Data Analysis. with Excel. An introduction for Physical scientists. LesKirkup university of Technology, Sydney CAMBRIDGE UNIVERSITY PRESS

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1 Data Analysis with Excel An introduction for Physical scientists LesKirkup university of Technology, Sydney CAMBRIDGE UNIVERSITY PRESS

2 Contents Preface xv 1 Introduction to scientific data analysis Introduction Scientific experimentation Aim of an experiment Experimental design Units and standards Units Standards Prefixes and scientific notation Significant figures Picturing experimental data Histograms Relationships and the x-y graph Logarithmic scales Key numbers summarise experimental data The mean and the median Variance and standard deviation Population and sample Population parameters True value and population mean Sample statistics Which standard deviation do we use? Approximating Experimental error Random error 33

3 1.7.2 Systematic error Repeatability and reproducibility Modern tools of data analysis - the computer based spreadsheet Review 35 Problems 36 2 Excel and data analysis Introduction What is a spreadsheet? Introduction to Excel Starting with Excel Worksheets and Workbooks Entering and saving data Rounding, range and the display of numbers Entering formulae Cell references and naming cells Operator precedence and spreadsheet readability Verification and troubleshooting Auditing tools Built in mathematical functions Trigonometrical functions Built in statistical functions SUMO, MAX() and MINO AVERAGE(),MEDIAN()andMODE Other useful functions Presentation options Charts in Excel Thex-ygraph Plotting multiple sets of data on an x-y graph Data analysis tools Histograms Descriptive statistics Review 80 Problems 81 3 Data distributions I Introduction Probability Rules of probability Probability distributions Limits in probability calculations 93

4 3.4 Distributions of real data The normal distribution Excel 's NORMDISTO function The standard normal distribution ExceP's NORMSDIST0 function xand s as approximations to ^ and o Confidence intervals and confidence limits The 68% and 95% confidence intervals Excel 's NORMINV0 function Excel 's NORMSINV0 function Distribution of sample means The central limit theorem Standard error of the sample mean Approximating cr Excel 's CONFIDENCE0 function The t distribution ExceFsTDISTO and TINVO functions The lognormal distribution Assessing the normality of data The normal quantile plot Population mean and continuous distributions Population mean and expectation value Review 133 Problems Data distributions II Introduction The binomial distribution Calculation of probabilities using the binomial distribution Probability of a success, p Excel 's BINOMDIST0 function Mean and standard deviation of binomially distributed data Normal distribution as an approximation to the binomial distribution The Poisson distribution Applications of the Poisson distribution Standard deviation of the Poisson distribution Excel 's POISSON0 function Normal distribution as an approximation to the Poisson distribution 156

5 4.4 Review 157 Problems Measurement, error and uncertainty Introduction The process of measurement True value, error and uncertainty Calculation of uncertainty, u Precision and accuracy Random and systematic errors Random errors Common sources of error Absolute, fractional and percentage uncertainties Combining uncertainties caused by random errors Equations containing a single variable Equations containing more than one variable Most probable uncertainty Review of combining uncertainties Coping with extremes in data variability Outliers Chauvenet's criterion Dealing with values that show no variability Uncertainty due to systematic errors Calibration errors and specifications Offset and gain errors Loading errors Dynamic effects Zero order system First order system Combining uncertainties caused by systematic errors Combining uncertainties due to random and systematic errors Type A and Type B categorisation of uncertainties Weighted mean Standard error in the weighted mean Should means be combined? Review 208 " Problems Least squares I Introduction The equation of a straight line The 'best' straight line through x-y data 215

6 6.2.2 Unweighted least squares Trendline in Excel Uncertainty in a and b Least squares, intermediate calculations and significant figures Confidence intervals for a and / ExcePs LINESTO function Using the line of best fit Comparing a 'physical' equation to y = a + bx Uncertaintiesinparameterswhicharefunctionsofaandfo Estimating y for a given x Uncertainty in prediction of y at a particular value of x Estimating x for a given y Fitting a straight line to data when random errors are confined to the x quantity Linear correlation coefficient, r Calculating r using Excel Is the value of r significant? Residuals Standardised residuals Data rejection Transforming data for least squares analysis Consequences of data transformation Weighted least squares Weighted uncertainty in a and b Weighted standard deviation, a w Weighted least squares and Excel Review 272 Problems Least squares II Introduction Extending linear least squares Formulating equations to solve for parameter estimates Matrices and Excel The MINVERSEO function The MMULTO function Fitting the polynomial y= a +bx+ ex 2 to data Multiple least squares Standard errors in parameter estimates Confidence intervals for parameters Weighting the fit 297

7 7.8 Coefficients of multiple correlation and multiple determination The LINESTO function for multiple least squares Choosing equations to fit to data Comparing equations fitted to data Non-linear least squares Review 309 Problems Tests of significance Introduction Confidence levels and significance testing Hypothesis testing Distribution of the test statistic, z Using Excel to compare sample mean and hypothesised population mean One tailed and two tailed tests of significance Type I and type II errors Comparing xwith n 0 when sample sizes are small Significance testing for least squares parameters Comparison of the means of two samples ExceP's TTEST ?test for paired samples ExceP's TTEST0 for paired samples Comparing variances using the Ftest The F distribution TheFtest ExceP's FINV0 function Robustness of the F test Comparing expected and observed frequencies using the x 2 test The x 2 distribution TheFtest Is the fit too good? Degrees of freedom in x 2 test ExceP's CHIINV0 function Analysis of variance Principle of ANOVA Example of ANOVA calculation Review 359 Problems 360

8 xiii 9 Data Analysis tools in Excel and the Analysis ToolPak Introduction Activating the Data Analysis tools General features Anova: Single Factor Correlation Ftest Two-Sample for Variances Random Number Generation Regression Advanced linear least squares using ExceP's Regression tool t tests Other tools Anova: Two-Factor With Replication and Anova: Two-Factor Without Replication Covariance Exponential Smoothing Fourier analysis Moving average Rank and percentile Sampling Review 380 Appendix 1 Statistical tables 381 Appendix 2 Propagation of uncertainties 390 Appendix 3 Least squares and the principle of maximum likelihood 392 A3.1 Mean arid weighted mean 392 A3.2 Best estimates of slope and intercept 394 A3.3 The line of best fit passes through x,y 396 A3.4 Weighting the fit 397 Appendix 4 Standard errors in mean, intercept and slope 398 A4.1 Standard error in the mean and weighted mean 398 A4.2 Standard error in intercept a and slope b for a straight line 399 Appendix 5 Introduction to matrices for least squares analysis 403 Appendix 6 Useful formulae 409 Answers to exercises and problems 413 References 438 Index 441

Index. Cambridge University Press Data Analysis for Physical Scientists: Featuring Excel Les Kirkup Index More information

Index. Cambridge University Press Data Analysis for Physical Scientists: Featuring Excel Les Kirkup Index More information χ 2 distribution, 410 χ 2 test, 410, 412 degrees of freedom, 414 accuracy, 176 adjusted coefficient of multiple determination, 323 AIC, 324 Akaike s Information Criterion, 324 correction for small data