Chapter 3: Experimental Error and Uncertainty
Chapter 3: Experimental Error and Uncertainty
Introduction to Experimental Error and Uncertainty
Error: Defined as the difference between a measured value and the “true” value. It is inherent to every measurement owing to the inability to measure the true value of a chemical sample directly.
Confidence in Measurements: An agreement of measurements made by different methods provides confidence that the results are accurate.
Uncertainty: Refers to the dispersion of measured values due to both random and systematic effects. The width of the uncertainty interval gives a general idea of how large the error might be.
Fitness for Purpose: Describes the degree to which data produced by a measurement enables correct decisions for a stated purpose.
Significant Figures
Definition of Significant Figures: The minimum number of digits necessary to express a given value in scientific notation without loss of precision. For example, the number 142.7 can be expressed as 1.427 × 10², which has four significant figures.
Examples:
The expression 1.4270 × 10² implies knowledge of the digit after 7, making it five significant figures, while in scientific notation 6.302 × 10⁻⁶ still has four significant figures irrespective of the zeroes preceding it.
Ambiguity in Significant Figures: The notation 92500 is ambiguous regarding the number of significant figures it holds. By writing it as:
9.25 × 10⁴ (three significant figures),
9.250 × 10⁴ (four significant figures), and
9.2500 × 10⁴ (five significant figures), the specific number of significant figures can be clearly indicated.
Rules of Significant Figures in Calculations
Rounding: Rounding should be done only on the final answer after completing all calculations.
Use of Spreadsheets: It is recommended to use spreadsheets for calculations as they retain all digits automatically.
Addition and Subtraction
The answer should be in the same decimal place as the number with the least significant digits. If numbers added or subtracted have different decimal places, the result is limited by the least certain value.
Example:
1.362 × 10⁻⁴ + 3.111 × 10⁻⁴ = 4.473 × 10⁻⁴
5.345 + 6.728 = 12.073
Uniform Exponent: All numbers should be expressed with the same exponent to maintain clarity.
Multiplication and Division
The final answer is limited to the number of significant figures present in the measurement with the fewest significant figures; the exponent does not affect the digit count retained.
Logarithms
The logarithm of a number n is represented as $n = 10^a$ and thus log n = a.
Composed of a characteristic (integer part) and a mantissa (decimal part): the number of digits in the mantissa corresponds to the number of significant figures in x. Similarly, the antilog involves translating the number of significant figures from the mantissa of x.
Types of Error
Every measurement incurs some uncertainty, categorized into three types:
Systematic Error (Determinate Error): Arising from a flaw in equipment or experiment design; results in a consistent positive or negative bias when repeated.
Random Error (Indeterminate Error): Arises from uncontrolled variables and has no bias; it can only be reduced by replication and improved techniques.
Blunders (Gross Errors): Significant deviations resulting from mistakes in procedure; may require data rejection or retesting.
Systematic Error
Detection:
Analyze a known sample (certified reference material).
Use different methods to measure the same quantity.
Compare results between laboratories with identical samples.
Analyze blank samples and check for nonzero responses indicating unintended method responses.
Minimization Methods:
Calibration of instruments.
Optimization of sample preparation to minimize analyte loss.
Use of standard addition or internal standards to correct for matrix effects.
Random Error
Characterized by variability in measurements, which can be reduced but not eliminated. Common examples include:
Subjective readings of scales.
Electrical noise from instruments.
Blunders
Resulting from procedural, instrumental, or clerical mistakes. Examples include calculation errors, overshooting titration endpoints, and sample contamination.
Precision, Trueness, and Accuracy
Precision: Describes how reproducible results are, indicated by agreement among repeated measurements.
Trueness: Refers to the closeness of the average value obtained from multiple measurements to the true value if precision were perfect.
Accuracy: Indicates how close a measured value is to the known true value from a limited number of replicates.
Absolute and Relative Uncertainty
Absolute Uncertainty: Expressed as a margin of uncertainty associated with a measurement, typically conveyed as a ± value. Example: Burette reading of 12.35 ± 0.02 mL indicates the true value is between 12.33 mL and 12.37 mL.
Relative Uncertainty: Expresses the size of the absolute uncertainty relative to its measurement size.
Example: The calculation for relative uncertainty in the preceding situation would be:
ext{Relative uncertainty} = rac{0.02 ext{ mL}}{12.35 ext{ mL}} imes 100 ext{%} = 0.162 ext{%}\
Propagation of Uncertainty from Random Error
Arithmetic Operations Impact: The final result’s uncertainty must be considered from individual errors, where random errors of individual measurements may cancel one another. There are specific equations for combining uncertainties based on the operation type.
Addition and Subtraction
Total uncertainty corresponds to the absolute uncertainty.
Equation:
u{ ext{total}} = u1 + u2 + u3 + … + u_n
Multiplication and Division
The percentage uncertainties must be considered.
Equation:
ext{Percent } u{ ext{total}} = ext{Percent } u1 + ext{Percent } u2 + … + ext{Percent } un
Examples of Propagation of Uncertainty
For addition/subtraction: If you calculate measurements such as 1.76 m (±0.03 m) + 1.89 m (±0.02 m) – 0.59 m (±0.02 m), you would sum the absolute uncertainties to find:
3.06 ext{ m} ext{ (±0.041 m)}
For multiplication/division, work out using relative uncertainties on measurements like 0.0247 ext{ M} ± 0.0002 ext{ M}; convert relative to absolute to find final values.
Matrix Effects
Analytical responses can be affected by components other than the analyte within a sample, termed as matrix effects. Understanding sample composition helps in eliminating any systematic errors that could integrate through matrix interactions.
Recommended Exercises and Problems
Exercises: Recommended exercises include 3-A, 3-B, 3-D.
Problems: Include significant figures (3-1, 3-2, 3-3, 3-4, 3-6), types of error (3-7, 3-8, 3-10, 3-12, 3-14), and propagation of uncertainty (3-15, 3-16, 3-18, 3-20, 3-21) for practice and reinforcement of concepts discussed in the chapter.