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Text of the page (random words):
alculators see all calculators n sample size power analysis determine optimal sample size for your study n normality test test if your data follows a normal distribution q tukey s hsd test post hoc test for anova comparisons z interactive z table standard normal distribution lookup f frequency table frequency distributions and histograms σ bell curve generator customized normal distribution curves trusted across academia 20 000 monthly users cited in 15 peer reviewed papers apa 7 formatted output nih pmc springer elsevier wiley bmj arxiv biorxiv eu jrc hypothesis testing calculators choose the right statistical test for your data our interactive selector helps you find the appropriate test based on your research question and data characteristics or browse the comprehensive table of tests with their assumptions and formulas prefer a visual guide view our interactive flow chart test table test selector test name check test statistic assumptions normality test parametric diagnostic shapiro wilk k s anderson darling text shapiro wilk k s anderson darling shapiro wilk k s anderson darling independence outlier detection parametric diagnostic grubbs dixon s q isolation forest text grubbs dixon s q isolation forest grubbs dixon s q isolation forest independence ancova parametric mean f m s a d j m s e a d j f frac ms_ adj mse_ adj f ms e a d j m s a d j normality independence equal variance linearity of covariate homogeneity of regression dunnett s test parametric mean d x ˉ i x ˉ c 2 m s e n d frac bar x _i bar x _c sqrt 2mse n d 2 mse n x ˉ i x ˉ c normality independence equal variance friedman test non parametric mean χ 2 d f 12 b k k 1 r j 2 3 b k 1 chi 2 df frac 12 bk k 1 sum r_j 2 3b k 1 χ 2 df bk k 1 12 r j 2 3 b k 1 dependent groups ordinal data games howell test parametric mean q x ˉ i x ˉ j s i 2 n i s j 2 n j q frac bar x _i bar x _j sqrt frac s_i 2 n_i frac s_j 2 n_j q n i s i 2 n j s j 2 x ˉ i x ˉ j normality independence equal variance kruskal wallis test non parametric mean h 12 n n 1 i 1 k r i 2 n i 3 n 1 h frac 12 n n 1 sum_ i 1 k frac r_i 2 n_i 3 n 1 h n n 1 12 i 1 k n i r i 2 3 n 1 independence ordinal data manova parametric mean λ e h e lambda frac mathbf e mathbf h mathbf e λ h e e normality independence equal variance one sample z test parametric mean z x ˉ μ 0 σ n z frac bar x mu_0 sigma sqrt n z σ n x ˉ μ 0 normality independence known σ one sample t test parametric mean t x ˉ μ 0 s n t frac bar x mu_0 s sqrt n t s n x ˉ μ 0 normality independence known σ one way anova parametric mean f m s g m s e f frac msg mse f mse msg normality independence equal variance paired t test parametric mean t d ˉ s d n t frac bar d s_d sqrt n t s d n d ˉ normality paired data permutation test non parametric mean p i 1 n i t i t o b s n p frac sum_ i 1 n i t _i geq t_ obs n p n i 1 n i t i t o b s independence normality equal variance repeated measures anova parametric mean f m s g m s e f frac msg mse f mse msg normality dependent groups sphericity scheffé test parametric mean f s x ˉ i x ˉ j 2 m s e 1 n i 1 n j f_s frac bar x _i bar x _j 2 mse frac 1 n_i frac 1 n_j f s mse n i 1 n j 1 x ˉ i x ˉ j 2 normality independence equal variance three way anova parametric mean f m s g m s e f frac msg mse f mse msg normality independence equal variance tukey s hsd test parametric mean q x ˉ i x ˉ j m s e n q frac bar x _i bar x _j sqrt mse n q mse n x ˉ i x ˉ j normality independence equal variance two sample z test parametric mean z x ˉ 1 x ˉ 2 μ 1 μ 2 σ 1 2 n 1 σ 2 2 n 2 z frac bar x _1 bar x _2 mu_1 mu_2 sqrt frac sigma_1 2 n_1 frac sigma_2 2 n_2 z n 1 σ 1 2 n 2 σ 2 2 x ˉ 1 x ˉ 2 μ 1 μ 2 normality independence known σ two sample t test pooled variance parametric mean t x ˉ 1 x ˉ 2 s p 1 n 1 1 n 2 t frac bar x _1 bar x _2 s_p sqrt frac 1 n_1 frac 1 n_2 t s p n 1 1 n 2 1 x ˉ 1 x ˉ 2 normality independence known σ equal variance two sample t test welch s parametric mean t x ˉ 1 x ˉ 2 s 1 2 n 1 s 2 2 n 2 t frac bar x _1 bar x _2 sqrt frac s_1 2 n_1 frac s_2 2 n_2 t n 1 s 1 2 n 2 s 2 2 x ˉ 1 x ˉ 2 normality independence known σ equal variance two way anova parametric mean f m s g m s e f frac msg mse f mse msg normality independence equal variance welch s anova parametric mean w w i x ˉ i μ 2 k 1 1 2 k 2 k 2 1 1 w i w j 2 n i 1 w frac sum w_i bar x _i hat mu 2 k 1 1 frac 2 k 2 k 2 1 sum frac 1 w_i sum w_j 2 n_i 1 w 1 k 2 1 2 k 2 n i 1 1 w i w j 2 w i x ˉ i μ 2 k 1 normality independence equal variance chi square goodness of fit test parametric proportion χ 2 o e 2 e chi 2 sum frac o e 2 e χ 2 e o e 2 normality independence chi square test of independence parametric proportion χ 2 o e 2 e chi 2 sum frac o e 2 e χ 2 e o e 2 normality independence fisher s exact test non parametric proportion p a b c d a c b d n a b c d p frac a b c d a c b d n a b c d p n a b c d a b c d a c b d independence fixed row column totals one sample proportion test parametric proportion z p p 0 p 0 1 p 0 n z frac hat p p_0 sqrt frac p_0 1 p_0 n z n p 0 1 p 0 p p 0 independence binomial data two sample proportion test parametric proportion z p 1 p 2 p 1 p 1 n 1 1 n 2 z frac hat p _1 hat p _2 sqrt hat p 1 hat p frac 1 n_1 frac 1 n_2 z p 1 p n 1 1 n 2 1 p 1 p 2 independence binomial data dunn s test non parametric rank z i j r ˉ i r ˉ j n n 1 12 1 n i 1 n j z_ ij frac bar r _i bar r _j sqrt frac n n 1 12 frac 1 n_i frac 1 n_j z ij 12 n n 1 n i 1 n j 1 r ˉ i r ˉ j independence ordinal data mann whitney u test non parametric rank z u μ u σ u z frac u mu_u sigma_u z σ u u μ u independence ordinal data wilcoxon signed rank test non parametric rank z w μ w σ w z frac w mu_w sigma_w z σ w w μ w paired data ordinal data descriptive statistics calculators choose from our comprehensive collection of descriptive statistics calculators for both quantitative and qualitative data analysis quantitative data qualitative data central tendency calculator formula description mean x ˉ i 1 n x i n bar x frac sum_ i 1 n x_i n x ˉ n i 1 n x i calculate arithmetic average of numerical data median middle value when ordered text middle value when ordered middle value when ordered find the middle value in ordered data mode most frequent value text most frequent value most frequent value identify most common value s in dataset geometric mean x 1 x 2 x n n sqrt n x_1 times x_2 times times x_n n x 1 x 2 x n calculate mean for multiplicative relationships harmonic mean n i 1 n 1 x i frac n sum_ i 1 n frac 1 x_i i 1 n x i 1 n calculate mean for rates and speeds variability calculator formula description standard deviation s x i x ˉ 2 n 1 s sqrt frac sum x_i bar x 2 n 1 s n 1 x i x ˉ 2 measure average deviation from mean mean absolute deviation mad x i x ˉ n text mad frac sum x_i bar x n mad n x i x ˉ measure average deviation from mean variance s 2 x i x ˉ 2 n 1 s 2 frac sum x_i bar x 2 n 1 s 2 n 1 x i x ˉ 2 measure spread of data points range max x min x text max x text min x max x min x calculate difference between largest and smallest values iqr q 3 q 1 q_3 q_1 q 3 q 1 calculate spread of middle 50 of data coefficient of variation c v s x ˉ 100 cv frac s bar x times 100 c v x ˉ s 100 compare variability between datasets position calculator formula description percentiles p k value at k th percentile p_k text value at k text th percentile p k value at k th percentile find value at specified percentile z score z x x ˉ s z frac x bar x s z s x x ˉ calculate standardized scores shape calculator formula description skewness x i x ˉ 3 n 1 s 3 frac sum x_i bar x 3 n 1 s 3 n 1 s 3 x i x ˉ 3 measure asymmetry of distribution kurtosis x i x ˉ 4 n 1 s 4 frac sum x_i bar x 4 n 1 s 4 n 1 s 4 x i x ˉ 4 measure tailedness of distribution relationships calculator formula description correlation coefficient r x i x ˉ y i y ˉ x i x ˉ 2 y i y ˉ 2 r frac sum x_i bar x y_i bar y sqrt sum x_i bar x 2 sum y_i bar y 2 r x i x ˉ 2 y i y ˉ 2 x i x ˉ y i y ˉ measure linear relationship strength with pearson s r probability distribution calculators access our suite of probability distribution calculators for both discrete and continuous random variables calculate probabilities find critical values and visualize distributions discrete distributions continuous distributions common discrete distribution probability function description binomial p x k n k p k 1 p n k p x k binom n k p k 1 p n k p x k k n p k 1 p n k model number of successes in fixed trials poisson p x k λ k e λ k p x k frac lambda k e lambda k p x k k λ k e λ model rare events in fixed interval geometric p x k p 1 p k 1 p x k p 1 p k 1 p x k p 1 p k 1 model trials until first success negative binomial p x k k 1 r 1 p r 1 p k r p x k binom k 1 r 1 p r 1 p k r p x k r 1 k 1 p r 1 p k r model trials until r successes hypergeometric p x k k k n k n k n n p x k frac binom k k binom n k n k binom n n p x k n n k k n k n k model sampling without replacement chart makers choose the right visualization based on your data type and analysis goals our tools help you create clear effective visual representations of your data one variable two variables multiple variables specialized categorical data chart type description best used for example bar chart display frequencies or counts for categories comparing categories showing distributions product sales by category pie chart show part to whole relationships displaying proportions percentages market share by company funnel chart visualize conversion rates and stages in a process process flows conversion funnels sales funnel conversion rates tree map display hierarchical data using nested rectangles hierarchical composition proportional sizes market share by product category waterfall chart visualize cumulative values gains and losses financial analysis breakdown visualization profit and loss breakdown numerical data chart type description best used for example histogram display distribution of continuous data examining data distribution shape age distribution of customers box plot show data distribution and outliers identifying outliers comparing distributions test scores distribution violin plot combine box plot with kernel density detailed view of data distribution income distribution by department dot plot visualize individual data points and their distribution small datasets simple distributions daily temperature readings stem and leaf plot show distribution while preserving individual values small to medium datasets retaining data values test scores with individual values density plot visualize data distribution using kernel density estimation smooth distribution visualization income distribution analysis bell curve graph generator visualize normal distribution with mean and standard deviation normal distribution visualization iq score distribution q q plot assess normality and compare data distributions testing normality assumptions checking if data follows normal distribution confidence interval calculators choose from our comprehensive collection of confidence interval calculators for estimating population parameters and analyzing differences between groups single parameter comparisons relationships population parameters interval type description best used for example mean estimate population mean continuous data normal distribution average customer spending margin of error proportion estimate population proportion binary outcomes categorical data customer satisfaction rate margin of error standard deviation estimate population variability process variation quality control manufacturing tolerance limits regression analysis calculators choose from our collection of regression analysis tools for modeling relationships between variables and making predictions from your data linear regression model name description example simple linear regression model relationship between two continuous variables height vs weight relationship multiple linear regression model with multiple predictors house price prediction using area location age non linear regression model name description example quadratic regression model curved relationships with quadratic terms projectile motion optimal pricing models exponential regression model exponential growth or decay patterns population growth radioactive decay compound interest classification models model name description example logistic regression model binary outcomes customer churn prediction limited dependent variable models model name description example censored regression tobit fit tobit models for censored continuous outcomes hours worked when many observations are zero multivariate analysis calculators analyze multiple variables simultaneously with our multivariate tools reduce dimensions classify groups and quantify relationships across variable sets dimensionality reduction method description example principal component analysis pca reduce dimensionality by finding directions of maximum variance compress 50 survey items into 3 latent dimensions exploratory factor analysis identify latent factors that explain observed correlations discover underlying personality traits from items confirmatory factor analysis test a hypothesized factor structure against your data confirm a 3 factor structure for a published instrument classification method description example discriminant analysis lda qda classify observations and project to maximize group separation predict species from morphological measurements relationships method description example canonical correlation analysis examine relationships between two sets of variables relate test battery to job performance metrics survival analysis calculators analyze time to event data model survival curves compare groups and quantify covariate effects on hazard rates with apa formatted output time to event analysis method description example kaplan meier survival analysis estimate survival probabilities over time with censoring patient survival after treatment over 5 years log rank test compare survival curves between two or more groups new drug vs standard treatment survival cox proportional hazards regression model hazard rates with covariates and produce hazard ratios effect of age and stage on cancer survival statistical resources access our collection of statistical tables interactive simulations and reference materials to support your statistical analysis statistical tables simulations data tools distribution tables resource description z table standard normal distribution critical values for z tests and standardized scores t table student s t distribution critical values ideal for small sample tests with unknown population σ chi square table chi square distribution critical values used in goodness of fit and independence tests f table f distribution critical values for anova and variance comparisons wilcoxon signed rank table critical values for wilcoxon signed rank test and paired sample comparisons analyze your data with confidence fast accurate statistical analysis trusted by students and 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