Analysis of physical measurements

For students of "Molecular biophysics". Mathematical skills on the upper secondary school level plus some skills in calculation of partial derivatives are assumed.

This subject introduces students to the problem of experiment planning, analyzing as well as presenting and interpreting its results. It combines an introductory lecture on data analysis with experimental activities in the Introductory Laboratory.

The aim of the lecture (2 hours a week in 10 weeks) is to prepare students for individual work in Students' Laboratories. It contains a short introduction into a broad spectrum of topics important in everyday laboratory practice: planning of experiments, data acquisition, presentation and analysis of the results with particular attention paid to statistical description of experimental uncertainties.

Here an error is understood as a deviation of the actual outcome of the measurement from the exact (unknown) value of the measured quantity. The task of an experimentalist is to: (a) avoid gross errors, (b) identify and eliminate systematic errors, and where this is not possible, introduce proper corrections that account for systematic effects, and (c) construct a reasonable mathematical model for the distribution of random errors and estimate its relevant parameters after taking a series of repeated measurements. It is then widely agreed (cf e.g. Guide to the Expression of Uncertainty in Measurement, ISO, 1993) to take the best estimate of the mean-square deviation as a measure of the measurement's "uncertainty". As a result an estimate of the exact value of the measured quantity is obtained together with thus defined uncertainty that, roughly speaking, defines the precision of the measurement.

Simultaneously, in parallel to the lecture, students make their own simple measurements in an Introductory Students' Laboratory. Experimental data obtained by each student are then used to exercise the application of statistical methods introduced in the lecture. Laboratory activities and the accompanying classes in data analysis follow in turn in blocks of 3 hours a week (for first 6 weeks), then students make measurements every week (3 hours a week in 4 weeks). Classes in data analysis prepare students to write laboratory reports on the experiments performed. All reports are then evaluated.

The plan of the lecture goes as follows:

1. Introduction: measurement, types and sources of experimental errors, measurement uncertainties. Graphic presentation and analysis of data: histograms, linear-linear, linear--logarithmic, and log-log plots.

2. Quantitative characteristics of data-sets: mean-value, median, mean square-deviation.

3. Graphic presentation and analysis of data: histograms, linear-linear, linear--logarithmic, and log-log plots.

4. Elements of the theory of probability: axioms, important definitions, and some important theorems; probability distributions: binomial, Poisson, and the normal or Gauss distribution, Gauss distribution as a standard model for the distribution of random errors.

5. Systematic errors, calibration and precision of measuring instruments.

6. The "law of propagation of small errors".

7. "Least-squares" and curve fitting (case studies: weighted-averages, proportionality and a straight-line).

8. Hypotheses testing (discussed tests: 3σ and χ².).

List of experiments in the Introductory Students' Laboratory:

1. Multiple measurements of the period of a pendulum - investigation of the distribution of results.

2. Different methods of density measurements.

3. Electrical circuits - Laws of Ohm and Kirchhoff.

4. Determination of water's heat of fusion or specific heat capacity of some substances.

5. Study of light absorption by matter.

6. Study of optical spectra by spectrometer.

7. Study of radon concentration in air.

Assessment form:

1. Positive notes for laboratory reports.

2. Positive note for a test in solving typical, simple problems in data-analysis.

3. The final note is equal to the weighted average of notes for laboratory reports (weight 2/3) and of the note for the test (1/3).

Description prepared by Andrzej Majhofer and Anna Modrak-Wójcik, November 2009

1. J. R. Taylor, An Introduction to Error Analysis, Oxford University Press, Oxford, 1992.

2. G. L. Squires, Practical Physics, McGraw-Hill, London, 1976.

Further reading:

1. W. T. Eadie, D. Drijard, F. E. James, M. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics, North-Holland, Amsterdam, 1971

2. S. Brandt, Data Analysis, Springer Verlag, New York, 1999.

After completing the course student:

KNOWLEDGE

1. Knows what are the standards for estimating and presenting uncertainties of measurements accepted in natural sciences.

2. Knows the standards for describing experimental set-ups and presenting experimental results accepted in natural sciences.

SKILLS

1. Presents experimental results in the form of graphs or histograms.

2. Uses graphs and histograms to detect relations between measured quantities.

3. Estimates uncertainties of measurements.

4. Uses least-squares method to estimate parameters of linear relations between measured quantities.

5. Uses statistical tests: the 3σ test and the χ² test.

Assessment form:

1. Positive notes for laboratory reports.

2. Positive note for a test in solving typical, simple problems in data-analysis.

3. The final note is equal to the weighted average of notes for laboratory reports (weight 2/3) and of the note for the test (1/3).

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