4: Evaluating Analytical Data
- Page ID
- 127232
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)When we use an analytical method we make three separate evaluations of experimental error. First, before we begin the analysis we evaluate potential sources of errors to ensure they will not adversely effect our results. Second, during the analysis we monitor our measurements to ensure that errors remain acceptable. Finally, at the end of the analysis we evaluate the quality of the measurements and results, and compare them to our original design criteria. This chapter provides an introduction to sources of error, to evaluating errors in analytical measurements, and to the statistical analysis of data.
- 4.1: Characterizing Measurements and Results
- This page focuses on determining the mass of a circulating United States penny and explores methods for analyzing data related to this measurement. It discusses the importance of defining the problem precisely and collects preliminary data from seven pennies to illustrate concepts. Measures of central tendency, including the mean and the median, are introduced as ways to summarize data, with the mean being more sensitive to extreme values than the median.
- 4.2: Characterizing Experimental Errors
- This text explores the concepts of accuracy and precision in experimental measurements, focusing on analyzing errors that impact these factors. It discusses absolute and relative errors as measures of accuracy and defines different types of determinate errors, such as sampling, method, measurement, and personal errors. It also explains identifying constant and proportional determinate errors.
- 4.3: Propagation of Uncertainty
- The page discusses the mathematical technique of propagation of uncertainty, which helps estimate the overall uncertainty in a result derived from several measurements, each with its own uncertainty. It elaborates on how to handle uncertainties in various mathematical operations, such as addition, subtraction, multiplication, and division, with relevant examples from analytical chemistry.
- 4.4: The Distribution of Measurements and Results
- The page discusses determining the mass of a U.S. penny through two different experiments, highlighting differences in their means and standard deviations. It explores how to estimate population properties from sample analyses, emphasizing the distinction between populations and samples. The page introduces key statistical concepts such as probability distributions, specifically binomial and normal distributions, confidence intervals, the central limit theorem, and degrees of freedom.
- 4.5: Statistical Analysis of Data
- This page discusses the concept of confidence intervals and significance testing in statistical analysis. Confidence intervals are used to estimate the range of values within which a population mean may lie, providing a probabilistic basis for making inferences. Significance testing, or hypothesis testing, involves comparing sample results to determine if differences are due to random error or significant factors.
- 4.6: Statistical Methods for Normal Distributions
- This page discusses the use of statistical tests to compare means and variances in analytical chemistry. Key methods include the t-test for comparing sample means, the F-test for variances, and significance tests for outliers like Dixon's Q-test, Grubb's test, and Chauvenet's criterion. The tests help determine if observed differences are significant or due to chance, aiding in validating analytical methods or identifying errors in analyses.
- 4.7: Detection Limits
- The page discusses the IUPAC's definition of a method's detection limit as the smallest signal indicating the presence of an analyte, distinct from a blank signal. It delves into statistical concepts like type 1 and type 2 errors, explaining the probability associated with detecting an analyte.
- 4.8: Using Excel and R to Analyze Data
- This chapter discusses using Excel and R for statistical calculations. Both tools offer functions for descriptive statistics, probability distributions, and significance tests. Excel provides built-in functions for calculating means, variances, and t-tests. R, a programming environment, offers similar capabilities and additional functions for detecting outliers using Dixon's Q-test and Grubb's test.
- 4.9: Problems
- The page outlines various statistical analysis exercises involving data sets from different scientific experiments. The tasks include calculating descriptive statistics such as mean, median, standard deviation, and variance; performing hypothesis tests to determine statistical significance; analyzing data distributions and detecting outliers. It also includes problems on data uncertainty, accuracy verification of equipment, and chemical analysis through different methods.
- 4.10: Additional Resources
- This document provides an extensive list of resources and experiments relevant to statistical analysis in an analytical chemistry laboratory setting. It offers references to papers discussing significant figures, accuracy, precision, data analysis, statistical concepts, error and uncertainty, detection of outliers, detection limits, limitations of statistical analysis based on significance testing, and the use of software tools like Excel and R for data analysis.
- 4.11: Chapter Summary and Key Terms
- The page discusses the characterization of data by central tendency and spread, involving measures like mean, median, range, and standard deviation. Errors affecting accuracy and precision are addressed through propagation of uncertainty. The page covers probability distributions, normal distribution confidence intervals, and statistical analysis techniques such as t-tests and F-tests for comparing data sets.