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- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/03%3A_Distributions_Probability_and_Expected_Values/3.12%3A_The_Normal_DistributionThe normal distribution is very important. The central limit theorem says that if we average enough values from any distribution, the distribution of the averages we calculate will be the normal distr...The normal distribution is very important. The central limit theorem says that if we average enough values from any distribution, the distribution of the averages we calculate will be the normal distribution.
- https://chem.libretexts.org/Courses/Los_Angeles_Trade_Technical_College/Analytical_Chemistry/2%3A_Analytical_Chemistry_2.0_(Harvey)/05%3A_Evaluating_Analytical_Data/5.04%3A_The_Distribution_of_Measurements_and_ResultsA population is the set of all objects in the system we are investigating. For our experiment, the population is all United States pennies in circulation. This population is so large that we cannot an...A population is the set of all objects in the system we are investigating. For our experiment, the population is all United States pennies in circulation. This population is so large that we cannot analyze every member of the population. Instead, we select and analyze a limited subset, or sample of the population.
- https://chem.libretexts.org/Courses/Los_Angeles_Trade_Technical_College/Analytical_Chemistry/2%3A_Analytical_Chemistry_2.0_(Harvey)/05%3A_Evaluating_Analytical_Data/5.06%3A_Statistical_Methods_for_Normal_DistributionsThe normal distribution is the most common distribution used for experimental results. Because the area between any two limits of a normal distribution is well defined, constructing and evaluating sig...The normal distribution is the most common distribution used for experimental results. Because the area between any two limits of a normal distribution is well defined, constructing and evaluating significance tests is straightforward.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/03%3A_Distributions_Probability_and_Expected_Values/3.13%3A_The_Expected_Value_of_a_Function_of_Several_Variables_and_the_Central_Limit_Theorem\[\sigma^2_N=\left\langle {\left({\overline{u}}_N-\mu \right)}^2\right\rangle =\left\langle {\left[\left(\frac{1}{N}\sum^N_{i=1}{u_i}\right)-\mu \right]}^2\ \right\rangle =\left\langle {\left[\left(\f...\[\sigma^2_N=\left\langle {\left({\overline{u}}_N-\mu \right)}^2\right\rangle =\left\langle {\left[\left(\frac{1}{N}\sum^N_{i=1}{u_i}\right)-\mu \right]}^2\ \right\rangle =\left\langle {\left[\left(\frac{1}{N}\sum^N_{i=1}{u_i}\right)-\frac{N\mu }{N}\right]}^2\right\rangle =\ \frac{1}{N^2}\ \left\langle \ {\left[\left(\sum^N_{i=1}{u_i}\right)-N\mu \right]}^2\right\rangle =\ \frac{1}{N^2}\ \left\langle {\left[\left(\sum^N_{i=1}{\left(u_i-\mu \right)}\right)\right]}^2\ \right\rangle =\frac{1}{N^2\…
- https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Instrumental_Analysis_(LibreTexts)/35%3A_Appendicies/35.02%3A_Single-Sided_Normal_DistributionFor example, the proportion of the area under a normal distribution to the right of a deviation of 0.04 is 0.4840 (see entry in red in the table), or 48.40% of the total area (see the area shaded blue...For example, the proportion of the area under a normal distribution to the right of a deviation of 0.04 is 0.4840 (see entry in red in the table), or 48.40% of the total area (see the area shaded blue in Figure \PageIndex1). This divides the normal distribution curve into three regions: the area that corresponds to our answer (shown in blue), the area to the right of this, and the area to the left of this.
- https://chem.libretexts.org/Courses/Lakehead_University/Analytical_I/4%3A_Evaluating_Analytical_Data/4.06%3A_Statistical_Methods_for_Normal_DistributionsThe normal distribution is the most common distribution used for experimental results. Because the area between any two limits of a normal distribution is well defined, constructing and evaluating sig...The normal distribution is the most common distribution used for experimental results. Because the area between any two limits of a normal distribution is well defined, constructing and evaluating significance tests is straightforward.
- https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/04%3A_Evaluating_Analytical_Data/4.06%3A_Statistical_Methods_for_Normal_DistributionsThis 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 signif...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.
- https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/16%3A_Appendix/16.16%3A_Countercurrent_SeparationsIn 1949, Lyman Craig improved the separation of analytes with similar distribution ratios through countercurrent liquid-liquid extraction, offering a foundational understanding of chromatographic sepa...In 1949, Lyman Craig improved the separation of analytes with similar distribution ratios through countercurrent liquid-liquid extraction, offering a foundational understanding of chromatographic separations. Unlike sequential extractions, countercurrent extraction involves serial extraction of both sample and extracting phases. While outdated due to chromotography's efficiency, it remains instructive theoretically.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/32%3A_Math_Chapters/32.02%3A_Probability_and_StatisticsThis page covers random variables and probability, distinguishing between discrete and continuous distributions with examples like coin flips and spherical dice. It explains how discrete distributions...This page covers random variables and probability, distinguishing between discrete and continuous distributions with examples like coin flips and spherical dice. It explains how discrete distributions assign specific probabilities to outcomes, while continuous ones use probability density functions. The text highlights the importance of distribution moments, where the first moment indicates the average and the second reflects long-term expectations.
- https://chem.libretexts.org/Courses/Lakehead_University/Analytical_I/4%3A_Evaluating_Analytical_Data/4.04%3A_The_Distribution_of_Measurements_and_ResultsA population is the set of all objects in the system we are investigating. For our experiment, the population is all United States pennies in circulation. This population is so large that we cannot an...A population is the set of all objects in the system we are investigating. For our experiment, the population is all United States pennies in circulation. This population is so large that we cannot analyze every member of the population. Instead, we select and analyze a limited subset, or sample of the population.
- https://chem.libretexts.org/Courses/Los_Angeles_Trade_Technical_College/Analytical_Chemistry/2%3A_Analytical_Chemistry_2.0_(Harvey)/05%3A_Evaluating_Analytical_Data/5.05%3A_Statistical_Analysis_of_DataA confidence interval is a useful way to report the result of an analysis because it sets limits on the expected result. In the absence of determinate error, a confidence interval indicates the range ...A confidence interval is a useful way to report the result of an analysis because it sets limits on the expected result. In the absence of determinate error, a confidence interval indicates the range of values in which we expect to find the population’s expected mean. When we report a 95% confidence interval for the mass of a penny as 3.117 g ± 0.047 g, for example, we are claiming that there is only a 5% probability that the expected mass of penny is less than 3.070 g or more than 3.164 g.
