{"product_id":"understanding-statistical-error-isbn-9781119106913","title":"Understanding Statistical Error","description":"\u003cp\u003eThis accessible introductory textbook provides a straightforward, practical explanation of how statistical analysis and error measurements should be applied in biological research.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eUnderstanding Statistical Error - A Primer for Biologists:\u003c\/i\u003e\u003c\/p\u003e \u003cul\u003e \u003cli\u003eIntroduces the essential topic of error analysis to biologists\u003c\/li\u003e \u003cli\u003eContains mathematics at a level that all biologists can grasp\u003c\/li\u003e \u003cli\u003ePresents the formulas required to calculate each confidence interval for use in practice\u003c\/li\u003e \u003cli\u003eIs based on a successful series of lectures from the author’s established course\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003eAssuming no prior knowledge of statistics, this book covers the central topics needed for efficient data analysis, ranging from probability distributions, statistical estimators, confidence intervals, error propagation and uncertainties in linear regression, to advice on how to use error bars in graphs properly. Using simple mathematics, all these topics are carefully explained and illustrated with figures and worked examples. The emphasis throughout is on visual representation and on helping the reader to approach the analysis of experimental data with confidence.\u003c\/p\u003e \u003cp\u003eThis useful guide explains how to evaluate uncertainties of key parameters, such as the mean, median, proportion and correlation coefficient. Crucially, the reader will also learn why confidence intervals are important and how they compare against other measures of uncertainty.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eUnderstanding Statistical Error - A Primer for Biologists \u003c\/i\u003ecan be used both by students and researchers to deepen their knowledge and find practical formulae to carry out error analysis calculations. It is a valuable guide for students, experimental biologists and professional researchers in biology, biostatistics, computational biology, cell and molecular biology, ecology, biological chemistry, drug discovery, biophysics, as well as wider subjects within life sciences and any field where error analysis is required.\u003c\/p\u003e \u003cp\u003eIntroduction 1\u003c\/p\u003e \u003cp\u003eWhy would you read an introduction? 1\u003c\/p\u003e \u003cp\u003eWhat is this book about? 1\u003c\/p\u003e \u003cp\u003eWho is this book for? 2\u003c\/p\u003e \u003cp\u003eAbout maths 2\u003c\/p\u003e \u003cp\u003eAcknowledgements 3\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 1 Why do we need to evaluate errors? 4\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 2 Probability distributions 7\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Random variables 8\u003c\/p\u003e \u003cp\u003e2.2 What is a probability distribution? 9\u003c\/p\u003e \u003cp\u003eProbability distribution of a discrete variable 9\u003c\/p\u003e \u003cp\u003eProbability distribution of a continuous variable 10\u003c\/p\u003e \u003cp\u003eCumulative probability distribution 11\u003c\/p\u003e \u003cp\u003e2.3 Mean, median, variance and standard deviation 11\u003c\/p\u003e \u003cp\u003e2.4 Gaussian distribution 13\u003c\/p\u003e \u003cp\u003eExample: estimate an outlier 15\u003c\/p\u003e \u003cp\u003e2.5 Central limit theorem 16\u003c\/p\u003e \u003cp\u003e2.6 Log-normal distribution 18\u003c\/p\u003e \u003cp\u003e2.7 Binomial distribution 20\u003c\/p\u003e \u003cp\u003e2.8 Poisson distribution 23\u003c\/p\u003e \u003cp\u003eClassic example: horse kicks 25\u003c\/p\u003e \u003cp\u003eInter-arrival times 26\u003c\/p\u003e \u003cp\u003e2.9 Student’s t-distribution 28\u003c\/p\u003e \u003cp\u003e2.10 Exercises 30\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 3 Measurement errors 32\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Where do errors come from? 32\u003c\/p\u003e \u003cp\u003eSystematic errors 33\u003c\/p\u003e \u003cp\u003eRandom errors 34\u003c\/p\u003e \u003cp\u003e3.2 Simple model of random measurement errors 35\u003c\/p\u003e \u003cp\u003e3.3 Intrinsic variability 38\u003c\/p\u003e \u003cp\u003e3.4 Sampling error 39\u003c\/p\u003e \u003cp\u003eSampling in time 39\u003c\/p\u003e \u003cp\u003e3.5 Simple measurement errors 41\u003c\/p\u003e \u003cp\u003eReading error 41\u003c\/p\u003e \u003cp\u003eCounting error 43\u003c\/p\u003e \u003cp\u003e3.6 Exercises 46\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 4 Statistical estimators 47\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Population and sample 47\u003c\/p\u003e \u003cp\u003e4.2 What is a statistical estimator? 49\u003c\/p\u003e \u003cp\u003e4.3 Estimator bias 52\u003c\/p\u003e \u003cp\u003e4.4 Commonly used statistical estimators 53\u003c\/p\u003e \u003cp\u003eMean 53\u003c\/p\u003e \u003cp\u003eWeighted mean 54\u003c\/p\u003e \u003cp\u003eGeometric mean 55\u003c\/p\u003e \u003cp\u003eMedian 56\u003c\/p\u003e \u003cp\u003eStandard deviation 57\u003c\/p\u003e \u003cp\u003eUnbiased estimator of standard deviation 59\u003c\/p\u003e \u003cp\u003eMean deviation 62\u003c\/p\u003e \u003cp\u003ePearson’s correlation coefficient 63\u003c\/p\u003e \u003cp\u003eProportion 65\u003c\/p\u003e \u003cp\u003e4.5 Standard error 66\u003c\/p\u003e \u003cp\u003e4.6 Standard error of the weighted mean 70\u003c\/p\u003e \u003cp\u003e4.7 Error in the error 71\u003c\/p\u003e \u003cp\u003e4.8 Degrees of freedom 72\u003c\/p\u003e \u003cp\u003e4.9 Exercises 73\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 5 Confidence intervals 74\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Sampling distribution 75\u003c\/p\u003e \u003cp\u003e5.2 Confidence interval: what does it really mean? 77\u003c\/p\u003e \u003cp\u003e5.3 Why 95%? 79\u003c\/p\u003e \u003cp\u003e5.4 Confidence interval of the mean 80\u003c\/p\u003e \u003cp\u003eExample 83\u003c\/p\u003e \u003cp\u003e5.5 Standard error versus confidence interval 84\u003c\/p\u003e \u003cp\u003eHow many standard errors are in a confidence interval? 84\u003c\/p\u003e \u003cp\u003eWhat is the confidence of the standard error? 85\u003c\/p\u003e \u003cp\u003e5.6 Confidence interval of the median 86\u003c\/p\u003e \u003cp\u003eSimple approximation 89\u003c\/p\u003e \u003cp\u003eExample 89\u003c\/p\u003e \u003cp\u003e5.7 Confidence interval of the correlation coefficient 90\u003c\/p\u003e \u003cp\u003eSignificance of correlation 93\u003c\/p\u003e \u003cp\u003e5.8 Confidence interval of a proportion 95\u003c\/p\u003e \u003cp\u003e5.9 Confidence interval for count data 99\u003c\/p\u003e \u003cp\u003eSimple approximation 102\u003c\/p\u003e \u003cp\u003eErrors on count data are not integers 102\u003c\/p\u003e \u003cp\u003e5.10 Bootstrapping 103\u003c\/p\u003e \u003cp\u003e5.11 Replicates 105\u003c\/p\u003e \u003cp\u003eSample size to find the mean 108\u003c\/p\u003e \u003cp\u003e5.12 Exercises 109\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 6 Error bars 112\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Designing a good plot 112\u003c\/p\u003e \u003cp\u003eElements of a good plot 113\u003c\/p\u003e \u003cp\u003eLines in plots 115\u003c\/p\u003e \u003cp\u003eA digression on plot labels 116\u003c\/p\u003e \u003cp\u003eLogarithmic plots 117\u003c\/p\u003e \u003cp\u003e6.2 Error bars in plots 118\u003c\/p\u003e \u003cp\u003eVarious types of errors 119\u003c\/p\u003e \u003cp\u003eHow to draw error bars 120\u003c\/p\u003e \u003cp\u003eBox plots 121\u003c\/p\u003e \u003cp\u003eBar plots 123\u003c\/p\u003e \u003cp\u003ePie charts 128\u003c\/p\u003e \u003cp\u003eOverlapping error bars 128\u003c\/p\u003e \u003cp\u003e6.3 When can you get away without error bars? 130\u003c\/p\u003e \u003cp\u003eOn a categorical variable 130\u003c\/p\u003e \u003cp\u003eWhen presenting raw data 130\u003c\/p\u003e \u003cp\u003eLarge groups of data points 130\u003c\/p\u003e \u003cp\u003eWhen errors are small and negligible 131\u003c\/p\u003e \u003cp\u003eWhere errors are not known 131\u003c\/p\u003e \u003cp\u003e6.4 Quoting numbers and errors 132\u003c\/p\u003e \u003cp\u003eSignificant figures 132\u003c\/p\u003e \u003cp\u003eWriting significant figures 133\u003c\/p\u003e \u003cp\u003eErrors and significant figures 135\u003c\/p\u003e \u003cp\u003eError with no error 137\u003c\/p\u003e \u003cp\u003eComputer-generated numbers 138\u003c\/p\u003e \u003cp\u003eSummary 140\u003c\/p\u003e \u003cp\u003e6.5 Exercises 140\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 7 Propagation of errors 142\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 What is propagation of errors? 142\u003c\/p\u003e \u003cp\u003e7.2 Single variable 143\u003c\/p\u003e \u003cp\u003eScaling 144\u003c\/p\u003e \u003cp\u003eLogarithms 144\u003c\/p\u003e \u003cp\u003e7.3 Multiple variables 146\u003c\/p\u003e \u003cp\u003eSum or difference 146\u003c\/p\u003e \u003cp\u003eRatio or product 147\u003c\/p\u003e \u003cp\u003e7.4 Correlated variables 149\u003c\/p\u003e \u003cp\u003e7.5 To use error propagation or not? 150\u003c\/p\u003e \u003cp\u003e7.6 Example: distance between two dots 151\u003c\/p\u003e \u003cp\u003e7.7 Derivation of the error propagation formula for one variable 153\u003c\/p\u003e \u003cp\u003e7.8 Derivation of the error propagation formula for multiple variables 155\u003c\/p\u003e \u003cp\u003e7.9 Exercises 157\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 8 Errors in simple linear regression 158\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Linear relation between two variables 158\u003c\/p\u003e \u003cp\u003eMean response 159\u003c\/p\u003e \u003cp\u003eTrue response and noise 160\u003c\/p\u003e \u003cp\u003eData linearization 161\u003c\/p\u003e \u003cp\u003e8.2 Straight line fit 161\u003c\/p\u003e \u003cp\u003e8.3 Confidence intervals of linear fit parameters 164\u003c\/p\u003e \u003cp\u003eExample 168\u003c\/p\u003e \u003cp\u003e8.4 Linear fit prediction errors 170\u003c\/p\u003e \u003cp\u003e8.5 Regression through the origin 173\u003c\/p\u003e \u003cp\u003eExample 174\u003c\/p\u003e \u003cp\u003e8.6 General curve fitting 175\u003c\/p\u003e \u003cp\u003e8.7 Derivation of errors on fit parameters 178\u003c\/p\u003e \u003cp\u003e8.8 Exercises 179\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 9 Worked example 181\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 The experiment 181\u003c\/p\u003e \u003cp\u003e9.2 Results 182\u003c\/p\u003e \u003cp\u003eSasha 183\u003c\/p\u003e \u003cp\u003eLyosha 186\u003c\/p\u003e \u003cp\u003eMasha 189\u003c\/p\u003e \u003cp\u003e9.3 Discussion 190\u003c\/p\u003e \u003cp\u003e9.4 The final paragraph 192\u003c\/p\u003e \u003cp\u003eSolutions to exercises 193\u003c\/p\u003e \u003cp\u003eAppendix A 206\u003c\/p\u003e \u003cp\u003eBibliography 209\u003c\/p\u003e \u003cp\u003eIndex 211\u003c\/p\u003e \"This volume highlights and promotes these high standards and practices, and should serve as an important starting point for biologists, data scientists, or anyone interested in effectively assessing and presenting uncertainty in data\" \u003cb\u003eMarc J. Lajeunesse, Integrative Biology, University of South Florida, Tampa, Florida on behalf of The Quarterly Review of Biology, Sept 17\u003c\/b\u003e \u003cp\u003e\u003cb\u003eDr Marek Gierlinski\u003c\/b\u003e is a bioinformatician at College of Life Science, University of Dundee, UK. He attained his PhD in astrophysics and studied X-ray emission from black holes and neutron stars for many years. In 2009 he started a new career in bioinformatics, bringing his knowledge and skills in statistics and data analysis to a biological institute. He works on a variety of topics, including proteomics, DNA and RNA sequencing, imaging and numerical modelling.\u003c\/p\u003e","brand":"Wiley-Blackwell","offers":[{"title":"Default Title","offer_id":47990433087717,"sku":"NP9781119106913","price":44.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119106913.jpg?v=1761787808","url":"https:\/\/k12savings.com\/products\/understanding-statistical-error-isbn-9781119106913","provider":"K12savings","version":"1.0","type":"link"}