This article outlines core skills in experimental design, data collection, uncertainty propagation, graphical analysis, and evaluation for IB Physics. For assessed work, the current guide, question instructions, and markscheme remain authoritative.

1. Classification of Experimental Errors and Basic Measurement Uncertainties

Every physical measurement carries uncertainty. In IB Physics, experimental errors are commonly classified as random or systematic and discussed in terms of their effects on precision and accuracy.

• Random Errors: These cause readings to scatter around the mean and affect precision. Their effect can be reduced by repeating measurements and calculating a mean.
• Systematic Errors: These cause readings to be consistently displaced from the true value and affect accuracy. Common causes include poor calibration, zero error, and flaws in the experimental design. Repetition does not reduce systematic errors; calibration or changes to the method are required.

Determining Uncertainty in Single Measurements

• Analog Instruments: The absolute uncertainty is estimated by the experimenter based on the readability of the scale and the experimental environment, typically taken as $\pm 0.5$ or $\pm 1$ of the smallest scale division.
• Digital Instruments: The absolute uncertainty is normally taken as the smallest readable unit, or resolution, of the display.
• Distance and Interval Measurements: For a quantity obtained from two position readings, such as $L=x_2-x_1$, the absolute uncertainties of the two readings are added.
  • Standard Case: For a standard millimeter ruler where each position reading has an uncertainty of $\pm 0.5\text{ mm}$, the total absolute uncertainty of the measured interval is the sum of the two individual uncertainties, yielding $\pm 1\text{ mm}$.

Determining Uncertainty in Repeated Measurements

When a variable is measured across multiple independent trials to mitigate random variations, the absolute uncertainty in the mean value ($\bar{x}$) is estimated using half the range of the repeated readings:

$\Delta x = \frac{x_{\text{max}} - x_{\text{min}}}{2}$

Resolution Constraint: If half the range is smaller than the instrument resolution, use the instrument resolution as the absolute uncertainty.

Quantitative Validation of Theoretical Models

To assess consistency with an accepted value, compare the experimental result $E\pm\Delta E$ with the literature value $A$. If the uncertainty in $A$ is not negligible, include it in the comparison:

• Consistent within uncertainty: If $|E-A|\leq\Delta E$ and the uncertainty in $A$ is negligible, the accepted value lies within $[E-\Delta E,E+\Delta E]$. The result is consistent with the model, but this alone neither proves the model nor shows that the entire discrepancy is random.
• Outside the reported uncertainty: If $|E-A|>\Delta E$, the result is not consistent with the accepted value under the present uncertainty estimate. Possible causes include systematic bias, underestimated uncertainty, a processing problem, failure to meet the model assumptions, or a limitation of the model in that regime; further evidence is needed to distinguish them.

2. Propagation of Uncertainties

In IB Physics, algebraic rules are used to propagate uncertainties when calculating derived quantities.

Rules for Propagating Uncertainties

• Addition and Subtraction ($y = a \pm b$): Absolute uncertainties add directly. $\Delta y = \Delta a + \Delta b$
• Multiplication and Division ($y = \frac{ab}{c}$): Fractional or percentage uncertainties add directly. $\frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b} + \frac{\Delta c}{c}$
• Power Functions ($y = a^n$): The fractional uncertainty is multiplied by the absolute value of the power index. $\frac{\Delta y}{y} = |n| \frac{\Delta a}{a}$

Logarithmic Transformations

For non-linear relationships, equations can be linearized by applying natural logarithms. The uncertainties on logarithmic axes are determined via linear approximation.

• Logarithmic Uncertainty Equation: The absolute uncertainty in a logarithmic quantity is taken as the fractional uncertainty of its argument: $\Delta(\ln x) = \frac{\Delta x}{x}$

3. Data Table Standards and Conventions

Data tables must reflect both the resolution of the apparatus used and mathematical consistency during data processing.

• Logarithmic Table Headers: Because the argument of a logarithmic function must be a dimensionless ratio, variables on logarithmic axes or table headers must be divided by their respective units, written strictly as: $\ln(V / \text{V})$ or $\ln(t / \text{s})$. Notations with isolated units, such as $\ln V\ (\text{V})$, are mathematically incorrect.
• Significant Figures and Decimal Alignment: * For raw data, all data points within a single column must be recorded to the same number of decimal places, aligning precisely with the resolution of the measuring instrument.
  • For processed data, results are normally recorded to the same number of significant figures as, or one more than, the least precise relevant raw data, while remaining consistent with the stated uncertainty.

Treatment of Anomalous Data

An anomaly (outlier) is a data point that deviates significantly from the trend established by the remaining data set.

• Procedural Standard: An anomalous point may be excluded when it can be attributed to an identifiable human error or instrument malfunction, and the reason must be stated. A point must not be removed simply because it does not agree with the theoretical model.

4. Linearization of Non-Linear Equations

Linearization Principle

For a relationship that is to be linearized, rearrange the equation into the form $Y=mX+C$ and plot compound variables so that physical constants can be determined from the gradient $m$ or intercept $C$.

Linearization of Common Physical Models

Equation	Dependent Variable (Y-Axis)	Independent Variable (X-Axis)	Gradient ($m$)
$T = 2\pi\sqrt{\frac{L}{g}}$	$T^2$	$L$	$\frac{4\pi^2}{g}$
$P = \frac{k}{V}$	$P$	$\frac{1}{V}$	$k$
$v^2 = u^2 + 2as$	$v^2$	$s$	$2a$

*Note: For the kinematical model $v^2 = u^2 + 2as$, the vertical intercept represents $u^2$. For the other two proportional models, the theoretical vertical intercept is strictly zero.

Error Bars for Derived Quantities

When plotting calculated or compound variables, the lengths of the error bars must be determined via fractional propagation rules prior to plotting:

$$\begin{aligned} \frac{\Delta(T^2)}{T^2} &= 2\frac{\Delta T}{T}\\ \frac{\Delta(1/V)}{1/V} &= \frac{\Delta V}{V} \end{aligned}$$

5. Graphical Analysis: Best-Fit and Limiting Lines

Error Bars and Error Boxes

Graphs should include horizontal ($\pm\Delta x$) and vertical ($\pm\Delta y$) error bars to show the uncertainty in the plotted coordinates. Together they define an error box around each point.

Best-Fit and Limiting Lines

• Line of Best Fit (LOBF): A line or curve determined from the overall trend. The points should be reasonably balanced around it, or it should follow a stated fitting criterion; it need not cross every point or every error box.
• Limiting Lines: Draw the steepest acceptable gradient $m_{\text{max}}$ and the shallowest acceptable gradient $m_{\text{min}}$ that still pass through most or all of the error boxes. These are used to estimate uncertainty in the gradient and intercept. Do not mechanically join extreme corners of the first and last error boxes, because a local anomaly can distort the limiting range.

Calculating Parameter Uncertainties

$\Delta m = \frac{m_{\text{max}} - m_{\text{min}}}{2} \quad \text{and} \quad \Delta c = \frac{c_{\text{max}} - c_{\text{min}}}{2}$

Critical Constraint: The intercept uncertainty $\Delta c$ must be calculated algebraically using coordinates ($c = y - mx$), never estimated visually from a truncated or broken axis.

6. Data Rounding Conventions

Significant Figure and Decimal Alignment

• Uncertainty Rounding: Calculated absolute uncertainties ($\Delta m, \Delta c$) must be rounded to one significant figure using standard rounding rules. If the leading digit of the uncertainty is 1, it is acceptable to retain two significant figures.
• Precision Alignment: The final values of the parameters ($m, c$) must be rounded so that their last significant digits align strictly with the decimal place value of their corresponding absolute uncertainty.

Intercept Evaluation for Systematic Shifts

If a model predicts $y=kx$ but the experimental intercept range $c\pm\Delta c$ does not include zero, the data are inconsistent with the zero-intercept assumption under the present uncertainty estimate. Systematic bias is one possible cause, but underestimated uncertainty, an unsuitable fit range, or unmet model conditions should also be considered.

Analysis of Non-Linear Graphs

• Instantaneous Rate: Extracted by calculating the gradient of a geometric tangent line constructed to touch the curve evenly at that specific continuous point.
• Rate Uncertainty: Determined by constructing the steepest and shallowest possible acceptable tangents that touch the data point without crossing the curve.

7. Experimental Methodology: Variable Management

A valid scientific investigation should explain clearly how variables are changed, measured, and controlled:

• Independent Variable: State precisely how the independent variable is altered and define a suitable physical range. Normally use at least five distinct values to provide enough data to identify a trend.
• Quantification of Qualitative Variables: Qualitative properties must be converted into quantifiable, measurable metrics (e.g., replacing descriptive labels like “rough/smooth” with “sandpaper grit size rating”).
• Dependent Variable: Specify the apparatus, resolution, and measurement procedure. Normally take at least three independent repeats for each value of the independent variable so that a mean and random uncertainty can be determined.
• Controlled Variables: General descriptions like “keep temperature constant” are insufficient. The exact physical method of control must be specified (e.g., “submerging the container in a thermostatically controlled water bath and monitoring with a digital thermometer”).
• Limitations of Control Variables: Identify the most important uncontrolled variable, explain how it affects the dependent variable, and propose a practical method of monitoring or control.

8. Safety and Protocol Improvements

Risk Assessment Structure

Safety and environmental statements must follow a clear causal chain:

$\text{Hazard} \rightarrow \text{Associated Risk} \rightarrow \text{Specific Control Measure}$

• Example: Laser beam (Hazard) $\rightarrow$ Direct viewing may damage the eye (Associated Risk) $\rightarrow$ Do not look into the beam and keep the beam path below eye level (Specific Control Measures).

Specificity of Experimental Improvements

Suggested modifications must directly address the specific physical mechanism causing the systematic error, rather than merely proposing the use of more precise instruments:

• Thermal Experiments: Add an insulation jacket and apply a cooling curve correction to determine and add back lost heat energy.
• Mechanical Experiments: Replace manual stopwatches with photogates and digital data loggers to eliminate human reaction-time error.
• Electrical Experiments: Use a pulsed-current switch to restrict current flow to active measurement intervals, preventing resistance drift due to Joule heating.

Appendix A: Mathematical Foundations of Uncertainty Propagation (Extension)

This appendix explains the mathematical origin of the linear uncertainty rules used in IB Physics. It is not required for the written examinations. Coursework and examination answers should continue to use the IB algebraic rules stated in the main text.

If a derived quantity $y$ depends on several measured quantities, write

$y=f(x_1,x_2,\dots,x_n).$

Its total differential is

$\mathrm{d}y=\sum_{i=1}^{n}\frac{\partial f}{\partial x_i}\mathrm{d}x_i.$

For small uncertainties, a first-order approximation can be used to estimate the effect of changes in the input quantities. Taking the absolute value of each term and adding linearly gives

$\Delta y=\sum_{i=1}^{n}\left|\frac{\partial f}{\partial x_i}\right|\Delta x_i.$

For addition, $y=a+b$:

$\Delta y=\Delta a+\Delta b.$

For division, $y=a/b$, this gives the fractional uncertainty rule used in the main text:

$\frac{\Delta y}{y}=\frac{\Delta a}{a}+\frac{\Delta b}{b}.$

In more advanced experimental data analysis, root-sum-square (RSS) propagation may be used when the input uncertainties can be treated as independent random quantities:

$\Delta y_{\mathrm{RSS}}=\sqrt{\sum_{i=1}^{n}\left(\frac{\partial f}{\partial x_i}\Delta x_i\right)^2}.$

RSS and the linear addition used in the IB main text rely on different conventions and assumptions. It is included here only as an extension comparison and does not replace the standard IB method for assessed work.

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