Uncertainty Analysis in Experimental Measurements

Uncertainty Analysis in Experimental Measurements

Introduction to Uncertainty Analysis

Uncertainty analysis is a critical aspect of experimental science, including chemistry, as it allows us to quantify the reliability of measured values. All measurements are subject to uncertainties, which can arise from various sources such as human error, limitations of measuring instruments, and inherent variability in the measurement process.

Types of Uncertainties

Personal Uncertainties

• Definition: Uncertainties that arise from the observer's actions or perceptions during measurement.

• Examples:

  • Reaction time delay when starting or stopping a stopwatch.

  • Parallax error when reading a scale from an incorrect angle.

  • Judgment in determining when an image is focused or the exact position of a ruler.

• Minimization: Experience and careful technique can reduce personal uncertainties.

Random Uncertainties

• Definition: Uncertainties that cause measured values to scatter randomly around the true value due to unpredictable variations.

• Identification: Repeating measurements reveals random uncertainties, as values will be distributed above and below the true value.

• Note: Systematic uncertainties are not detected by repeated measurements, as they shift all values in the same direction.

Statistical Treatment of Uncertainties

Arithmetic Mean

The arithmetic mean (average) provides the best estimate of a measured quantity when multiple measurements are taken.

• Formula:

$f_\text{average} = \frac{f_1 + f_2 + \ldots + f_n}{n} = \frac{1}{n} \sum_{i=1}^{n} f_i$

• All measured values are included unless an anomaly (outlier) is identified and justifiably discarded.

Random Uncertainty (Range Error)

The range error estimates the uncertainty in the mean value due to random variation.

• Formula:

$\text{uncertainty} = \frac{\text{range}}{\text{number of repeat measurements}} = \frac{f_\text{max} - f_\text{min}}{n}$

• Uncertainty is quoted to one significant figure.

• The final result is expressed as:

$f_\text{average} \pm \Delta f \; \text{(units)}$

This means the true value is likely between $f_\text{average} - \Delta f$ and $f_\text{average} + \Delta f$.

Absolute, Fractional, and Percentage Uncertainty

• Absolute Uncertainty ($\Delta x$): The uncertainty quoted with units (e.g., $2.00 \pm 0.04$ mm).

• Fractional Uncertainty: The ratio of the absolute uncertainty to the measured value (unitless).

• Percentage Uncertainty: The fractional uncertainty expressed as a percentage.

Formulas:

$\text{Fractional uncertainty} = \frac{\Delta x}{x}$

$\text{Percentage uncertainty} = \left( \frac{\Delta x}{x} \right) \times 100\%$

• Example: An absolute uncertainty of 0.04 mm in 2.00 mm is a fractional uncertainty of 0.02 and a percentage uncertainty of 2%.

• Absolute uncertainties have units; fractional and percentage uncertainties do not.

Combining Uncertainties in Calculations

When calculated quantities depend on measured values, their uncertainties must be combined according to specific rules.

Rule 1: Multiplication by a Known Constant

• If $c = k \times a$ (where $k$ is exact), then $\Delta c = k \times \Delta a$.

• Example: If radius = $(1.45 \pm 0.03)$ m, then diameter = $2 \times 1.45 = 2.90$ m and $\Delta d = 2 \times 0.03 = 0.06$ m, so diameter = $(2.90 \pm 0.06)$ m.

Rule 2: Addition or Subtraction

• If $c = a + b$ or $c = a - b$, then $\Delta c = \Delta a + \Delta b$.

• Example: $a = (2.31 \pm 0.03)$ m, $b = (1.24 \pm 0.02)$ m, so $a - b = 1.07 \pm 0.05$ m.

Rule 3: Multiplication or Division

• If $c = a \times b$ or $c = \frac{a}{b}$, then the fractional uncertainties add:

$\frac{\Delta c}{c} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$

• Alternatively, percentage uncertainties add.

• Example: $R = \frac{V}{I}$, $V = 10 \pm 0.1$ V, $I = 1.3 \pm 0.05$ A

$R = \frac{10}{1.3} = 7.7 \, \Omega$

$\frac{\Delta R}{R} = \frac{0.1}{10} + \frac{0.05}{1.3} = 0.01 + 0.038 = 0.048$

$\Delta R = 0.048 \times 7.7 = 0.37 \approx 0.05 \, \Omega$

So $R = (7.7 \pm 0.05) \, \Omega$

Rule 4: Raising a Quantity to a Power

• If $c = a^n$, then $\frac{\Delta c}{c} = |n| \times \frac{\Delta a}{a}$

• Or, percentage uncertainty in $c$ is $|n|$ times the percentage uncertainty in $a$.

• Example: $E = \frac{9 \times 10^9 q}{d^2}$, $q = 0.0035 \pm 0.00005$ C, $d = 92 \pm 0.5$ m

$E = \frac{9 \times 10^9 \times 0.0035}{92^2} = 3700 \, \text{N/C}$

$\frac{\Delta E}{E} = \frac{0.00005}{0.0035} + 2 \times \frac{0.5}{92} = 0.014 + 0.011 = 0.025$

$\Delta E = 0.025 \times 3700 = 92.5 \approx 70 \, \text{N/C}$

So $E = (3700 \pm 70) \, \text{N/C}$

Summary Table: Rules for Combining Uncertainties

Operation	How to Combine Uncertainties	Formula	Example
Multiplication by constant	Multiply uncertainty by constant	$\Delta c = k \times \Delta a$	Diameter = $2 \times$ radius
Addition/Subtraction	Add absolute uncertainties	$\Delta c = \Delta a + \Delta b$	$(2.31 \pm 0.03) - (1.24 \pm 0.02)$
Multiplication/Division	Add fractional (or percentage) uncertainties	$\frac{\Delta c}{c} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$	$R = \frac{V}{I}$
Power	Multiply fractional uncertainty by power	$\frac{\Delta c}{c} = |n| \times \frac{\Delta a}{a}$	$E = \frac{q}{d^2}$

Best Practices

• Always quote uncertainties to one significant figure.

• Express final results as: measured value ± uncertainty (with units).

• When combining uncertainties, use the appropriate rule based on the mathematical operation.

• Tabulate values, uncertainties, and percentage uncertainties for clarity when dealing with multiple quantities.

Additional info: These uncertainty analysis skills are foundational for all laboratory sciences, including general chemistry, and are essential for reporting reliable and meaningful experimental results.