. Systematic (Determinate) Error
2. Random (Indeterminate) Error
– Natural Limitations
– Uncontrollable
– Random
– Ultimate Limitation
– Use Statistics

8. Detect
Known samples
Blank samples
Different people/laboratories
Different Methods

9. Types of Error
Clicker Question
Which of the following represents a systematic error
a. Using a pH meter that has not been calibrated
b. Missing the end point in a titration due to color blindness
c. Missing the end point in a titration due to slow reactions
d. All of the above
e. None of the above

10. Evaluate Uncertainty in the Result
Significant Figures
Absolute / Relative Uncertainty
Propagation of Error
Statistics
Accept/Reject Data Point (Q-test)
Define interval around a mean with given
probability
Types of Error

11. Clicker question
How many significant figures does 0.00110 have
a. 5
b. 4
c. 3
d. 2

12. Absolute / Relative Uncertainty
Absolute
margin of uncertainty associated with a measurement

13. Absolute / Relative Uncertainty/Error
Absolute
margin of uncertainty associated with a measurement
ex. Tolerance on volumetric flask, pipet, buret,...
100.00 mL (±0.08) mL
ex. Standard deviation
ex. Xi – Xt (where Xt is true value)

14. Relative
absolute uncertainty / magnitude of measurement
think about it as being “relative” to something
Xi – Xt (where Xt is true value)
Xt
Relative Standard Deviation
S
___
X
x 100%

15. Clicker questions
If the average of three repetitive measurements is 3.05 grams
and the standard deviation is 0.06, what is the relative percent
standard deviation
a. 2.0%
b. 0.06%
c. 6.0%
d. 3.05%
A student uses a gravimetric technique to determine
how much nickel is in an unknown. She obtains a
value of 3.06 %. The “true” value determined by the
manufacturer through many trials is 3.03%. What is
the percent relative error in the students data?
a. 0.03%
b. 0.09%
c. 1%
d. 3.03%

16. Propagation of Error
• Used to deal with Random Error
• Takes into account error assoc. with all individual measurements
• Error of Computed Results
– What is the uncertainty associated with the final result?

18. Propagation of Error
Used to deal with Random Error
Takes into account error assoc. with all individual measurements
Error of Computed Results
What is the uncertainty associate with the
Final Result?
Addition-Subtraction
y = a + b + c
Ea, Eb, Ec : uncertainty associated with indiv. #s (a, b,c)
Ey = (Ea
2
+ Eb
2
+ Ec
2
)1/2 Memorize
Absolute Error

19. Propagation of Error
Used to deal with Random Error
Takes into account error assoc. with all individual measurements
Error of Computed Results
What is the uncertainty associate with the
Final Result?
Addition-Subtraction
y = a + b + c
Ea, Eb, Ec : uncertainty associated with indiv. #s (a, b,c)
Ey = (Ea
2
+ Eb
2
+ Ec
2
)1/2
Example: 0.05 (±0.02) + 4.10 (±0.03) - 1.97 (±0.05)
Ey = ((0.02)2
+ (0.03)2
+ (0.05)2
)1/2
Ey = 0.062
Memorize
Absolute Error

20. Propagation of Error
Used to deal with Random Error
Takes into account error assoc. with all individual measurements
Error of Computed Results
What is the uncertainty associate with the
Final Result?
Addition-Subtraction
y = a + b + c
Ea, Eb, Ec : uncertainty associated with indiv. #s (a, b,c)
Ey = (Ea
2
+ Eb
2
+ Ec
2
)1/2
Example: 0.05 (±0.02) + 4.10 (±0.03) - 1.97 (±0.05)
Ey = ((0.02)2
+ (0.03)2
+ (0.05)2
)1/2
Ey = 0.062
Answer: 2.63 (±0.06)
NOT: 2.63 (± 0.0620)

21. Propagation of Error
1. Addition and Subtraction
2. Division and Multiplication
y = (a * b) / c
Ea, Eb, Ec : uncertainty associated with indiv. #s (a, b,c)
(Ea)r, (Eb)r, (Ec)r : relative uncertainties
ex. (Ea)r = Sa / a
(Ey )r= ((Ea)r
2
+ (Eb)r
2
+ (Ec)r
2
)1/2
Ey = (Ey)r * y
Memorize
Absolute Error Relative Error

22. Example Problem
Calculate the Molarity and its uncertainty of a sodium hydroxide
solution prepared by dissolving 0.0210 (±0.0002) g in
a 100.00 (± 0.02) mL volumetric flask?

23. Statistical Treatment of Random Errors
Evaluate random error
Assumption: random errors follow a gaussian (normal)
distribution
A. Infinite number of Data
Gaussian Error Curve
Concentration
-40 -20 0 20 40 60 80
Occurrence
15
20
25
30
35
40
45
u = population mean
σ = population std. dev

24. Gaussian Error Curve
68.3% of all data
within ±1s
95.4% of all data
within ± 2 s
99.7% of all data
within ± 3s

25. Sample Mean
Sample std dev (s)
sample variance (s2
)
relative std deviation (rsd)
N: # measurements
Use calculator
s / Mean * 100 (or 1000)
Define:
Reality: Finite data set
Sum pts divide by Nx
When N → ∞ , → u and s → σx

26. Important points:
1. In most situations, you will have a finite sample of data,
therefore use
standard deviation (s), not σ
average
students t
2. Use your calculator to calculate mean and std dev.
Quicker and less error
Useful advice: Know how to use your calculator
x

27. Statistical Treatment of Random Errors
B. Finite sample size
Use statistics to set limits around a mean since
can not determine the population mean
Confidence intervals (4-2)
1. σ is not known (have a finite data set)
t = Student’s t (Table 4-2)
Memorize
Degrees of Freedom: N-1
n
ts
x ±=µ

28. What does a confidence interval mean?

30. Statistical Treatment of Random Errors
• Rejection of Data
• Use caution
Q-test
• Simple and widely used
Qexp = d / w
= gap /range
= l Xquest – Xnearest l / range
Qexp > Qtable reject
Memorize

32. If N = 4, mean = 5.0, and s = 1.0, what is confidence
Interval (95%)
Can you statistically reject 56.23 ?
55.95
56.00
56.04
56.08
56.23
Questions:

33. Method of Least Squares
Calibration Curves
y = mx + b Equation for a straight line
memorize
Useful advice: Examine your data for sensibility

34. Selectivity
Sensitivity
Specificity
Linearity
Blank
Detection limit
Standard addition
Matrix effect
Internal standard
Important Terms