Università Roma Tre

ERROR ANALYSIS AND STATISTICS: MEASUREMENT S AND ERRORS; PROPAGATION OF ERRORS; FIT OF EXPERIMENTAL DATA; STATISTICAL DESCRIPTION OF DATA (MEDIA, STANDARD DEVIATION, FREQUENCY ...); PROBABILITY DISTRIBUTIO; STATISTICAL CORRELATION BETWEEN VARIABLES (PEARSON AND SPEARMAN COEFFICIENTS); HYPOTHESIS SIGNIFICANCE TESTING (STUDENT T-TEST, ANALYSIS OF VARIANCE, BONFERRONI TEST, CHI-SQUARE TEST); CONFIDENCE INTERVALS (BLAND-ALTMANN TEST); RELIABILITY AND VALIDITY OF A CLINICAL TEST
THE ACTUAL EXERCITATIONS WILL BE SCHEDULED ACCORDING TO THE ANSWER OF THE CLASSROOM FROM THE FOLLOWING LIST:.

1. PRACTICE WITH PRECISION INSTRUMENTS (VENIER CALIPER DECIMAL AND CENTESIMAL, SPHEROMETER) FOR PRECISION MEASUREMENTS OF GEOMETRICAL QUANTITIES AND THE STATISTICAL ANALYSIS OF THE MEASUREMENT ERROR.
2. CHECKOUT OF THE LIGHT REFLECTION LAW BY A PLANE AND A CURVED MIRROR, DEFINITION OF THE REFLECTION CAUSTIC.
3. APPLICATION OF THE SNELL’S LAW OF THE REFRACTION TROUGH A THIN OPTICAL PLEXIGLAS SLAB.
4. CHECKOUT OF THE KEPLERO’S LAW ON THE PROPAGATION OF THE LIGHT WITH DISTANCE.
5. CHECKOUT OF THE LAMBERT-BEER’S LAW ON THE ATTENUATION OF LIGHT PASSING THROUGH AN OPAQUE MEDIUM.
6. MEASUREMENT OF THE FOCAL LENGTH AND OF THE DIOPTRIC POWER OF A THIN LENS AND OF A SIMPLE SYSTEM OF THIN LENS, BOTH THEORETICALLY AND THROUGH STRAIGHTFORWARD OPTICAL MEASUREMENT.
7. MEASUREMENT OF THE FOCAL LENGTH AND OF THE DIOPTRIC POWER OF ASTIGNMATIC LENS USING THE GENEVE GAUGE (OPTICS SPHEROMETER) USED AS A FRONTIFOCOMETER TO ESTIMATE THE ASTIGMATISMUS OF SEVERAL LENSES.
8. EVALUATION OF FOCAL LENGTHS FOR CONTACT LENSES AND COUPLES OF DISTANT LENSES: CHECKOUT OF THE GULLTRASND’S LAW AND EVALUATION OF THE POSITION OF THE PRINCIPAL PLANES OF A SIMPLE OPTICAL SYSTEM.
9. DISPERSION OF LIGHT THROUGH A PRISM AND CHECKOUT OF THE CAUCHY’S LAW AND EVALUATION OF CAUCHY’S COEFFICIENTS;
10. AFOCAL SYSTEMS: CHECKOUT OF THE SMITH-HELMHOLTZ EQUATION (MEASUREMENTS OF LAGRANGIAN’S INVARIANTS.);
11. OPTICAL SYSTEMS MAGNIFICATION;
12. DIOPTRIC CONTRIBUTION OF AN OPTIC STOP TO THE IMAGES FOCUSING;
13. EVALUATION OF THE ANSWER CURVE AT THE FIRST AND AT THE SECOND ORDER OF A LUXMETER HAVING SPECTRAL FEATURES COMPARABLE TO THAT OF THE HUMAN EYE.

PROFESSORS F. OFFI AND F. FONTANA LECTURE NOTES
(TAYLOR, J.R.) INTRODUZIONE ALL’ANALISI DEGLI ERRORI [ZANICHELLI, BOLOGNA, 2010]
(GETTYS, W.E. KELLER, F.J. AND SKOVE, M.J.) FISICA 2, ELETTROMAGNETISMO – ONDE – OTTICA [MAC GRAW-HILL, MILANO, 2007 ]
(OBSTFELD H.) OPTICS IN VISION, [BUTTERWORTHS, BOSTON, 1982]
(KEATING M. P.) GEOMETRICAL, PHYSICAL AND VISUAL OPTICS [ BUTTERWORTHS, BOSTON, 2002]
(TUNNACLIFFE A. H.) INTRODUCTION TO VISUAL OPTICS [ABDO, LONDON 2001 ]
(MEYER-ARENDT J. R.) INTRODUCTION TO CLASSICAL AND MODERN OPTICS [PRENTICE HALL, ENGLEWOOD CLIFFS, 1972]
(BLAND M.) STATISTICA MEDICA [APOGEO, 2009]
(DERENIAK L. DERENIAK T. D.) GEOMETRICAL AND TRIGONOMETRIC OPTICS. [CAMBRIDGE UNIVERSITY PRESS, CAMBRIDGE, 2008]

The theoretical lessons will focus on the following topics:

Tools and measures. Measurement units - Trigonometry angles and definitions - Sizes and errors - Systematic error and random error - Instrument error - Standard deviation and standard error - Error propagation - Representing data: significant figures, comparison of measurements, tables and graphs, histograms

Descriptive statistics. Indices of central tendency: average, median and mode - Percentiles - Indices of variability: variance and standard deviation - Limit distributions

Correlation between variables. Linear regression, least squares method - Linearization of a function - Verification of the acceptability of a fit: Chi square test and determination coefficient - Pearson r coefficient, Spearman correlation coefficient for ranks

Probability. Definition and fundamental properties - Probability distributions (binomial, Poisson and uniform distribution - Gaussian and error integral, justification of sum in quadrature and standard deviation of the mean, standard normal distribution) - Average and variance of a random variable - Confidence intervals - Standard deviation as acceptability percentage - Applications and verifications: correlation index and chi-square test - Student's t test for comparison between measurements

Inferential statistics. Hypothesis test and significance - The null hypothesis and the alternative hypothesis - Types of studies and sampling - Sample distributions - Criterion of significance and critical values - Errors and decisions of a statistical test - Sign test - Chi square test as test of significance - z-test and t-test - Distribution F - Analysis of variance (ANOVA) with one and two ways

The topics of the laboratory exercises will be chosen from the following list

1) Use of precision instruments (decimal and twentieth caliber, spherometer) for the measurement of quantities. Statistical treatment of the error.
2) Verification of the law of light reflection from a plane mirror.
3) Verification of the law of refraction in a plexiglass plate.
4) Verification of Kepler's law on the reduction of light intensity as a function of distance.
5) Verification of the Lambert-Beer law on the attenuation of luminous intensity on passing through an opaque medium.
6) Determination of the focal distance and dioptric power of a thin lens through an optical measurement.
7) Determination of the focal distance and dioptric power of a lens system with the Bessel method.
8) Determination of the focal distance and dioptric power of an ophthalmic lens by measuring with a spherometer and with a dioptrometer and characterization of an astigmatic lens
9) Verification of Gullstrand's law and search for the main planes of an optical system consisting of two thin lenses.
10) Dispersion of light through a prism: verification of Cauchy's law and determination of the number of Abbe
11) Estimate of the magnification of an optical system: the magnifying lens
12) Use of Student's t test to compare experimental data and the chi-square test to assess how well a theoretical trend reproduces the behavior of experimental data; construction of a histogram and its limit function.

Professors F. Offi and F. Fontana lecture notes

- S. Walker, Fondamenti di Fisica (Pearson, 2010)

- R. Meyer-Arendt, Introduction to classical and modern optics (Prentice Hall, 1995)

- R. Taylor, Introduzione all’analisi dell’errore (Zanichelli, 2010)

- M. Bland, Statistica medica (Maggioli Editore, 2014)

- M. M. Triola e M. F. Triola, Fondamenti di Statistica (Pearson, 2013)

- J. Welkowitz, B. Cohen e R. Ewen, Statistica per le scienze del comportamento (Apogeo)