Excel Sample

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Charles Zaiontz, 14 June 2015

Copyright 2013 - 2015 Charles ZaiontzConcise Table of ContentsExcel EnvironmentReal Statistics EnvironmentProbability FunctionsDescriptive StatisticsHypothesis TestingNormal DistributionSampling DistributionsBinomial and Related DistributionsOther Key DistributionsStudent's t DistributionChi-square DistributionF DistributionTesting for Normality and SymmetryNon-parametric TestsCorrelationOne-way Analysis of Variance (ANOVA)Factorial ANOVALinear RegressionMultiple RegressionLogistic RegressionMultinomial and Ordinal Logistic RegressionLog-Linear RegressionANOVA with Random or Nested FactorsANOVA with Repeated MeasuresAnalysis of Covariance (ANCOVA)ReliabilityMissing Data and Multiple ImputationAppendicesMathematical NotationExcel CapabilitiesMatrices and Iterative ProceduresTables

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Samples Table of Contents

Excel EnvironmentExcel spreadsheetsSample Excel worksheet (Figure 1)ChartsBar Chart (Example 1)Line Chart (Example 2)Line Chart (Example 3)Array formulas and functionsArray formula (Example 1)Array function (Example 2)

Sorting and FilteringSorting (Example 1)Remove Duplicates (Example 2)Filter (Example 3)

Goal SeekSquare root example (Example 1)

Reformatting capabilities

Reformatting dataSorting, removing duplicates, etc. (Example 1)Sort and extract unique (Example 2)Sort using data analysis tool (Example 3)

Sorting and removing duplicates by rowsSort by rows (Example 1)Sort rows removing duplicates (Example 2)

Extracting columns from a data rangeExtract columns from data range (Example 1)Select columns from data range (Example 2)

Miscellaneous functionsExamples of miscellaneous Real Statistics functions

Probability Functions

Basic probability conceptsProbability calculation (Example 6)Discrete probability distributionsFrequency function (Example 1)Frequency function (Example 2)Frequency function using Real Statistics function (Example 3)Frequency function using Real Statistics data analysis tool

Descriptive Statistics

Measures of central tendencyMean, median, etc. (Examples 1, 3-6)Column means (Example 2)Geometric Mean (extra example)

Measures of variabilityVariance, standard deviation, etc. (Examples 1,2, 5-9)Column variance and standard deviation (Examples 3)Combined sample (Example 4)

Measures of shapeSkewness and kurtosis (Examples 1, 2)Chi-square distribution (Figure 1)Kurtosis and skewness (Figure 2)RankingVarious ranking and percentile functions (Examples 1-4, 6, 8, 10)Rank and Percentile data analysis tool (Example 5)PERCENTRANK vs. PERCENTRANK.EXC (Example 7)PERCENTILE vs. PERCENTILE.EXC (Example 9)Descriptive statistics toolsDescriptive Statistics data analysis tool (Example 1)Descriptive Statistics data analysis tool (Example 2)Real Statistics Descriptive Statistics data analysis tool (Example 3)Frequency tablesMean and variance based on frequency tables (Example 1)Frequency function (Example 1)Calculations from a frequency table with intervals (Example 2)Excel FREQUENCY function (Example 3)Real Statistics FREQTABLE function (Example 4)Descriptive statistics for data in a frequency table (Example 5)Real Statistics data analysis tool for obtaining desc stats for frequency table (Example 6)

HistogramHistogram data analysis tool (Example 1)Real Statistics Frequency Table data analysis tool (Example 2)Box plotsBox Plot (Example 1)Box Plot (Example 1 using .EXC)Box Plot with negative data

ROCClassification Table (Example 1)ROC Plot (Example 2)Classification and ROC data analysis tool (Example 3)

Outliers and RobustnessTrimmed and Winsorized data (Example 1)

Dealing with missing dataRemove missing data (Example 1)Remove missing data via data analysis tool (Example 2)

Hypothesis TestingStatistical powerPower table (Figure 1)

Normal DistributionBasic characteristicsChart of normal distribution (Example 1)Extra picture (two normal distributions)Probability using normal distribution (Example 2)

Standard normal distributionChart of standard normal distribution (Figure 1)

Log-normal distributionChart of log-normal distribution (Figure 1)Equivalent formulas

Sampling DistributionsSingle sample hypothesis testing with known varianceOne sample hypothesis testing with known variance (Example 1)One sample hypothesis testing with known variance (Example 2)

Standardized effect size

Cohen's d effect size (Example 1)

Confidence intervalsConfidence interval (Example 1)Confidence interval (Example 2)

One sample hypothesis testing using the Central Limit TheoremOne sample hypothesis test via Central Limit Theorem (Example 1)

Comparing two means when variances are knownTwo sample hypothesis testing with known variance (Example 1)

SimulationSimulation of Central Limit Theorem (Example 1)Sample data for Example 1, using Uniform distributionSimulation of Central Limit Theorem (extra example)Sample data for extra example, using Poisson distributionNormal distribution sample (Figure 2)

SamplingSampling using data analysis tool (Example 1)Sampling using Real Statistics functions (Figure 3)Sampling using Real Statistics data analysis tool (Example 2)

PowerPower for normal distribution (Example 1)Power curve for Example 1 (Figure 2)What if analysis regarding power for normal distribution (Example 2)Summary of results for what if analysis regarding power (Example 2)Alternative approach for calculating power (Figure 8)Alternative approach for determining sample size (Figure 9)

Outliers and Missing DataReal statistics data analysis tool (Example 1)Real statistics STANDARD function (Example 2)

Binomial and Related Distributions

Binomial distributionChart of binomial distribution (Figure 1)Binomial distribution (Example 1)

Hypothesis testingHypothesis testing binomial distribution (Example 1)Hypothesis testing binomial distribution (Example 2)Hypothesis testing binomial distribution (Example 3)

Relationship with normal distributionBinomial vs. normal distribution (Example 1)

Proportion distributionProportion distribution (Example 1)Proportion distribution - confidence interval (Example 2)Proportion distribution - sample size (Example 3)Two sample hypothesis testing (Example 4)

Negative binomial and geometric distributionsGeometric distribution (Example 1)Negative binomial distribution (Example 2)Inverse negative binomial distribution (Example 3)

Hypergeometric distributionHypergeometric distribution (Example 1)Hypergeometric distribution (Example 2)Hypergeometric distribution, what-if (Example 3)Hypergeometric distribution (Example 4)

Beta distributionBeta distribution chart (Figure 1)Beta distribution (Example1)Multinomial distributionMultinomial distribution (Example 1)Poisson distributionPoisson distribution chart (Figure 1)Poisson distribution (Example 1)Inverse Poisson distribution (Example 2)

RunsProbability of a run of at least 6 heads (Example 1)Probability of a run of exactly 6 heads (Figure 2)Probability of a run of at least 6 heads or tails (Example 2)

Power of binomial distributionPower of one-tailed test (Example 1)Power curve for Example 1Power of two-tailed test (Example 2)Sample size requirement (Example 3)

Other Key Distributions

Gamma distributionTime to k events (Example 1)

Exponential distributionChart of exponential distribution (Figure 1)Time to first event (Example 1)Time to first event + MTTF (Example 2)Product reliability acceptance test (Example 3)

Uniform distributionUniform distribution (Example 2)

Weibull distributionWeibull distribution (Example 1)

Student's t DistributionBasic concepts of the t distributionChart of t distribution (Figure 1)Examples of equivalent functions (Figure 5)One sample hypothesis testing of the mean when population variance is unknownOne sample t test (Example 1)Test for symmetry with box plot (Figure 2)One sample t test (Example 2)Test for symmetry with histogram and descriptive stats (Figure 4)Real Statistics one sample t test data analysis tool (Figure 5)Confidence interval for t distribution (Example 3)Cohen's effect size (Example 4)Power for given mean, one-tailed test (Example 5)Power curve (Figure 8)Power, one-tailed test, given mean (Figure 14)Power, based on effect size, alpha and sample size (Example 6a)Effect size needed to achieve power goal, based on alpha and sample size (Example 6b)Sample size needed to achieve power goal, based on effect size and alpha (Example 6c)Power for two-tailed test

Two sample hypothesis testing of the means when variances are unknown but equalTwo independent sample t test, equal variance (Example 1)Two sample data analysis tool with equal variance (Figure 2)Two independent sample t test, equal variance (Example 2)Effect size (example 3)

Two sample hypothesis testing of the means when variances are unknown but unequalTwo independent sample t test, unequal variance (Example 1)Two independent samples, unequal variance (Example 2)Two independent sample t test, unequal variance (Example 2)Two sample data analysis tool with unequal variance (Figure 2)Two sample Real Statistics data analysis tool (Example 3)

Two sample hypothesis testing of the means with paired samplesTwo paired samples t test (Example 1)Two paired samples Real Statistics data analysis tool (Figure 4)Two paired samples data analysis tool (Figure 5)Two paired samples data analysis tool with missing data (Example 2)Comparing paired and independent approaches (Figure 7 and 8)One sample data analysis (Example 3)

Noncentral t distribution and powerChart of the distribution (Figure 1)Chart of the distribution (Figure 2)One sample test (Example 1)Paired samples (Example 3)Two independent samples (Example 4)Calculating sample size (Example 5)Confidence interval for effect size (Example 6)Confidence interval for power (Example 7)

Testing for OutliersGrubbs' Test (Example 1)Extension to Grubbs' Test (Example 2)

Chi-square DistributionChi-square distributionChart of chi-square distribution (Figure 1)Examples of equivalent chi-square functionsSingle sample hypothesis testing of varianceOne sample testing of variance (Example 1)One sample testing of variance (Example 2)Power (Example 3)Sample size required (Example 4)Goodness of fitChi-square goodness of fit (Example 1)Chi-square goodness of fit (Example 2)Chi-square goodness of fit (Example 3)Chi-square goodness of fit (Example 4)Index of dispersion (Example 5)Independence testingChi-square independence testing (Example 1)Chi-square independence testing (Example 2)Chi-square Real Statistics data analysis tool (Figure 3)Chi-square independence testing using standard format (Example 4)

Fisher exact testFisher exact test (Example 1)

Effect size for chi-squareCramer's V effect size (Figure 1)Odds ratio effect size (Example 1)

Noncentral chi-square distribution and PowerChart of the noncentral chi-square distribution (Figure 1)Inverse functionPower for goodness of fit (Example 1)Sample size for independence testing (Example 2)Confidence interval for effect size and power (Example 3)

F Distribution

Basic concepts of the F distributionExample of the various functions (Figure 1)

Two sample hypothesis testing to compare variancesF-Test two-sample for variances (Example 1)F-Test Two-Sample for Variances data analysis tool (Figure 2)Power for two sample variance test, 1 tail (Example 2)Power for two sample variance test, 2 tail (Example 3)Sample size for two-sample variance test (Example 4)Examples 2-4 using Excel 2007

Noncentral F distribution and PowerChart of the noncentral F distribution (Figure 1)Noncentral distribution and inverse functions

Testing for Normality and Symmetry

Graphical testsTesting for normality via histogram (Example 1)Testing for normality via QQ plot (Example 2)Testing for normality via QQ plot (Example 3)Testing for normality/symmetry via Box plot (Example 4)

Analysis of skewness and kurtosisTesting for normality using skewness and kurtosis (Example 1)

Statistical tests

Chi-square test for normalityChi-square Test (Example 1)Chi-square Test (Example 2)

Kolmogorov-Smirnov testKS Test (Example 1)KS Test (Example 2)Lilliefors Test (Example 3)

Shapiro-Wilk original testSW Test (Example 1)SW Test (Example 2)

Shapiro-Wilk expanded testSW Test (Example 1)SW Test using Real Statistics formulas (Example 1)SW Test using Real Statistics formulas (Example 2)SW Test using Real Statistics data analysis tool (Example 3)

TransformationsLog transformation (Example 1)

Non-parametric Tests

Sign testOne-sample sign test (Example 1)Confidence interval for sign testMood's median test (Example 2)

Wilcoxon rank-sum test for independent samplesWilcoxon Rank Sum Test for Independent Samples (Example 1)Wilcoxon Rank Sum Test and assumptions (Figure 2)Wilcoxon Rank Sum Test unequal sample sizes (Example 2)Wilcoxon Rank Sum Test normal approximation (Example 3)Permutation distributionPermutation distribution via Real Statistics functionsExact test (Example 4)Mann-Whitney testMann-Whitney U Test (Example 1)Mann-Whitney U Test normal approximation (Example 2)Exact test (modifying Permutation distribution)Exact test

Wilcoxon signed-ranks testWilcoxon Signed-Rank Test for Paired Samples (Example 1)Signed-Rank test for paired samples normal approximation (Example 2)Example 2 using Real Statistics function with two argumentsSigned Rank test for one sample (Example 3)Permutation distributionExact test

McNemar's testMcNemars Test (Example 1)McNemars Test (Example 2)

Runs testOne Sample Runs Test (Example 1)One Sample Runs Test (Example 2)One Sample Runs Test - exact test (Example 3)

Kolmogorov-Smirnov one sample testKS Test for exponential distribution (Example 1)

Kolmogorov-Smirnov two sample testKS test (Example 1)

ResamplingOne sample case (Example 1)One sample case using Real Statistics data analysis tool (Example 1)Two independent samples case (Example 2)Two independent samples case using Real Statistics data analysis tool (Example 2)Two paired samples case (Example 3)Two paired samples case using Real Statistics data analysis tool (Example 3)

Data analysis tools (Non-parametric analyses)Two sample KS test (freq)Two sample KS test (raw)McNemar's Test

Data analysis tools (Resampling)One sample caseTwo independent samples case (Example 1)Two paired samples case

One-way Analysis of Variance (ANOVA)Basic conceptsOne factor ANOVA (Example 1)One factor ANOVA (Example 2)Error terms for Example 2 (Figure 4)One factor ANOVA with unequal sample sizes (Example 3)One factor ANOVA using Real Statistics functions

Confidence intervalConfidence interval using Real Statistics Anova data analysis tool (Example 1)Real Statistics data analysis tool with data in standard format (Example 2)Converting from Anova Excel format to standard format

Planned comparisonsPairwise comparisons (Example 1)Pairwise contrasts (Example 1)Non-pairwise contrasts (Example 2)Contrasts using Bonferroni or Dunn/Sidk corrections (Example 3)Real Statistics Anova data analysis tool for contrasts (Example 4)

Unplanned comparisonsTukey HSD test (Example 1)Tukey HSD Real Statistics data analysis tool (Example 1)Tukey-Kramer data analysis tool (Example 2)Games-Howell data analysis tool (Example 3)REGWQ test (Example 4)Scheffe test (Example 5)

Homogeneity of variancesLevene's test on means (Example 1)Real Statistics function LEVENE and data analysis toolLevene's test on medians and trimmed means (Example 1, extra)Levene's test on means (extra example)Bartlett's test (Example 1)Fligner Killeen Test (Example 1)Log transform (Example 1)

OutliersTesting for outliers (Example 1)

Effect sizeEffect size for omnibus ANOVA (Example 1)Effect size via Real Statistics Anova data analysis tool (Example 2)Effect size for contrasts via Contrast data analysis tool (Example 3)

PowerPower of ANOVA (Example 1)Sample size required (Example 2)Confidence interval for effect size and power (Example 3)

Kruskal-Wallis testKruskal-Wallis test (Example 1)Kruskal-Wallis test using Anova on ranksKruskal-Wallis test using Real Statistics functionsKruskal-Wallis data analysis toolPairwise comparison tests (Example 2)

Welch's TestWelch's Test (Example 1)Welch's Test standard format (Example 1)Extra examples of Welch's test

Brown-Forsythe F* testBrown-Forsythe F* Test (Example 1)Brown-Forsythe F* Test (data from Example 3 of Basic Concepts of ANOVA)

Mood's Median TestMood's Median Test (Example 1)

ResamplingANOVA Resampling (Example 1)ANOVA Resampling of Error Terms (Example 2)

Factorial ANOVA

Two factor ANOVA without replicationTwo factor ANOVA without replication (Example 1)Using Real Statistics functions (Example 1)Using Real Statistics data analysis tool (Example 1)

Two factor ANOVA with replicationTwo factor ANOVA with replication (Example 1)Two factor ANOVA with replication - Interaction plots for Example 1 (Figure 4)Two factor ANOVA with replication - alternative data format (Example 2)

Real Statistics capabilities for two factor ANOVATwo factor ANOVA with replication using Real Statistics functionsTwo factor ANOVA using Real Statistics data analysis toolTwo factor ANOVA with replication - standard data format

Contrasts for two factor ANOVATwo factor ANOVA with replication (Example 1)Main effect for data in Example 1Simple effect for data in Example 1Interaction Table for data in Example 1 (Figure 5)Contrasts for data in Example 3 (Example 2)

ANOVA with more than 2 factorsThree factor ANOVA (Example 1)

Real Statistics capabilities for three factor ANOVAStandard format by rowsStandard format by columns (equivalent to the above example)Standard format by columns, including conversion (Example 1)

Correlation

Basic conceptsCovariance and independence (Figure 1)Correlation coefficient (Example 1)

Scatter diagramsScatter diagrams (Figure 1)Scatter diagrams (Example 1)

One sample hypothesis testingOne sample testing of correlation coefficient (Example 1)One sample testing of correlation coefficient (Example 2)Hypothesis testing using Fisher transformation (Example 3)Hypothesis testing using Fisher transformation (Example 4)Hypothesis testing (extra example)Power of test of correlation coefficient (Example 5)Sample size required to test correlation coefficient (Example 5)

Two sample hypothesis testingComparing the correlation coefficients of two independent samples (Example 1)Comparing the correlation coefficients of two dependent samples (Example 1 of Detail)

Multiple correlationCorrelation data analysis tool (Example 1)Multiple correlation coefficient (Example 1)Partial and semi-partial correlation (Example 2)Observation about partial and semi-partial correlation (Figure 4 and 5)

Spearman's rank correlationSpearman's rho (Example 1)Alternative way of calculating Spearman's rho (Example 2)Hypothesis testing using Spearman's rho (Example 3)Hypothesis testing using Spearman's rho (extra example corresponding to Example 1)

Kendall's tau correlationKendall's tau (Example 1)Real Statistics Kendall's tau function (Example 1)Real Statistics data analysis tool (Example 1)Kendall's tau with ties (Example 2)Kendall's tau with ties (Example 3)

Correlation and the t-testDummy variables and two sample t test (Example 1)Effect size (Observation)Point biserial correlation coefficient

Correlation and the chi-square test for independencePoint-biserial correlation coefficient (Example 1)Alternative approach for performing chi-square test (Example 2)Calculation of point-biserial correlation coefficient (Figure 3)

ResamplingResampling (Example 1)Data analysis toolExtra example

Real Statistics data analysis toolPearson's correlation (Example 1)Spearman's (rho) correlation (Example 2)Kendall's (tau) correlation (Example 3)Bootstrapping (Example 4)Randomization (Example 5)

Linear Regression

Method of least squaresRegression line (Example 1)

Regression analysisUsing regression line for prediction (Example 1)Significance vs effect size (re correlation coefficient)

Hypothesis testing whether the regression line is a good fit for the dataTesting fit of regression line (Example 1)

Hypothesis testing of the significance of the slope of the regression lineTesting slope of regression line (Example 1)LINEST function (Figure 2)Regression data analysis tool (Figures 3 and 4)Comparing the slopes of two independent samples (Example 1 of Detail)

Confidence and prediction intervals for forecasted valuesConfidence/prediction Intervals (Example 1)Testing intercept of regression line (Example 2)

Exponential regressionExponential Regression - linear regression (Example 1)LOGEST and GROWTH functions (Figure 4)Nonlinear regression via Solver, before (Example 1)Nonlinear regression via Solver, after (Example 1)Nonlinear regression via Newton's method (Example 1)Data analysis tool (Example 1)

Power regressionLog-log Regression (Example 1)Linear regression models for comparing meansRegression to Compare Means (Example 1)Full results from data analysis tool for Example 1Regression to Compare Means (Example 2)Full results from data analysis tool for Example 2

Multiple RegressionMethod of least squaresMethod of least squares (Example 1)Method of least squares using covariance matrix (Example 2)Method of least squares using hat matrix (Example 3)Method of least squares using Real Statistics functions (Example 3)

Multiple regression analysisSample size requirements for multiple regression (Figure 1)TREND and LINEST function (Example 1)Data for Example 1 is normal via QQ plot (extra worksheet)Regression data analysis tool (Example 2)Real Statistics regression data analysis tool (Example 2)Formulas for regression analysis (for Figure 5 and 6)Real Statistics functions (for Figure 5)Alternative approach to multiple regression (Example 1 of Detail)Coding categorical data (Example 4)

Confidence and prediction intervalsConfidence and prediction intervals (Example 1)Polynomial regressionPolynomial regression (Example 1)Full data analysis for quadratic model (Example 1)Full data analysis for linear model (Example 1)

Multiple regression with log transformationsLog-level transformation (Example 1)Log-log transformation (Example 2)

InteractionRegression with interaction (Example 1)

ANOVA using regressionOne factor ANOVA via Regression model (Example 1)One factor ANOVA via Regression model (Example 1 alternative coding)Group means and group effect sizes (Figure 5)Two factor ANOVA via Regression model (Example 2)

Unbalanced factorial ANOVAUnbalanced ANOVA via Regression model (Example 1)Unbalanced ANOVA via Regression model using Real Statistics analysis tool (Example 1)

Three factor ANOVA using regressionBalanced model, including data format conversion (Example 1)Unbalanced model (Example 2)

Other measures of effect size for ANOVAOmega square effect size for 1 factor ANOVA (Example 3 of Basic Concepts for ANOVA)Omega square effect size for 2 factor ANOVA (Example 2 of ANOVA using Regression)

ResidualsStudentized residuals and hat matrix (Example 1)Plot of studentized residuals (Figure 2)

Outliers and influencersOutliers and influencers: Cook's distance (Example 1)Outliers and influencers: Cook's distance (Example 2)

AutocorrelationDurbin-Watson test (Example 1)

CollinearityCollinearity (Figure 1)Tolerance and VIF (Example 1)

Testing the significance of extra variablesTesting significance of extra varaiables (Example 1)Testing significance of extra varaiables using R Square (Figure 2)Akaikes Information Criterion (Example 2)

Multiple CorrelationPartial correlation coefficient (Example 1)Partial correlation matrix (Example 2)Coefficient of determination

Statistical Power and Sample SizeStatistical Power (Example 1)Statistical Power (other example)Statistical Power (other example)Sample Size Requirement (Example 2)Confidence interval for effect size and power (Example 3)

Logistic Regression

Basic conceptsSigmoid curve (Figure 1)Logistic Regression (Example 1)

Finding coefficients using Excel's SolverSolution using Excel's Solver (Example 1)Logistic Regression data analysis tool using Solver (Example 1)

Testing coefficientsLogistic Regression data analysis tool - coefficient table (Example 1)

Fit of the modelLog-linear ratio (Example 2)

Finding coefficients using Newton's methodSolution using Newton's Method (Example 1)Logistic Regression data analysis tool using Newton's method (Example 1)Logistic Regression data analysis tool using raw data (Example 3)Logistic Regression with raw data, alternative approachCategorical coding: before (Example 4)Categorical coding: after (Example 4)

Comparing modelsBasic model (Example 4)Reduced models (Example 4)Interaction model (Example 4)

Hosmer-Lemeshow TestHosmer-Lemeshow Test (Example 1)Real Statistics Functions (Example 1)

Classification Table and ROC CurveClassification Table (Example 1)ROC Table and ROC Curve (Example 1)

Real Statistics functionsLogistic Regression Real Statistics functions, summary data (Figure 1)Logistic Regression Real Statistics functions, rawdata (Figure 2)

Multinomial and Ordinal Logistic Regression

Finding multinomial logistic regression coefficients using binary logistic regressionUsing binary logistic regression

Finding coefficients using SolverUsing Solver

Finding coefficients using Newton's methodOne step using Newton's methodUsing Newton's method

Real Statistics capabilitiesUsing supplemental formulas (Figure 1)Using Real Statistics data analysis toolUsing raw data (Figure 3)Using Real Statistics data analysis tool for input with raw data

Ordinal logistic regressionUsing binary logistic regressionUsing SolverProportional odds modelMultinomial logistic regression model

Log-Linear Regression

Two-way contingency tablesTwo-way log-linear saturated model (Example 1)

Saturated model for two-way contingency tablesMarginal averages for two-way models (Figure 3)Alternative coding for coefficients of two-way models

Independence model for two-way contingency tablesTwo-way log-linear independence model (Example 2)Residuals and chi-square for two-way model (Figure 2)Coefficients via Excel regression data analysis tool (Figure 3)

Other models for two-way contingency tablesOther two-way log-linear models (Figure 3)

Best fit model for two-way contingency tablesSummary of all two-way log-linear models (Figure 1)

Three-way contingency tablesThree-way contingency tables (Example 1)

Independence and non-comprehensive modelsExpectation for 3-way models (various figures)Residuals and chi-square for three-way models (Figure 2 of Conditional Independence Model)

Homogeneous association modelIterative proportional fitting for homogeneous case (Figure 1)

Best fit modelSummary of all three-way log-linear models (Figure 1)Odds ratio for three-way log-linear models (Figure 2)Coefficients for three-way log-linear model (Figure 3)

ANOVA with Random Factors and Nested Models

One random factor ANOVAOne-way ANOVA with random factor (Example 1)

Two factor mixed modelTwo factor mixed model (Example 1)Two factor mixed Anova data analysis tool (Example 1)

Nested modelsNested model (Example 1)Nested Anova data analysis tool (Example 1)Nested Anova using data in standard form (Example 2)

ANOVA with Repeated Measures

One within subjects factorANOVA with repeated measures, 1 treatment variable (Example 1)Contrasts for ANOVA with repeated measures (Example 2)

SphericityTests for sphericity (Example 1)Calculation of GG and HF epsilon and lower bound correction (Figure 3)ANOVA with repeated measures corrected for sphericity (Example 2)Effect size for ANOVA

Additional information about sphericityGG epsilon calculation (Example 1)GG epsilon calculation using eigenvalues (Example 2)Mauchly's test for sphericity (Example 3)More powerful test for sphericity, John, Nagao and Sugiura's test (Example 4)

Two within subjects factorsANOVA with repeated measures, 2 within-subjects factors (Example 1)Chart comparing means for intersection (Figure 9)Comparisons of means for intersection (Example 2)Comparisons of means for intersection (Example 3)

One within subjects factor and one between subjects factorANOVA with repeated measures, 1 within and 1 between subjects (Example 1)ANOVA with repeated measures, use of data analysis toolsAssumptions between subjects: Boxplot (Figure 5)Assumptions between subjects: Levene's test (Figure 6)Assumptions within subjects: Boxplot (Figure 7)Assumptions within subjects: Levene's test (Figure 8)Within subjects covariance matrices (Figure 9)Within subjects GG and HF epsilon factors (Figure 10)Within subjects ANOVA corrected for sphericity (Figure 11)Within subjects simple effects ANOVA (Figure 12)Within subjects GG and HF epsilon factors for Young (Figure 13)Summary of within subjects GG and HF epsilon factors (Figure 14)Corrected within subjects simple effects ANOVA (Figure 15)Between subjects simple effects ANOVA (Figure 16)Corrected between subjects simple effects ANOVA (Figure 18)Box's test for equivalence of covariance matrices (Example 1 of Detail)

Friedman testFriedman's test (Example 1)

Cochran's Q testCochran's Q test, raw data (Example 1)Cochran's Q test, summarized data (Example 2)Analysis of Covariance (ANCOVA)

Basic conceptsANCOVA data (Example 1)

Using a regression approachANCOVA regression approach (Figure 1)ANCOVA regression models (Figure 2)ANCOVA results (Figure 3 and 4)ANCOVA full model (Figure 5)ANCOVA adjusted means (Figure 6 and 7)

AssumptionsANCOVA assumptions (Example 1, Figure 1 and 2)ANCOVA complete model (Figure 3)ANCOVA assumptions - homogeneous regression slopes (Figure 4)

ANOVA approachANOVA approach (Example 1)ANCOVA data analysis tool (Example 1)

ContrastsANCOVA Contrasts (Example 1)

Effect sizeANCOVA effect sizes - omnibus testANCOVA effect sizes - contrasts

Reliability

Split-half methodologySplit-half methodology for reliability (Example 1)Split-half methodology for reliability (Example 2)Kuder and Richardson formula 20Kuder and Richardson Formula 20 (Example 1)Cronbach's alphaCronbach's alpha (Example 1)Cronbach's alpha with one question removed (Example 2)Cronbach's alpha using ANOVA (Example 3)Cronbach's alpha using ANOVA (Example 4)Real Statistics functions and data analysis tool (Example 4)Hypothesis testing and confidence interval (Example 5)Sample size to achieve confidence width (Example 6)Statistical power (Example 7)Sample size (Example 8)

Cohen's kappaCohen's kappa (Example 1)Standard error and confidence interval (Figure 5)Real Statistics function and data analysis tool (Figure 7)

Weighted kappaWeighted kappa (Example 1)Unweighted kappa using a weighted approach (Figure 2)Standard error and confidence interval (Example 2)

Fleiss' kappaFleiss' kappa (Example 1)Extra example (Fleiss' kappa for Example 1 of Cohen's kappa)Extra example (Fleiss' kappa for Example 1 of Intraclass correlation)

Intraclass correlationIntraclass Correlation - case 1ICC(1,1) Power and Sample SizeIntraclass Correlation - case 2Intraclass Correlation - case 3Real Statistics data analysis tool

Kendall's WReal Statistics function and data analysis tool (Example 1)Kendall's W (Example 1)Kendall's W with ties (Example 2)

Item AnalysisDifficulty and discrimination index (Example 1)Item analysis for multiple choice tests (Example 2)Real Statistics function (Example 3)Partial credit (Example 4)Real Statistics data analysis tool (Example 5)

Missing Data and Multiple Imputation

Traditional approachesListwise Deletion (Example 1)Mean Imputation (Example 2)Regression Imputation (Example 3)Stochastic Regression Imputation (Example 4)

Multiple ImputationFrequency and patterns of missing dataSimple imputation (with and w/o constraints)One step of the FCS algorithmOne step of the FCS algorithm detailsFully conditional specification (FCS)Summary of FCS resultsMultiple FCS imputationsCombining multiple imputationsMI data analysis toolMultiple imputations from MI data analysis toolMI data analysis tool with constraintsMultiple imputations from MI data analysis tool with constraints

Full Information Maximum Log-Likelihood (FIML)FIML statisticsFIML initial conditionsFIML after using SolverFIML with modified degrees of freedomFIML data analysis toolFIML complete data: initial conditionsFIML complete data: after using SolverFIML complete data: data analysis tool

Mathematical Notation

Functions, polynomials, limits and graphsGraph of a function (Example 1)

Excel Capabilities

Built-in functionsUnique counting (Figure 7)SUMIF and COUNTIF (Figure 8)Table lookup functions (esp. MATCH)

Sorting and eliminating duplicatesSort and remove duplicates in standard Excel (Examples 1, 2, 3, 4)Reformatting toolsReformatting (Example 1)Remove missing data (Example 2)

Frequency table conversionConversion from frequency tableDescriptive statistics for frequency table (Example 1)Conversion to raw data using Real Statistics data analysis tool (Example 2)Conversion to frequency table

Coding of categorical variablesCoding of categorical variables (Example 1)

Table lookupTable Lookup (Example 1)Table Lookup (Example 2)Table Lookup (Example 3)Table Lookup (Example 4)

Special charting capabilitiesBox Plot (Example 1)Box Plot for negative data (Example 2)Plot of sample means with std error intervals (Example 3)

Matrices and Iterative Procedures

Basic conceptsLength of a vector (Example 2)Diagonal of a square matrix (Example 3)

Matrix operationsInverse of a matrix (Example 1)Real Statistics Matrix data analysis tool (Example 2)

Determinants and simultaneous linear equationsDeterminant (Example 1)Determinant using Gaussian elimination (Example 2)Solution to linear equations via Cramer's rule (Example 3)Solution to linear equations via Gaussian elimination (Example 4)Solution to homogeneous linear equations (Example 5)Solution to homogeneous linear equations (Example 6)Inverting a matrix via Gaussian elimination (Example 10)Gaussian elimination via Real Statistics functions (Examples 7,8,9 + extra examples)

Newton's methodNewton's Method for one equation (Example 1)Newton's Method for multiple equations (Example 2)Newton's Method for one equation (Example 2)

Goal seeking and SolverGoal Seek to find eigenvalues (Example 1)Solver to find regression coefficients (Example 2)

Interative proportional fitting procedureIterative Proportional Fitting Procedure, 2-way (Example 1)Real Statistics function, 2-wayRepresentation of a 3-way contingency table (Figure 5)Iterative Proportional Fitting Procedure, 3-way (Example 3)Real Statistics function, 3-way

TablesWilcoxon's Rank Sum Table for Independent SamplesMann-Whitney TableWilcoxon's Rank Sum Table for Paired SamplesRuns TableKolmogorov-Smirnov TableKolmogorov-Smirnov Table (alternative)Lilliefors Test Table - original tableLilliefors Test Table - newer, enlarged tableShapiro Wilk TablesStudentized Q Table (table 1)Studentized Q Table (table 2)Pearson's Correlation TableSpearman's Rho TableKendall's Tau TableDurbin-Watson Table (alpha .01)Durbin-Watson Table (alpha .05)

Bar ChartResults of Marketing Campaign(sales in millions of euros)Brand ABrand BBrand CLondon23.512.315Paris13.88.15.5Rome17.34.56.9Madrid14.86.82.7Vienna7.24.21.6Berlin29.521.411.6Total106.157.343.3

Line ChartAverage Income by AgeAgeIncome3123500322400033250003426700352750036292003733000383510039374004039500

Line Chart 2Average Income and Rent by AgeAgeIncomeRent312350060003224000650033250007000342670045003527500600036292007500373300080003835100900039374007000403950012000AgeIncomeRent312350060003224000650033250007000342670045003527500600036292007500373300080003835100900039374007000403950012000

Array FormulaEquipment SalesUnit PriceQuantityRevenueDesks5004020000Lamps80302400Chairs15013019500Pen Sets60704200

Array FunctionTransposeArea CodePopulationArea Code345378678712815345230000Population230000340000145000235900195000378340000678145000712235900815195000

Sort FilterSort and FilterSort by IncomeSort by Income/PersonRemove Duplicates

PersonGenderAgeIncomePersonGenderAgeIncomePersonGenderAgeIncomeIncomeMaryF3545000JaneF4530000JaneF453000045000BobM4040000JimM5535000AlanM403500040000JimM5535000AlanM4035000JimM553500035000BettyF2580000SteveM3035000SteveM303500080000AlanM4035000BobM4040000BobM404000060000DebraF4045000MaryF3545000DebraF404500030000DaveM6060000DebraF4045000MaryF3545000SteveM3035000DaveM6060000DaveM6060000JaneF4530000BettyF2580000BettyF2580000FilterPersonGenderAgeIncomePersonGenderAgeIncomeMaryF3545000MaryF3545000BobM4040000BobM4040000JimM5535000JimM5535000BettyF2580000AlanM4035000AlanM4035000DebraF4045000DebraF4045000DaveM6060000SteveM3035000SteveM3035000JaneF4530000

SeekGoal Seekx^210.9999919532x^211.0000152484x3.3166235773x-3.3166270891

Reformat 1Reformat data in a rangeSort and Extract Unique

Input rangeReshapeReverseSortSort no dupesInput RangeSort UniqueExtract Unique358ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?RussiaItalyERROR:#NAME?ERROR:#NAME?15AERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?GermanyFranceERROR:#NAME?ERROR:#NAME?5A0ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ItalyRussiaERROR:#NAME?ERROR:#NAME?12B8ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FranceSpainERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?COUNTUERROR:#NAME?ERROR:#NAME?COUNTAUERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?

Reformat 2Reformat data in a range

Input rangeSort358ERROR:#NAME?ERROR:#NAME?15-13ERROR:#NAME?ERROR:#NAME?520ERROR:#NAME?ERROR:#NAME?12-48ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?

Sort RowsSort Rows without duplicatesSort Rows=QSORTRows(T3:W14,4)=QSORTRows(T3:W14)MaleFranceRich45=SortRowsUnique(A3:C14)=SortRowsCount(A3:C14)=SortRowsSum(A3:D14,"")MaleFranceRich45ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKPoor35FemaleUKPoor35ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MaleUSRich15ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MaleUSRich15ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MaleUSRich40ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MaleUSRich40ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKRich25ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKRich25ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MaleUSPoor20ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MaleUSPoor20ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleFranceRich10ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleFranceRich10ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKRich50ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKRich50ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKPoor35ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKPoor35ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MaleFrancePoor45ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MaleFrancePoor45ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUSPoor20ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUSPoor20ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKRich40FemaleUKRich40ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?=SortRowsUnique(A3:B14)ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?=CountRowsUnique(A3:B14)ERROR:#NAME?

ExtractExtract Columns from Data RangePovertyInfant MortWhiteCrimeDoctorsTraf DeathsUniversityUnemployIncomePovertyInfant MortDoctorsTraf DeathsIncomeAlabama15.79.071.0448218.21.8122.05.042,666Alabama15.79218.17340830571.8142666Alaska8.46.970.6661228.51.6327.36.768,460Alaska8.46.9228.45028196581.6368460Arizona14.76.486.5483209.71.6925.15.550,958Arizona14.76.4209.66657175721.6950958Arkansas17.38.580.8529203.41.9618.85.138,815Arkansas17.38.5203.42285988231.9638815California13.35.076.6523268.71.2129.67.261,021California13.35268.69056049811.2161021Colorado11.45.789.7348259.71.1435.64.956,993Colorado11.45.7259.68608874671.1456993Connecticut9.36.284.3256376.40.8635.65.768,595Connecticut9.36.2376.37526691550.8668595Delaware10.08.374.3689250.91.2327.54.857,989Delaware108.3250.94175668511.2357989Florida13.27.379.8723247.91.5625.86.247,778Florida13.27.3247.86359820581.5647778Georgia14.78.165.4493217.41.4627.56.250,861Georgia14.78.1217.44570184041.4650861Hawaii9.15.629.7273317.01.3329.13.967,214Hawaii9.15.6316.98289278791.3367214Idaho12.66.894.6239168.81.6024.04.947,576Idaho12.66.8168.8339031311.647576Illinois12.27.379.1533280.21.1629.96.556,235Illinois12.27.3280.15386787691.1656235Indiana13.18.088.0334216.91.2622.95.947,966Indiana13.18216.93969975991.2647966Iowa11.55.194.2295189.31.4224.34.148,980Iowa11.55.1189.2832242841.4248980Kansas11.37.188.7453222.51.3829.64.450,177Kansas11.37.1222.54770859751.3850177Kentucky17.37.589.9295232.31.8019.76.441,538Kentucky17.37.5232.34854500661.841538Louisiana17.39.964.8730262.72.1720.34.643,733Louisiana17.39.9262.66146237062.1743733Maine12.36.396.4118278.41.2225.45.446,581Maine12.36.3278.39482802921.2246581

Select ColSelect Columns=SelectCols(A3:D14,"2,3,1")=SelectCols(A3:D14,"3,2,1,4",1)MaleFranceRich45ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKPoor35ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MaleUSRich15ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MaleUSRich40ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKRich25ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MaleUSPoor20ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleFranceRich10ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKRich50ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKPoor35ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MaleFrancePoor45ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUSPoor20ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?FemaleUKRich40ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?

MiscMiscellaneous Real Statistics FunctionsERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?50ERROR:#NAME?

Prob 0Probability calculationn12p0.16666666671-p0.8333333333(1-p)^n0.11215665481-(1-p)^n0.8878433452

Prob 1Frequency/Distribution Functionsxf(x)F(x)10.120.12=B420.250.37=B5+C430.080.45=B6+C540.140.59=B7+C650.090.68=B8+C760.180.86=B9+C870.090.95=B10+C980.051.00=B11+C10P(3)0.08=PROB(A4:A11,B4:B11,3)P(x 5) 0.68=PROB(A4:A11,B4:B11,,5)P(3 x 5)0.31=PROB(A4:A11,B4:B11,3,5)

Prob 2Frequency FunctionFREQTABLE functionFrequency Table data analysis toolnodupe14xfreqf(x)ERROR:#NAME?xfreqf(x)ItemfreqprobFrequency Table181220.1666666667ERROR:#NAME?1220.1666666667ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?131330.25ERROR:#NAME?1330.25ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?itemfreqprob121410.0833333333ERROR:#NAME?1410.0833333333ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?0ERROR:#DIV/0!181620.1666666667ERROR:#NAME?1620.1666666667ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?0ERROR:#DIV/0!121840.3333333333ERROR:#NAME?1840.3333333333ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?0ERROR:#DIV/0!1312ERROR:#NAME?12ERROR:#NAME?0ERROR:#DIV/0!13ERROR:#NAME?ERROR:#NAME?0ERROR:#DIV/0!16ERROR:#NAME?0ERROR:#DIV/0!18ERROR:#NAME?18ERROR:#NAME?16ERROR:#NAME?

CentralMeasures of Central Tendency

Data 1Data 2Data 3Data 1Data 2Data 3555555222222-1-1-1-1-1-1333Data 4Data 5Data 6333777501.055777558801.0525580001.1-1000261.1026

AVERAGE32.8753.75HARMEAN61.53846153851.0744186047ERROR:#NUM!countERROR:#NAME?ERROR:#NAME?ERROR:#NAME?meanERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MEDIAN32.54GEOMEAN63.24555320341.074709263ERROR:#NUM!sumERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MODE55ERROR:#N/AMODE.SNGL55ERROR:#N/AMODE.MULT55ERROR:#N/A52ERROR:#N/A

Geo MeanGeometric MeanIf you earn 5% interest in year 1, 8% in year 2 and 10% interest in year 3, the average interest is given by the geometric meanInterest - year 10.05Int1Interest - year 20.08Int2Interest - year 30.10Int3

Return from year 11.05= 1 + Int1Return from year 21.08= 1 + Int2Return from year 31.10= 1 + Int3Return after year 11.0500= 1 + Int1Return after year 21.1340= (1 + Int1) * (1 + Int2)Return after year 31.2474= (1 + Int1) * (1 + Int2) * (1 + Int3)

Average annual return1.0765= SQRT((1 + Int1) * (1 + Int2) * (1 + Int3)) = GEOMEAN(1 + Int1, 1+ Int2, 1 + Int3)Average interest0.0765= 1 - Average annual returnTotal return1.2474= (1 + Average Interest) ^ 3 = Return after 3 years as calculated above0.0765= GEOMEAN(1.05,1.08,1.1) - 11.334025= (1.05)(1.05)(1.1)(1.1)0.074709263= 1.334025 ^ 0.25 -10.074709263= GEOMEAN(1.05,1.05,1.1,1.1) - 1

VarMeasures of Variability55Data 1Data 2Data 322555-1-122233-1-1-14433355777005582200026AVERAGE2.5MEDIAN2.5countERROR:#NAME?ERROR:#NAME?ERROR:#NAME?VAR.S = VAR4.8571428571MADERROR:#NAME?meanERROR:#NAME?ERROR:#NAME?ERROR:#NAME?VAR.P = VARP4.25varianceERROR:#NAME?ERROR:#NAME?ERROR:#NAME?MIN-1stdevERROR:#NAME?ERROR:#NAME?ERROR:#NAME?STDEV.S = STDEV2.2038926601MAX5STDEV.P = STDEVP2.0615528128RNGERROR:#NAME?DEVSQ34INCEXCQ11.50.5AVEDEV1.75Q34.254.75IQRERROR:#NAME?ERROR:#NAME?

CombinedCombined samplesample 1sample 2combined36341465677615sample 1sample 2combinedsize4377mean544.57142857144.5714285714var3.333333333374.28571428574.2857142857stdev1.82574185842.64575131112.0701966782.070196678

SKEWSkewness and Kurtosis25-134502SKEW-0.4270518649ERROR:#NAME?KURT-0.9397923875ERROR:#NAME?SKEW.PERROR:#NAME?ERROR:#NAME?n8ERROR:#NAME?SKEW.P-0.3424032353ERROR:#NAME?

Rank 0RankingPercentileQuartileDataRankReverse4433%-INC%-EXCQ-INCQ-EXC00420.00-1ERROR:#NUM!0-1ERROR:#NUM!-1-1510.250-0.510-0.577150.504424455240.75563561.007ERROR:#NUM!47ERROR:#NUM!MIN-1DataRankReverseMAX71ERROR:#NAME?ERROR:#NAME?0.453.22.8k5ERROR:#NAME?ERROR:#NAME?0.805.46.6SMALL1-15ERROR:#NAME?ERROR:#NAME?200ERROR:#NAME?ERROR:#NAME?348ERROR:#NAME?ERROR:#NAME?PercentRank4557DataRankReverseINCEXC1ERROR:#NAME?ERROR:#NAME?50.750.66666LARGEk5ERROR:#NAME?ERROR:#NAME?5.40.80.7175ERROR:#NAME?ERROR:#NAME?0.850.303120.36875250ERROR:#NAME?ERROR:#NAME?348ERROR:#NAME?ERROR:#NAME?408ERROR:#NAME?ERROR:#NAME?5-15ERROR:#NAME?ERROR:#NAME?

Rank 1Rank and PercentileData analysis toolPERCENTRANK and PERCENTRANK.EXCPERCENTILEPERCENTILE.EXC

Scores546734549455324587643960PointScoresRankPercentScoresRank%-INC%-EXCPercentileScorePercentileScore5941100.00%941100.00%92.30%0%320%ERROR:#NUM!987290.90%87290.90%84.60%10%34.510%32.6267381.80%67381.80%76.90%20%40.220%371064472.70%64472.70%69.20%30%47.730%44.41260563.60%60563.60%61.50%40%5440%54655654.50%55654.50%53.80%50%54.550%54.5154736.30%54736.30%38.40%60%5860%59454736.30%54736.30%38.40%70%62.870%64.3845927.20%45927.20%30.70%80%66.480%7511391018.10%391018.10%23.00%90%8590%91.9334119.00%34119.00%15.30%100%94100%ERROR:#NUM!732120.00%32120.00%7.60%

Desc 1Descriptive Statistics data analysis toolReal Statistics Descriptive StatisticsScoresScoresCalculations using Excel functionsCalculation of MAD23Scores38Mean30.8181818182Mean30.8181818182|xi - median(xi)|Mean30.818181818245Standard Error4.9319333878Standard Error4.93193338780Standard Error4.931933387821Median23Median2315Median2317Mode21Mode2122Mode2121Standard Deviation16.3573725385Standard Deviation16.35737253852Standard Deviation16.35737253858Sample Variance267.5636363636Sample Variance267.56363636366Sample Variance267.563636363661Kurtosis-0.5471310926Kurtosis-0.54713109262Kurtosis-0.547131092621Skewness0.6251387508Skewness0.625138750815Skewness0.625138750852Range53Range5338Range5332Minimum8Minimum82Minimum8Maximum61Maximum6129Maximum61Sum339Sum3399Sum339Count11Count11Count11Geometric mean26.7547165456Geometric Mean26.7547165456Harmonic mean22.6537343672Harmonic Mean22.6537343672INCEXCAAD13.4380165289AAD13.4380165289Q12121MAD9MADERROR:#NAME?Q341.545IQR20.5IQRERROR:#NAME?IQRERROR:#NAME?ERROR:#NAME?

Desc 2Descriptive StatisticsReal Statistics - IQR using QUARTILEReal Statistics - IQR using QUARTILE.EXCTwo sample example using Descriptive Statistics data analysis toolDescriptive StatisticsDescriptive Statistics

Sample 1Sample 2Sample 1Sample 2Sample 1Sample 2Sample 1Sample 21912Mean30.461538461530.6153846154Mean30.461538461530.61538461544127Mean30.4615384615Mean30.6153846154Standard Error4.67345941245.4473452268Standard Error4.67345941245.44734522682918Standard Error4.6734594124Standard Error5.4473452268Median2927Median29271823Median29Median27Mode2927Mode2927872Mode29Mode27Standard Deviation16.850397545319.6406825303Standard Deviation16.850397545319.64068253032927Standard Deviation16.8503975453Standard Deviation19.6406825303Sample Variance283.9358974359385.7564102564Sample Variance283.9358974359385.75641025641127Sample Variance283.9358974359Sample Variance385.7564102564Kurtosis-1.15073080480.0628405897Kurtosis-1.15073080480.06284058975953Kurtosis-1.1507308048Kurtosis0.0628405897Skewness0.26560133980.786979652Skewness0.26560133980.786979652413Skewness0.2656013398Skewness0.786979652Range5169Range51694845Range51Range69Minimum83Maximum59725353Minimum8Minimum3Maximum5972Minimum832913Maximum59Maximum72Sum396398Sum3963981125Sum396Sum398Count1313Count1313Count13Count13Geometric Mean25.608674447623.9814225796Geometric Mean25.608674447623.9814225796Harmonic Mean20.83719053815.9005529264Harmonic Mean20.83719053815.9005529264AAD13.79881656815.4674556213AAD13.79881656815.4674556213MADERROR:#NAME?ERROR:#NAME?MADERROR:#NAME?ERROR:#NAME?IQRERROR:#NAME?ERROR:#NAME?IQRERROR:#NAME?ERROR:#NAME?

FreqMean and Variance based on Frequency Tablesdataxfreqxffxfx2lowerupperfmidptfxfx222424816mean304326122313139var1.42857142864101774924242832SS821020215304502515152520301252562538248276811364Frequency Function4fxnxfxnx5xf(x)24833ERROR:#NAME?6879.714285714320.530.125fx2nx2varfx2nx2var40.2582891.42857142861.4285714286ERROR:#NAME?1136794.367346938879.238095238150.125

Freq 2Conversion from frequency tableReal Statistics data analysis toolReal Statistics data analysis tool

Frequency TableRaw Data (with duplicates)Frequency TableDescriptive StatisticsFrequency TableDescriptive StatisticsRaw Data

ScoreFreqCumScoreRowCountScoreFreqMean5.5ERROR:#NAME?ScoreFreqMean5.5ERROR:#NAME?5.0115.0105.01Standard Error0.0730296743ERROR:#NAME?5.01Standard Error0.0730296743ERROR:#NAME?5.1125.1205.11MedianERROR:#NAME?ERROR:#NAME?5.11MedianERROR:#NAME?ERROR:#NAME?5.2355.2325.23Mode5.7ERROR:#NAME?5.23Mode5.7ERROR:#NAME?5.5275.2315.52Standard Deviation0.2828427125ERROR:#NAME?5.52Standard Deviation0.2828427125ERROR:#NAME?5.6185.2305.61Sample Variance0.08ERROR:#NAME?5.61Sample Variance0.08ERROR:#NAME?5.74125.5415.74Kurtosis-1.3104395604ERROR:#NAME?5.74Kurtosis-1.3104395604ERROR:#NAME?5.83155.5405.83Skewness-0.5682108063ERROR:#NAME?5.83Skewness-0.5682108063ERROR:#NAME?5.650Range0.8ERROR:#NAME?Range0.8ERROR:#NAME?5.763Maximum5.8ERROR:#NAME?Maximum5.8ERROR:#NAME?5.762Minimum5.0ERROR:#NAME?Minimum5.0ERROR:#NAME?5.761Sum82.5ERROR:#NAME?Sum82.5ERROR:#NAME?5.760Count15ERROR:#NAME?Count15ERROR:#NAME?5.872Geometric Mean5.4930868808ERROR:#NAME?Geometric Mean5.4930868808ERROR:#NAME?5.871Harmonic Mean5.4860552296ERROR:#NAME?Harmonic Mean5.4860552296ERROR:#NAME?5.870AAD0.24ERROR:#NAME?AAD0.24MADERROR:#NAME?ERROR:#NAME?MADERROR:#NAME?Alternative VersionIQRERROR:#NAME?ERROR:#NAME?IQRERROR:#NAME?Frequency TableScoreFreqCum5.0115.1125.2355.5275.6185.74125.833Raw Data (with duplicates)ScoreRowCount5.02705.12805.22925.22915.22905.53015.53005.63105.73235.73225.73215.73205.83325.83315.8330

HistogramFrequency TableUsing FREQTABLE with bin size 15Using Frequency Table data analysis tool

ScoresBinsCountitemfreqcumFrequency Table3420206ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?45344010ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Bin size152329602ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?2212803ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?itemfreqcum772over 801ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?03423ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?1555910ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?308136614ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?45518292360018672475321449090122

Histogram data analysis tool

BinFrequency2064010602Using FREQTABLE with bin size 15Frequency Table803and max bin value 100More1Bin size15itemfreqcumERROR:#NAME?ERROR:#NAME?ERROR:#NAME?itemfreqcumERROR:#NAME?ERROR:#NAME?ERROR:#NAME?-5ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?1033ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?25811ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?40516ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?55218ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?70220ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?85121100122

Box Plot 1Boxplot (aka box and whiskers plot)Using QUARTILEBrand ABrand BBrand CBrand ABrand BBrand CBrand ABrand BBrand C10208401430Min380300430Min38030043015609401750Q1-Min142.5185342.5Q1-Min142.5185342.5560780870Med-Q1202.5220132.5Med-Q1202.5220132.5780650920Q3-Med237.5120332.5Q3-Med237.5120332.59907201300Max-Q3597.51025512.5Max-Q3597.51025512.567043089051018507404903007203803604308806901050Using QUARTILE.EXC and Real Statistics data analysis toolBox PlotBrand ABrand BBrand CMin380300430Q1-Min125112.5305Med-Q1202.5220132.5Q3-Med272.5160427.5Max-Q3562.5985417.5

Box Plot 2Boxplot - Alternative ApproachUsing QUARTILEBrand ABrand BBrand CBrand ABrand BBrand C10208401430Q1522.5485772.515609401750Med-Q1202.5220132.5560780870Q3-Med237.5120332.5780650920Q1-Min142.5785342.59907201300Max-Q3597.51025512.56704308905101850740490-3007203803604308806901050Using QUARTILE.EXCBrand ABrand BBrand CQ1505412.5735Med-Q1220292.5170Q3-Med272.5160427.5Q1-Min125712.5305Max-Q3562.5985417.5

Box Plot 3Boxplot - Approach for Negative DataUsing QUARTILEBrand ABrand BBrand CBrand ABrand BBrand C10208401430Min380-30043015609401750Q1-Min142.5785342.5560780870Med-Q1202.5220132.5780650920Q3-Med237.5120332.59907201300Max-Q3597.51025512.56704308905101850740490-3007203803604308806901050Using QUARTILE.EXCBrand ABrand BBrand CMin380-300430Q1-Min125712.5305Med-Q1445352.5505Q3-Med272.5160427.5Max-Q3562.5985417.5

ROCROC and Classification TableROC TableObservedCumulativeDosageLivesDiesLivesDiesFPRTPRAUCFail-ObsSuc-Obs00110.064516129Fail-Pred41358471less than 2.003433430.9354838710.98924731180.1182591663Suc-Pred1142213352.00 - 3.9963797100.81593927890.96415770610.16099787125272798064.00 - 5.998811185210.64895635670.92473118280.18424435336.00 - 7.9910514290350.449715370.87455197130.204117443Accuracy0.78368121440.79211469530.78660049638.00 - 9.9912323413580.21631878560.79211469530.142791074110.00 - 11.9995605081180.03605313090.57706093190.0098549305Cutoff512.00 - 13.999755171930.01897533210.30824372760.003509416314.00 - 15.996415232340.00759013280.16129032260.00122421516.00 -17.9943052726400.0537634409018.00 or more0155272790005272790.8895145988Estimating AUC using rectanglesROC TableDosageLivesDiesIndexLivesDiesFPRTPRAUCClassification Tableless than 2.00343000110.0645161292.00 - 3.9963713430.9354838710.98924731180.1182591663Fail-ObsSuc-Obs4.00 - 5.998811297100.81593927890.96415770610.1609978712Fail-Pred413584716.00 - 7.99105143185210.64895635670.92473118280.1842443533Suc-Pred1142213358.00 - 9.99123234290350.449715370.87455197130.20411744352727980610.00 - 11.9995605413580.21631878560.79211469530.142791074112.00 - 13.9997565081180.03605313090.57706093190.0098549305Accuracy0.78368121440.79211469530.786600496314.00 - 15.9964175171930.01897533210.30824372760.003509416316.00 -17.9943085232340.00759013280.16129032260.001224215Cutoff518.00 or more015952726400.05376344090527279105272790000.8895145988Estimating AUC using trapezoidsROC TableDosageLivesDiesIndexLivesDiesFPRTPRAUCClassification Tableless than 2.00343000110.06416926812.00 - 3.9963713430.9354838710.98924731180.116759503Fail-ObsSuc-Obs4.00 - 5.998811297100.81593927890.96415770610.1577060932Fail-Pred413584716.00 - 7.99105143185210.64895635670.92473118280.1792454755Suc-Pred1142213358.00 - 9.99123234290350.449715370.87455197130.194497153752727980610.00 - 11.9995605413580.21631878560.79211469530.123407670412.00 - 13.9997565081180.03605313090.57706093190.0075595275Accuracy0.78368121440.79211469530.786600496314.00 - 15.9964175171930.01897533210.30824372760.002672869416.00 -17.9943085232340.00759013280.16129032260.0008161433Cutoff518.00 or more015952726400.05376344090527279105272790000.846833704

ROC 1ROCROC TableClassification TableDosageLivesDiesDosageLivesDiesFPRTPRAUCFail-ObsSuc-Obs234300110.0641692681Fail-Pred41358471463723430.9354838710.98924731180.116759503Suc-Pred11422133568811497100.81593927890.96415770610.15770609325272798068105146185210.64895635670.92473118280.179245475510123238290350.449715370.87455197130.1944971537Accuracy0.78368121440.79211469530.786600496312956010413580.21631878560.79211469530.123407670414975125081180.03605313090.57706093190.0075595275Cutoff1016641145171930.01897533210.30824372760.002672869418430165232340.00759013280.16129032260.0008161433200151852726400.05376344090527279205272790000.846833704

WinsorTrimmed and Winsorized Data

DataTrimmedWinsorized3ERROR:#NAME?ERROR:#NAME?4ERROR:#NAME?ERROR:#NAME?6ERROR:#NAME?ERROR:#NAME?9ERROR:#NAME?ERROR:#NAME?9ERROR:#NAME?ERROR:#NAME?3ERROR:#NAME?ERROR:#NAME?4ERROR:#NAME?ERROR:#NAME?5ERROR:#NAME?ERROR:#NAME?7ERROR:#NAME?ERROR:#NAME?5ERROR:#NAME?ERROR:#NAME?40ERROR:#NAME?ERROR:#NAME?1ERROR:#NAME?ERROR:#NAME?0ERROR:#NAME?ERROR:#NAME?8ERROR:#NAME?ERROR:#NAME?30ERROR:#NAME?ERROR:#NAME?1ERROR:#NAME?ERROR:#NAME?7ERROR:#NAME?ERROR:#NAME?5ERROR:#NAME?ERROR:#NAME?4ERROR:#NAME?ERROR:#NAME?2ERROR:#NAME?ERROR:#NAME?7.65ERROR:#NAME?ERROR:#NAME?5.1428571429ERROR:#NAME?

MissingDeleting Missing Data ListwisePovertyWhiteCrimeDoctorsUniversityIncomeERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Alabama15.771.0448218.222.042,666ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Alaska8.470.6661228.527.368,460ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Arizona14.786.5483209.725.150,958ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Arkansas17.380.8203.418.838,815ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?California13.376.6523268.729.661,021ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Colorado89.7348259.735.656,993ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Connecticut9.384.3256376.435.668,595ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Delaware10.074.3689250.927.557,989ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Florida13.279.8723247.925.847,778ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Georgia14.765.4493217.427.550,861ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Hawaii9.129.7273317.029.167,214ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Idaho94.6239168.824.047,576ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Illinois12.279.1533280.229.956,235ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Indiana13.1216.947,966ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Iowa11.594.2295189.324.348,980ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Kansas11.388.7453222.529.650,177ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Kentucky17.389.9295232.319.741,538ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Louisiana17.364.8730262.720.343,733ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Maine12.396.4118278.425.446,581ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?Number of rows without missing dataERROR:#NAME?

PowerPowerSample size for t-test (single sample)AlphaPowerEffect Size0.050.010.800.21992960.800.534510.800.815220.900.22653760.900.544630.900.819270.950.23274490.950.554750.950.82332

Normal 1Normal distribution

IQ Scoref(x)500.0002 =NORM.DIST(A4,100,16,FALSE)510.0002520.0003530.0003540.0004550.0005560.0006570.0007580.0008590.0009600.0011610.0013620.0015630.0017640.0020650.0023IQ Example - mean = 100, std dev = 16660.0026670.0030680.0034690.0038700.0043710.0048720.0054730.0060740.0067750.0074760.0081770.0089780.0097790.0105800.0114810.0123820.0132830.0142840.0151850.0161860.0170870.0179880.0188890.0197900.0205910.0213920.0220930.0227940.0232950.0237960.0242970.0245980.0247990.02491000.02491010.02491020.02471030.02451040.02421050.02371060.02321070.02271080.02201090.02131100.02051110.01971120.01881130.01791140.01701150.01611160.01511170.01421180.01321190.01231200.01141210.01051220.00971230.00891240.00811250.00741260.00671270.00601280.00541290.00481300.00431310.00381320.00341330.00301340.00261350.00231360.00201370.00171380.00151390.00131400.00111410.00091420.00081430.00071440.00061450.00051460.00041470.00031480.00031490.00021500.0002

z-scores

-3.000.0044-2.750.0091-2.500.0175-2.250.0317-2.000.0540-1.750.0863-1.500.1295-1.250.1826-1.000.2420-0.750.3011-0.500.3521-0.250.38670.000.39890.250.38670.500.35210.750.30111.000.24201.250.18261.500.12951.750.08632.000.05402.250.03172.500.01752.750.00913.000.00443.250.0020

Comparing the IQ of two populations

One with standard IQ scores, the other with a mean 16 points higher

P1P2520.00030.0000560.00060.0000600.00110.0001640.00200.0001680.00340.0003720.00540.0006760.00810.0011800.01140.0020840.01510.0034880.01880.0054920.02200.0081960.02420.01141000.02490.01511040.02420.01881080.02200.02201120.01880.02421160.01510.02491200.01140.02421240.00810.02201280.00540.01881320.00340.01511360.00200.01141400.00110.00811440.00060.00541480.00030.00341520.00010.00201560.00010.00111600.00000.00061640.00000.00031680.00000.00011720.00000.00011760.00000.00001800.00000.0000

Normal 2Normal distributionmean150std dev25x1145x2155P(x < x1)0.4207402906P(x < x2)0.5792597094P(x1 < x < x2)0.1585194189

Log-Norm 1Log-normal DistributionLog-normal Distribution

mean000mean000stdev10.50.25stdev10.50.25xf(x)f(x)f(x)xf(x)f(x)f(x)0.020.00950.00000.0000ERROR:#NAME?0.020.00950.00000.0000ERROR:#NAME?0.040.05610.00000.00000.040.05610.00000.00000.060.12710.00000.00000.060.12710.00000.00000.080.20540.00000.00000.080.20540.00000.00000.100.28160.00020.00000.100.28160.00020.00000.120.35120.00080.00000.120.35120.00080.00000.140.41250.00250.00000.140.41250.00250.00000.160.46510.00600.00000.160.46510.00600.00000.180.50950.01240.00000.180.50950.01240.00000.200.54630.02240.00000.200.54630.02240.00000.220.57630.03700.00000.220.57630.03700.00000.240.60040.05660.00000.240.60040.05660.00000.260.61930.08140.00000.260.61930.08140.00000.280.63370.11150.00000.280.63370.11150.00000.300.64420.14650.00000.300.64420.14650.00000.320.65140.18580.00020.320.65140.18580.00020.340.65570.22890.00040.340.65570.22890.00040.360.65760.27480.00100.360.65760.27480.00100.380.65740.32280.00230.380.65740.32280.00230.400.65540.37210.00480.400.65540.37210.00480.420.65200.42170.00920.420.65200.42170.00920.440.64730.47100.01650.440.64730.47100.01650.460.64150.51930.02790.460.64150.51930.02790.480.63490.56600.04470.480.63490.56600.04470.500.62750.61050.06830.500.62750.61050.06830.520.61950.65240.10030.520.61950.65240.10030.540.61100.69140.14170.540.61100.69140.14170.560.60220.72730.19350.560.60220.72730.19350.580.59300.75990.25620.580.59300.75990.25620.600.58360.78910.32980.600.58360.78910.32980.620.57400.81480.41360.620.57400.81480.41360.640.56430.83710.50680.640.56430.83710.50680.660.55450.85590.60750.660.55450.85590.60750.680.54460.87140.71400.680.54460.87140.71400.700.53480.88380.82390.700.53480.88380.82390.720.52500.89300.93480.720.52500.89300.93480.740.51520.89941.04410.740.51520.89941.04410.760.50550.90301.14940.760.50550.90301.14940.780.49590.90411.24850.780.49590.90411.24850.800.48640.90281.33930.800.48640.90281.33930.820.47700.89931.42010.820.47700.89931.42010.840.46780.89381.48960.840.46780.89381.48960.860.45860.88651.54680.860.45860.88651.54680.880.44970.87751.59120.880.44970.87751.59120.900.44080.86711.62240.900.44080.86711.62240.920.43210.85531.64070.920.43210.85531.64070.940.42360.84231.64640.940.42360.84231.64640.960.41520.82841.64020.960.41520.82841.64020.980.40700.81351.62300.980.40700.81351.62301.000.39890.79791.59581.000.39890.79791.59581.020.39100.78161.55961.020.39100.78161.55961.040.38330.76481.51561.040.38330.76481.51561.060.37570.74761.46511.060.37570.74761.46511.080.36830.73011.40921.080.36830.73011.40921.100.36100.71231.34901.100.36100.71231.34901.120.35390.69431.28571.120.35390.69431.28571.140.34700.67631.22021.140.34700.67631.22021.160.34010.65821.15341.160.34010.65821.15341.180.33350.64011.08621.180.33350.64011.08621.200.32700.62211.01931.200.32700.62211.01931.220.32060.60430.95331.220.32060.60430.95331.240.31440.58660.88881.240.31440.58660.88881.260.30830.56910.82611.260.30830.56910.82611.280.30230.55180.76571.280.30230.55180.76571.300.29650.53480.70771.300.29650.53480.70771.320.29080.51810.65251.320.29080.51810.65251.340.28520.50170.60021.340.28520.50170.60021.360.27980.48560.55071.360.27980.48560.55071.380.27450.46980.50431.380.27450.46980.50431.400.26930.45440.46081.400.26930.45440.46081.420.26420.43940.42021.420.26420.43940.42021.440.25920.42470.38251.440.25920.42470.38251.460.25440.41040.34761.460.25440.41040.34761.480.24960.39640.31531.480.24960.39640.31531.500.24500.38290.28561.500.24500.38290.28561.520.24040.36970.25821.520.24040.36970.25821.540.23600.35680.23321.540.23600.35680.23321.560.23170.34440.21031.560.23170.34440.21031.580.22740.33230.18941.580.22740.33230.18941.600.22330.32060.17041.600.22330.32060.17041.620.21920.30920.15311.620.21920.30920.15311.640.21520.29820.13741.640.21520.29820.13741.660.21140.28760.12311.660.21140.28760.12311.680.20760.27720.11031.680.20760.27720.11031.700.20390.26730.09871.700.20390.26730.09871.720.20020.25760.08821.720.20020.25760.08821.740.19670.24830.07881.740.19670.24830.07881.760.19320.23920.07031.760.19320.23920.07031.780.18980.23050.06271.780.18980.23050.06271.800.18650.22210.05591.800.18650.22210.05591.820.18320.21400.04981.820.18320.21400.04981.840.18000.20610.04431.840.18000.20610.04431.860.17690.19860.03941.860.17690.19860.03941.880.17390.19130.03501.880.17390.19130.03501.900.17090.18420.03111.900.17090.18420.03111.920.16800.17740.02761.920.16800.17740.02761.940.16510.17090.02451.940.16510.17090.02451.960.16230.16460.02171.960.16230.16460.02171.980.15960.15850.01931.980.15960.15850.01932.000.15690.15260.01712.000.15690.15260.01712.020.15420.14700.01512.020.15420.14700.01512.040.15170.14150.01342.040.15170.14150.01342.060.14920.13630.01192.060.14920.13630.01192.080.14670.13120.01052.080.14670.13120.01052.100.14430.12640.00932.100.14430.12640.00932.120.14190.12170.00822.120.14190.12170.00822.140.13960.11720.00732.140.13960.11720.00732.160.13730.11280.00642.160.13730.11280.00642.180.13510.10860.00572.180.13510.10860.00572.200.13290.10460.00502.200.13290.10460.00502.220.13080.10070.00442.220.13080.10070.00442.240.12870.09700.00392.240.12870.09700.00392.260.12660.09340.00352.260.12660.09340.00352.280.12460.09000.00312.280.12460.09000.00312.300.12260.08660.00272.300.12260.08660.00272.320.12070.08340.00242.320.12070.08340.00242.340.11880.08030.00212.340.11880.08030.00212.360.11690.07740.00192.360.11690.07740.00192.380.11510.07450.00162.380.11510.07450.00162.400.11330.07180.00142.400.11330.07180.00142.420.11160.06910.00132.420.11160.06910.00132.440.10980.06660.00112.440.10980.06660.00112.460.10810.06410.00102.460.10810.06410.00102.480.10650.06180.00092.480.10650.06180.00092.500.10490.05950.00082.500.10490.05950.00082.520.10330.05740.00072.520.10330.05740.00072.540.10170.05530.00062.540.10170.05530.00062.560.10020.05320.00052.560.10020.05320.00052.580.09870.05130.00052.580.09870.05130.00052.600.09720.04940.00042.600.09720.04940.00042.620.09580.04760.00042.620.09580.04760.00042.640.09430.04590.00032.640.09430.04590.00032.660.09290.04420.00032.660.09290.04420.00032.680.09160.04260.00032.680.09160.04260.00032.700.09020.04110.00022.700.09020.04110.00022.720.08890.03960.00022.720.08890.03960.00022.740.08760.03820.00022.740.08760.03820.00022.760.08630.03680.00022.760.08630.03680.00022.780.08510.03550.00012.780.08510.03550.00012.800.08390.03420.00012.800.08390.03420.00012.820.08270.03300.00012.820.08270.03300.00012.840.08150.03180.00012.840.08150.03180.00012.860.08030.03070.00012.860.08030.03070.00012.880.07920.02960.00012.880.07920.02960.00012.900.07800.02850.00012.900.07800.02850.00012.920.07690.02750.00012.920.07690.02750.00012.940.07590.02650.00002.940.07590.02650.00002.960.07480.02560.00002.960.07480.02560.00002.980.07380.02470.00002.980.07380.02470.00003.000.07270.02380.00003.000.07270.02380.0000

Log-Norm 2Log-normal equivalences

xf(x)F(x)pxExcel 201030.01556456560.35842441240.050.0001052861Excel 200730.01556456560.35842441240.050.0001052861

1 Sample Z 1One-sample testing of the meanpop mean80808080pop std dev20202020sample size606060100alpha0.050.050.050.05left tail testright tail testtwo tail testtwo tail teststd error2.58198889752.58198889752.58198889752sample mean75757575p-value0.02640375570.97359624430.02640375570.0062096653crit value75.753006197284.246993802874.939394752576.0800720309sigyesnonoyes

z-score-1.9364916731-1.9364916731-1.9364916731-2.5p-value0.02640375570.97359624430.1664608040.1056497737crit value-1.6448536271.644853627-1.9599639845-1.9599639845sigyesnonoyesCalculation of confidence intervalmargin of err5.06060524753.9199279691lower69.939394752571.0800720309upper80.060605247578.9199279691Using CONFIDENCE5.06060524753.9199279691Effect sized0.25

1 Sample Z 2One-sample testing of the meanExcel 2010Real Statistics functionsExcel 2007137739911396135z test0.1351302521ERROR:#NAME?z test0.1351302521ERROR:#NAME?1118374681371077298847011698sample size48ERROR:#NAME?sample size48ERROR:#NAME?11513111363119128hyp mean100hyp mean1001387311512410197sample mean103.8125ERROR:#NAME?sample mean103.8125ERROR:#NAME?8213311113265132std dev23.9588798889ERROR:#NAME?std dev23.9588798889ERROR:#NAME?769210113411375std err3.4581664383ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?std err3.4581664383ERROR:#NAME?1009811414064133p-value0.1351302521ERROR:#NAME?p-value0.1351302521ERROR:#NAME?

alpha0.05alpha0.05margin of error6.7778816717ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?margin of error6.7778816717ERROR:#NAME?lower97.0346183283ERROR:#NAME?ERROR:#NAME?=NORM_LOWER(A3:F10,I12)lower97.0346183283ERROR:#NAME?upper110.5903816717ERROR:#NAME?ERROR:#NAME?ERROR:#NAME?upper110.5903816717ERROR:#NAME?

2 Sample ZTwo-sample testing of the meanExcel 2010

ControlNutrientControlNutrientControlNutrientz-Test: Two Sample for Means82.6789.03109.13106.1895.47117.51sample size303082.67106.1890.1194.5181.59100.86108.66115.64sample mean95.74106.6990.11100.86ControlNutrient89.2093.3294.99129.8583.3297.2289.20129.85Mean95.7373333333106.691119.1589.26101.34100.30117.64131.04pop variance256119.15100.30Known Variance25625683.01110.36104.8287.5696.90101.58alpha0.0583.0187.56Observations303093.6192.52106.9296.8766.46103.80pooled variance17.0666666667ERROR:#NAME?93.6196.87Hypothesized Mean Difference088.42112.8780.50112.5787.80111.99pooled std dev4.131182236ERROR:#NAME?88.42112.57z-2.651460536297.0264.05106.31148.36115.52119.34z-score-2.6514605362ERROR:#NAME?97.02148.36P(Z x0.1428765395

Poisson 2Poisson Distribution Iterativemean100100100100100x120130125123124prob x0.97733067090.99829315960.9932023510.98875646350.991226401using inverse functionERROR:#NAME?

RunsRunsA run if at least 6 headsA run of at least 7 headsA run of at least 60123456012345671234560000000100000000100000011000000.5110000000.51100000.51200000.250.512000000.250.5120000.250.5130000.1250.250.51300000.1250.250.513000.1250.250.514000.06250.1250.250.5140000.06250.1250.250.51400.06250.1250.250.51500.031250.06250.1250.250.515000.031250.06250.1250.250.5150.031250.06250.1250.250.5160.0156250.031250.06250.1250.250.51600.0156250.031250.06250.1250.250.5160.0468750.0781250.1406250.2656250.515625170.02343750.03906250.07031250.13281250.25781250.5078125170.00781250.0156250.031250.06250.1250.250.5170.06250.093750.156250.281250.5234375180.031250.0468750.0781250.1406250.2656250.51171875180.011718750.019531250.035156250.066406250.128906250.253906250.50390625180.0781250.1093750.1718750.292968750.53125190.03906250.05468750.08593750.14843750.2714843750.515625190.0156250.02343750.03906250.07031250.13281250.25781250.505859375190.093750.1250.1855468750.30468750.53906251100.0468750.06250.093750.15527343750.277343750.519531251100.019531250.027343750.042968750.074218750.136718750.26074218750.50781251100.1093750.13964843750.199218750.316406250.5468751110.05468750.07031250.10107421880.1621093750.2832031250.52343751110.02343750.031250.0468750.0781250.14013671880.2636718750.5097656251110.12451171880.1542968750.2128906250.3281250.55468751120.06250.07788085940.10839843750.16894531250.28906250.527343751120.027343750.035156250.050781250.08178710940.14355468750.26660156250.511718751120.13940429690.16870117190.22631835940.33959960940.56225585941130.07019042970.08544921880.11572265620.175781250.2949218750.531251130.031250.03906250.05456542970.08544921880.14697265620.269531250.5136718751130.15405273440.18286132810.23950195310.35083007810.56970214841140.07781982420.0929565430.12298583980.18255615230.30072021480.53509521481140.035156250.04290771480.05834960940.08911132810.1503906250.27246093750.5156251140.16845703120.19677734380.25244140620.36187744140.57702636721150.08538818360.1004028320.13018798830.18927001950.30645751950.53890991211150.03903198240.04675292970.06213378910.09277343750.15380859380.2753906250.5175781251150.18261718750.21044921880.26516723630.37274169920.58422851561160.09289550780.10778808590.13732910160.19592285160.31214904790.54269409181160.04289245610.05058288570.065902710.09642028810.15721130370.27830505370.51951599121160.19653320310.22389221190.27767944340.38342285160.59130859381170.10034179690.11511230470.14440917970.20252227780.31779479980.54644775391170.04673767090.0543975830.06965637210.10005187990.16059875490.28120422360.5214462281170.21021270750.23710632320.28997802730.39392089840.59826660161180.10772705080.12237548830.15143203740.20906829830.32339477540.55017089841180.0505676270.05819702150.07339477540.10366821290.16397094730.28409194950.52336883541180.22365951540.25009536740.3020668030.40423965450.60510635381190.11505126950.12957954410.15839767460.21556091310.32894897460.55386352541190.05438232420.06198120120.07711791990.10726928710.16732978820.28696823120.52528381351190.23687744140.26286315920.3139495850.41438293460.61182975771200.12231540680.1367244720.16530609130.22200012210.33445739750.55752563481200.05818176270.06575012210.08082580570.11085605620.17067527770.28983306880.52719116211

Bin Power 1Power of binomial distribution

One-tailed testPower (one-tailed)p0.35p0.35n24n24x130.05p-value0.0164185109crit12xf(x)F(x)p1-00.00003235340.00003235340.350.95774692520.042253074810.00041810610.00045045950.400.88573491770.114265082320.00258904150.00303950110.450.7579665180.24203348230.01022339480.01326289580.500.58059012890.419409871140.02890075060.04216364640.550.38490122410.615098775950.06224777050.10441141690.600.2130217990.78697820160.10614042910.2105518460.650.0942297570.90577024370.14696367110.35751551710.700.03139365410.968606345980.16816035440.52567587150.750.007199650.9928003590.16097401450.6866498860.800.00097796980.9990220302100.13001747320.81666735920.850.00005930980.9999406902110.08910288380.9057702430.900.00000084940.9999991506120.05197668220.9577469252130.02583456390.98358148910.5416666667ERROR:#NAME?ERROR:#NAME?140.01093000780.9945114969150.00392359250.9984350895160.00118839580.9996234853170.0003011320.9999246173180.00006305750.9999876748190.00001072230.9999983972200.00000144340.9999998406210.0000001480.9999999886220.00000001090.9999999995230.000000000512401

Bin Power 2Power of binomial distributionPower (two-tailed)p0.35n240.05-crit4+crit13p1-0.350.97031859330.02968140670.400.9430048630.0569951370.450.86509521920.13490478080.500.72923332450.27076667550.550.54606566940.45393433060.600.34975976020.65024023980.650.18333248130.81666751870.700.07423645740.92576354260.750.02133824160.97866175840.800.00378837810.99621162190.850.00032035730.99967964270.900.00000719430.99999280570.5416666667ERROR:#NAME?ERROR:#NAME?

Bin Power 3Sample size requirements, one-tailed testp00.35p00.35p00.35p10.80p10.80p10.80alpha0.01alpha0.01alpha0.01x-crit0.00=BINOM.INV(B7,B3,1-B5)x-crit10.00=BINOM.INV(G7,G3,1-G5)x-crit10.00=BINOM.INV(L7,L3,1-L5)nn16.030061346n161-0=1-BINOM.DIST(B6,B7,B4,TRUE)1-0.9183121115=1-BINOM.DIST(G6,G7,G4,TRUE)1-0.9183121115=1-BINOM.DIST(L6,L7,L4,TRUE)

Real Statistics Functionsone-tailERROR:#NAME?ERROR:#NAME?two-tailERROR:#NAME?ERROR:#NAME?one-tailERROR:#NAME?=BINOM_POWER(0.35,0.8,B13,1,0.01)two-tailERROR:#NAME?=BINOM_POWER(0.35,0.8,B14,2,0.01)

GammaGamma Distribution4100.25ERROR:#NAME?x3P(t Median38ERROR:#NAME?429 Median723ERROR:#NAME?452219 df23456789101112131415161718192018.92913.43716.35818.48820.15021.50422.64223.62124.47725.23725.91826.53627.10027.61828.09728.54228.95829.34729.71324.1295.7336.7727.5388.1398.6339.0499.4099.72510.00610.25910.48810.69810.89111.07011.23711.39211.53811.67633.3284.4675.1995.7386.1626.5116.8067.0627.2877.4877.6677.8317.9828.1208.2488.3688.4798.5848.68343.0153.9764.5865.0355.3885.6795.9266.1396.3276.4946.6456.7836.9097.0257.1327.2337.3267.4147.49752.8503.7174.2644.6644.9795.2385.4585.6485.8165.9656.1006.2236.3366.4396.5366.6266.7106.7886.86362.7483.5584.0654.4354.7264.9665.1685.3445.4995.6375.7625.8755.9796.0756.1646.2476.3256.3986.46672.6793.4513.9314.2804.5554.7804.9715.1375.2835.4135.5305.6375.7355.8265.9105.9886.0616.1306.19582.6303.3743.8344.1694.4314.6464.8294.9875.1265.2505.3625.4645.5585.6445.7245.7995.8695.9355.99792.5923.3163.7614.0844.3374.5454.7214.8735.0075.1265.2345.3335.4235.5065.5835.6555.7225.7865.845102.5633.2703.7044.0184.2644.4654.6364.7834.9135.0295.1345.2295.3165.3975.4725.5425.6075.6685.726112.5403.2343.6583.9654.2054.4014.5674.7114.8384.9515.0535.1455.2315.3095.3825.4505.5145.5735.630122.5213.2043.6213.9214.1564.3494.5114.6524.7764.8864.9865.0765.1605.2365.3085.3745.4365.4955.550132.5043.1793.5893.8854.1164.3044.4644.6024.7244.8324.9305.0195.1005.1755.2455.3105.3715.4295.483142.4913.1583.5633.8544.0814.2674.4244.5604.6794.7864.8824.9695.0505.1245.1925.2565.3165.3725.426152.4793.1403.5403.8284.0524.2354.3904.5244.6414.7464.8414.9275.0065.0795.1465.2095.2685.3245.376162.4693.1243.5203.8044.0264.2074.3604.4924.6084.7124.8054.8904.9685.0405.1065.1695.2275.2825.333172.4603.1103.5033.7844.0034.1824.3344.4644.5794.6814.7744.8574.9345.0055.0715.1335.1905.2445.295182.4523.0983.4873.7663.9844.1614.3104.4404.5534.6544.7464.8294.9054.9755.0405.1015.1585.2115.262192.4453.0873.4743.7513.9664.1424.2904.4184.5304.6304.7214.8034.8784.9485.0125.0725.1295.1825.232202.4393.0773.4623.7363.9504.1244.2714.3984.5104.6094.6994.7804.8554.9234.9875.0475.1035.1555.205212.4333.0693.4513.7243.9364.1094.2554.3804.4914.5904.6784.7594.8334.9014.9655.0245.0795.1315.180222.4283.0613.4413.7123.9234.0954.2394.3644.4744.5724.6604.7404.8144.8824.9445.0035.0585.1095.158232.4243.0543.4323.7013.9114.0824.2264.3504.4594.5564.6444.7234.7964.8634.9264.9845.0385.0895.138242.4203.0473.4233.6923.9004.0704.2134.3364.4454.5414.6284.7074.7804.8474.9094.9665.0205.0715.119252.4163.0413.4163.6833.8904.0594.2014.3244.4324.5284.6144.6934.7654.8314.8934.9505.0045.0555.102262.4123.0363.4093.6753.8814.0494.1914.3134.4204.5154.6014.6804.7514.8174.8784.9364.9895.0395.086272.4093.0303.4023.6673.8734.0404.1814.3024.4094.5044.5904.6674.7394.8044.8654.9224.9755.0255.072282.4063.0263.3963.6603.8654.0324.1724.2934.3994.4934.5794.6564.7274.7924.8534.9094.9625.0125.058292.4033.0213.3913.6543.8584.0244.1634.2844.3894.4844.5684.6454.7164.7814.8414.8974.9504.9995.046302.4003.0173.3863.6483.8514.0164.1554.2754.3814.4744.5594.6354.7064.7704.8304.8864.9394.9885.034312.3983.0133.3813.6423.8454.0094.1484.2684.3724.4664.5504.6264.6964.7604.8204.8764.9284.9775.023322.3963.0103.3763.6373.8394.0034.1414.2604.3654.4584.5414.6174.6874.7514.8114.8664.9184.9675.013332.3933.0063.3723.6323.8333.9974.1354.2534.3574.4504.5334.6094.6794.7434.8024.8574.9094.9575.003342.3913.0033.3683.6273.8283.9914.1294.2474.3514.4434.5264.6024.6714.7344.7944.8494.9004.9494.994352.3893.0003.3643.6233.8233.9864.1234.2414.3444.4364.5194.5944.6634.7274.7864.8414.8924.9404.986362.3882.9983.3613.6193.8193.9814.1174.2354.3384.4304.5124.5884.6564.7204.7784.8334.8844.9324.978372.3862.9953.3573.6153.8143.9764.1124.2304.3324.4244.5064.5814.6504.7134.7714.8264.8774.9254.970382.3842.9923.3543.6113.8103.9724.1074.2244.3274.4184.5004.5754.6434.7064.7654.8194.8704.9184.963392.3832.9903.3513.6083.8063.9674.1034.2204.3224.4134.4954.5694.6374.7004.7584.8124.8634.9114.956402.3812.9883.3483.6053.8023.9634.0994.2154.3174.4084.4904.5644.6324.6944.7524.8064.8574.9044.949482.3722.9733.3303.5833.7783.9374.0704.1854.2854.3754.4554.5284.5954.6564.7134.7664.8164.8634.907602.3632.9593.3123.5623.7553.9114.0424.1554.2544.3424.4214.4934.5584.6194.6754.7274.7754.8214.864802.3532.9453.2943.5413.7313.8854.0144.1254.2234.3094.3874.4574.5214.5814.6364.6874.7354.7804.8221202.3442.9303.2763.5203.7073.8593.9864.0964.1914.2764.3534.4224.4854.5434.5974.6474.6944.7384.7792402.3352.9163.2583.4993.6843.8343.9594.0664.1604.2444.3194.3864.4484.5054.5584.6074.6534.6964.737Inf2.3262.9023.2403.4783.6613.8083.9314.0374.1294.2114.2854.3514.4124.4684.5194.5684.6124.6544.694Alpha0.05k 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