Fundamental University Physical Chemistry - Errors and Statistical Treatment of Data

Measurements

• Laboratory Devices and Units: Chemistry is an experimental science utilizing specific devices to measure macroscopic properties. The international system of units ($S.I$) is standard.

  • Length: Measured by metre stick or scale in metres ($m$).

  • Volume: Measured by volumetric flasks, burettes, pipettes, or graduated cylinders in cubic centimetres ($cm^3$).

  • Mass: Measured by chemical balance in kilograms ($kg$).

  • Weight: Measured by spring balance in Newtons ($N$).

  • Electric Current: Measured by ammeter in amperes ($A$).

  • Time: Measured by stop clock or watch in seconds ($s$).

Theory of Errors

• Fundamental Concept: Scientific measurements are inherently imperfect; repetitions under controlled conditions rarely yield identical results.

• Classification of Errors:

  • Determinate (Systematic) Errors: Arise from actual mistakes by the analyst or faulty instruments. They are predictable and can be avoided or traced. They are further divided into:

    • Constant Errors: Remain identical across all measurements.

    • Systematic Errors: Vary between measurements due to biases or environmental variability.

    • Instrumental and Reagent Error: Uncalibrated equipment, insufficient sensitivity, or impure chemicals.

    • Operative Error: Caused by analyst inexperience (catastrophic failures, dust introduction, incorrect drying).

    • Personal Error: Inability of the observer to judge correctly (e.g., color change at endpoint).

    • Methodical Errors: Inherent in the chemical system/analysis method (e.g., solubility of precipitate, incomplete reactions, co-precipitation or post-precipitation).

  • Indeterminate (Random) Errors: Caused by inherent variability in the measuring process. They are small, unpredictable, numerous, and subject to probability analysis.

Minimization and Treatment of Errors

• Techniques to Minimize Determinate Errors:

  • Calibration: Adjusting apparatus and applying corrections.

  • Blank Determination: Carrying out experiments without the sample to account for external influences.

  • Control Determination: Using standard references like sodium oxalate or benzoic acid.

  • Standard Addition: Adding known amounts of the constituent to verify recovery rates.

  • Internal Standard: Adding reference material to plot ratios for quantification.

• Least Squares Method: A computational method to estimate the true value ($M$) by minimizing the sum of the squares of deviations:

  • Minimum of $(m<em>1 - M)^2 + (m</em>2 - M)^2 + … (m_n - M)^2$

  • This leads to the least squares estimate being the arithmetic mean: $\bar{x} = \frac{\sum x}{N}$

Statistical Estimation of Errors

• Average Deviation ($d$): The mean of absolute individual errors: $d = \frac{\sum |x - \bar{x}|}{N}$

• Standard Deviation ($s$): The root-mean-square error.

  • For $N \le 30$: $s = \sqrt{\frac{\sum (x - \bar{x})^2}{N-1}}$

  • For N > 30: $s = \sqrt{\frac{\sum (x - \bar{x})^2}{N}}$

• Relative Standard Deviation (R.S.D): Also called coefficient of variation.

  • $R.S.D = \frac{s}{\bar{x}}$

  • $R.S.D (\%) = \frac{s \times 100}{\bar{x}}$

  • $R.S.D (ppt) = \frac{s \times 1000}{\bar{x}}$

• Variance ($\sigma^2$): Equal to the square of the standard deviation ($s^2$).

• Probable Error ($r$): Defined by the relation: $r = 0.6745s = 0.7876d$

Precision and Accuracy

• Precision: The degree of agreement between replicate measurements; relates to the distribution of random errors (spread).

• Accuracy: Closeness of a measurement to the true or accepted value ($T$).

• Absolute Error ($E$): $E = x - T$ (for a single measurement) or $E = \bar{x} - T$ (for a set).

• Relative Error ($R$): $R = \frac{E}{T}$. Can be expressed in percentage or parts per thousand ($ppt$).

• Relative Accuracy ($RA$): $RA = \frac{measured value}{true value}$. Often expressed as $\frac{\bar{x} \times 100}{T}$.

Rejection of Results

• Objective: Removing outliers to improve the accuracy of estimates.

• Q-Test (Dixon and Massey):

  • Calculation: $r = \frac{|suspected value - nearest value|}{|suspected value - furthest value|}$

  • If the calculated ratio exceeds the critical value from Table 1.2 (e.g., $0.56$ for 5 observations at 90% confidence), the result is rejected.

• Trimmed Mean: The mean calculated after discarding the highest and lowest suspicious results via visual inspection or statistical testing.

Rules for Combination of Errors

• Addition and Subtraction ($Z = A + B - C$):

  • Maximum error: $E<em>Z = E</em>A + E<em>B + E</em>C$

  • Standard error: $\Delta Z = \sqrt{(\Delta A)^2 + (\Delta B)^2 + (\Delta C)^2}$

• Multiplication and Division ($Z = \frac{A \times B}{C}$):

  • Maximum relative error: $\frac{E<em>Z}{Z} = \frac{E</em>A}{A} + \frac{E<em>B}{B} + \frac{E</em>C}{C}$

  • Standard relative error: $\frac{\Delta Z}{Z} = \sqrt{(\frac{\Delta A}{A})^2 + (\frac{\Delta B}{B})^2 + (\frac{\Delta C}{C})^2}$

• Powers ($Z = \frac{A^n \times B^m}{C^p}$):

  • Maximum relative error: $\frac{E<em>Z}{Z} = n \frac{E</em>A}{A} + m \frac{E<em>B}{B} + p \frac{E</em>C}{C}$

  • Standard relative error: $\frac{\Delta Z}{Z} = \sqrt{(n \frac{\Delta A}{A})^2 + (m \frac{\Delta B}{B})^2 + (p \frac{\Delta C}{C})^2}$

Significant Figures and Rounding

• Rules for Counting:

  • All non-zero digits are significant.

  • Zeros between non-zero digits are significant (e.g., $60.007$ has 5).

  • Leading zeros are never significant ($0.00912$ has 3).

  • Terminal zeros to the right of the decimal are significant ($9.00$ has 3).

  • Scientific notation should be used for clarity ($9.00 \times 10^2$ has 3).

• Calculations:

  • Addition/Subtraction: The answer matches the least number of decimal places in the data.

  • Multiplication/Division: The answer matches the least number of significant figures in the data.

• Rounding Off:

  • If the leftmost digit dropped is > 5, round up.

  • If < 5, drop it.

  • If the digit is exactly $5$: round to make the last retained digit even (e.g., $1.225 \rightarrow 1.22$; $1.215 \rightarrow 1.22$).