Reporting Results: Significant Figures, Fitting Practice and the Lab Notebook

The Logic of Reporting

When we report a physical quantity, we are communicating two things: the best estimate of the true value and the range within which we believe the true value lies. The number of significant figures we report is not arbitrary; it is a direct consequence of the precision of our measurement. If your uncertainty is in the second decimal place, reporting the third is misleading because it implies a precision you do not possess. Conversely, rounding too aggressively discards valid information. The goal is to match the precision of the central value to the magnitude of the uncertainty. This is not about "rules" for the sake of rules, but about intellectual honesty. A result like \( 9.81 \pm 0.05 \ \mathrm{m/s^2} \) is clear, but \( 9.8123 \pm 0.05 \ \mathrm{m/s^2} \) is not, because the last two digits of the central value are noise. The lab notebook serves as the primary evidence for this process, documenting every step from raw data to final fit, allowing any peer to reconstruct your analysis and verify your claims.

Significant Figures and Uncertainty

The standard convention for reporting a measured quantity \( x \) with an uncertainty \( \sigma_x \) is to round the uncertainty to one or two significant figures, and then round the central value to the same decimal place. This ensures that the uncertainty dominates the precision of the reported number. For example, if a calculated uncertainty is \( 0.0473 \ \mathrm{kg} \), it is standard practice to round this to \( 0.05 \ \mathrm{kg} \) (one significant figure) or \( 0.047 \ \mathrm{kg} \) (two significant figures, if higher precision is warranted by the context). The central value must then be rounded to the hundredths place. If the raw central value is \( 1.2345 \ \mathrm{kg} \), the reported result should be \( 1.23 \pm 0.05 \ \mathrm{kg} \). Reporting it as \( 1.2345 \pm 0.05 \ \mathrm{kg} \) is incorrect because the digits '4' and '5' are not supported by the uncertainty. This practice prevents the false impression of higher precision and aligns the reported value with the statistical confidence interval.

When combining uncertainties from multiple measurements, the rules of propagation of error dictate the final precision. If \( z = x + y \), then \( \sigma_z = \sqrt{\sigma_x^2 + \sigma_y^2} \). The final reported uncertainty should follow the same rounding conventions. Avoid rounding intermediate results: rounding errors build up and cost accuracy in the final result. Keep extra digits in intermediate steps and only apply significant figure rules to the final reported value.

Least-Squares Fitting in Practice

In most experimental scenarios, we do not measure a single value but a set of data points \( (x_i, y_i) \) that we wish to fit to a model function \( f(x; \theta) \), where \( \theta \) represents the parameters of interest. The method of least squares minimizes the sum of the squared residuals, \( \chi^2 = \sum_{i=1}^{N} \frac{(y_i - f(x_i; \theta))^2}{\sigma_i^2} \). The optimal parameters \( \hat{\theta} \) are those that minimize this quantity. The goodness of fit is assessed by the reduced chi-squared, \( \chi^2_{\nu} = \frac{\chi^2}{N - k} \), where \( k \) is the number of fitted parameters. A value of \( \chi^2_{\nu} \) close to 1 indicates a good fit, assuming the uncertainties \( \sigma_i \) are correctly estimated. If \( \chi^2_{\nu} \) is significantly greater than 1, the model may be inadequate or the uncertainties underestimated. If it is significantly less than 1, the uncertainties may be overestimated.

The covariance matrix of the parameters provides the uncertainties on the fitted values. The diagonal elements give the variances \( \sigma_{\theta_j}^2 \), and the off-diagonal elements give the covariances, which indicate the correlation between parameters. High correlation can make the individual parameter uncertainties less meaningful, as a change in one parameter can be compensated by a change in another. Always report the correlation matrix alongside the parameter uncertainties to give a complete picture of the fit quality.

The Laboratory Notebook

The lab notebook is the legal and scientific record of your work. It must be complete, unambiguous, and chronological. Every entry should include the date, time, and a clear description of the experiment. Raw data must be recorded exactly as obtained, with units and conditions noted. Any deviations from the planned procedure, such as instrument malfunctions or environmental changes, must be documented immediately. The notebook should also contain the full analysis, including the fitting procedure, the values of the parameters, their uncertainties, and the goodness-of-fit statistics. It is not a place for speculation or informal notes; it is a formal record that stands in for the experimenter if the work is questioned later. A well-kept notebook allows a third party to reproduce the experiment and the analysis without any additional information from the original researcher.

When writing up results for a report or publication, the notebook serves as the source of truth. If there is a discrepancy between the notebook and the report, the notebook prevails. This is why it is critical to be meticulous in recording data and analysis steps. The notebook should also include a section for "Conclusions and Next Steps," where you summarize the key findings and suggest further experiments or improvements. This forward-looking section demonstrates scientific thinking and helps guide future work.