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grouped data wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 example 2 mean of grouped data 3 see also 4 references toggle the table of contents grouped data 4 languages فارسی ਪੰਜਾਬੀ simple english українська edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia organized raw data that has not been otherwise processed or transformed grouped data are data formed by aggregating individual observations of a variable into groups so that a frequency distribution of these groups serves as a convenient means of summarizing or analyzing the data there are two major types of grouping data binning of a single dimensional variable replacing individual numbers by counts in bins and grouping multi dimensional variables by some of the dimensions especially by independent variables obtaining the distribution of ungrouped dimensions especially the dependent variables example edit the idea of grouped data can be illustrated by considering the following raw dataset table 1 time taken in seconds by a group of students to answer a simple math question 20 25 24 33 13 26 8 19 31 11 16 21 17 11 34 14 15 21 18 17 the above data can be grouped in order to construct a frequency distribution in any of several ways one method is to use intervals as a basis the smallest value in the above data is 8 and the largest is 34 while the sample mean amounts to 19 7 seconds the interval from 8 to 34 is broken up into smaller subintervals called class intervals for each class interval the number of data items falling in this interval is counted this number is called the frequency of that class interval the results are tabulated as a frequency table as follows table 2 frequency distribution of the time taken in seconds by the group of students to answer a simple math question time taken in seconds frequency 5 t 10 1 10 t 15 4 15 t 20 6 20 t 25 4 25 t 30 2 30 t 35 3 another method of grouping the data is to use some qualitative characteristics instead of numerical intervals for example suppose in the above example there are three types of students 1 below normal if the response time is 5 to 14 seconds 2 normal if it is between 15 and 24 seconds and 3 above normal if it is 25 seconds or more then the grouped data looks like table 3 frequency distribution of the three types of students frequency below normal 5 normal 10 above normal 5 yet another example of grouping the data is the use of some commonly used numerical values which are in fact names we assign to the categories for example let us look at the age distribution of the students in a class the students may be 10 years old 11 years old or 12 years old these are the age groups 10 11 and 12 note that the students in age group 10 are from 10 years and 0 days to 10 years and 364 days old and their average age is 10 5 years old if we look at age in a continuous scale the grouped data looks like table 4 age distribution of a class of students age frequency 10 10 11 20 12 10 mean of grouped data edit an estimate x displaystyle bar x of the mean of the population from which the data are drawn can be calculated from the grouped data as x f x f displaystyle bar x frac sum f x sum f in this formula x refers to the midpoint of the class intervals and f is the class frequency note that the result of this will be different from the sample mean of the ungrouped data the mean for the grouped data in the above example can be calculated as follows class intervals frequency f midpoint x f x 5 and above below 10 1 7 5 7 5 10 t 15 4 12 5 50 15 t 20 6 17 5 105 20 t 25 4 22 5 90 25 t 30 2 27 5 55 30 t 35 3 32 5 97 5 total 20 405 thus the mean of the grouped data is x f x f 405 20 20 25 displaystyle bar x frac sum f x sum f frac 405 20 20 25 the mean for the grouped data in example 4 above can be calculated as follows age group frequency f midpoint x f x 10 10 10 5 105 11 20 11 5 230 12 10 12 5 125 total 40 460 thus the mean of the grouped data is x f x f 460 40 11 5 displaystyle bar x frac sum f x sum f frac 460 40 11 5 see also edit aggregate data censoring statistics data binning partition of a set level of measurement frequency distribution discretization of continuous features logistic regression minimum chi squared estimator for grouped data this article includes a list of general references but lacks sufficient corresponding inline citations please help improve this article by introducing more precise citations june 2010 learn how and when to remove this message references edit newbold p carlson w thorne b 2009 statistics for business and economics seventh ed pearson education isbn 978 0 13 507248 6 v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduction dimensionality reduction principal component analysis factor analysis time series preprocessing differencing detrending seasonal adjustment stationarity transformation data collection study design effect size missing data optimal design population replication sample size determination statistic statistical power survey methodology sampling cluster stratified opinion poll questionnaire standard error controlled experiments blocking factorial experiment interaction random assignment randomized controlled trial randomized experiment scientific control adaptive designs adaptive clinical trial stochastic approximation up and down designs observational studies cohort study cross sectional study natural experiment quasi experiment statistical inference statistical theory population statistic probability distribution sampling distribution order statistic empirical distribution density estimation statistical model model specification l p space parameter location scale shape parametric family likelihood monotone location scale family exponential family completeness sufficiency statistical functional bootstrap u v optimal decision loss function efficiency statistical distance divergence asymptotics robustness sensitivity analysis frequentist inference point estimation estimating equations maximum likelihood method of moments m estimator minimum distance unbiased estimators mean unbiased minimum variance rao blackwellization lehmann scheffé theorem median unbiased plug in interval estimation confidence interval pivot likelihood interval prediction interval tolerance interval resampling bootstrap jackknife testing hypotheses 1 2 tails power uniformly most powerful test permutation test randomization test multiple comparisons parametric tests likelihood ratio g test score lagrange multiplier wald z test normal specific tests parametric student s t test f test goodness of fit chi squared kolmogorov smirnov anderson darling lilliefors jarque bera normality shapiro wilk model selection cross validation aic bic rank statistics sign sample median signed rank wilcoxon hodges lehmann estimator rank sum mann whitney nonparametric anova 1 way kruskal wallis 2 way friedman ordered alternative jonckheere terpstra van der waerden test bayesian inference bayesian probability prior posterior credible interval bayes factor bayesian estimator maximum posterior estimator correlation regression analysis correlation pearson product moment partial correlation confounding variable coefficient of determination regression analysis errors and residuals regression validation mixed effects models simultaneous equations models multivariate adaptive regression splines mars template least squares and regression analysis linear regression simple linear regression ordinary least squares general linear model bayesian regression non standard predictors nonlinear regression nonparametric semiparametric isotonic robust homoscedasticity and heteroscedasticity generalized linear model exponential families logistic bernoulli binomial poisson regressions partition of variance analysis of variance anova analysis of variance ancova manova degrees of freedom categorical multivariate time series survival analysis categorical cohen s kappa contingency table graphical model log linear model mcnemar s test cochran mantel haenszel statistics multivariate regression manova principal components canonical correlation discriminant analysis cluster analysis classification structural equation model factor analysis multivariate distributions elliptical distributions normal time series general decomposition trend stationarity seasonal adjustment exponential smoothing cointegration structural break granger causality specific tests dickey fuller johansen q statistic ljung box durbin watson breusch godfrey time domain autocorrelation acf partial pacf cross correlation xcf arma model arima model box jenkins autoregressive conditional heteroskedasticity arch vector autoregression var autoregressive model ar frequency domain spectral density estimation fourier analysis least squares spectral analysis wavelet whittle likelihood survival survival function kaplan meier estimator product limit proportional hazards models accelerated failure time aft model first hitting time hazard function nelson aalen estimator test log rank test applications biostatistics bioinformatics clinical trials studies 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