::Free Statistics and Forecasting Software::

All rights reserved. The non-commercial (academic) use of this software is free of charge. The only thing that is asked in return is to cite this software when results are used in publications.

This free online software (calculator) computes the Box-Cox Normality Plot. This analysis identifies the lambda (Box-Cox parameter) value that results in the (quasi-)optimal fit against the normal distribution. The software uses two computational algorithms to find the value for lambda. The first method maximizes the correlation from the normal probability plot (for all values between a user-specified minimum and maximum). The second method uses Maximum Likelihood Estimation (without limitation) and a Likelihood Ratio test against the nulhypotheses: lambda = 0 and lambda = 1.

Enter (or paste) your data delimited by hard returns.

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Source code of R module

library(car) par2 <- abs(as.numeric(par2)*100) par3 <- as.numeric(par3)*100 if(par4=="") par4 <- 0 par4 <- as.numeric(par4) numlam <- par2 + par3 + 1 x <- x + par4 n <- length(x) c <- array(NA,dim=c(numlam)) l <- array(NA,dim=c(numlam)) mx <- -1 mxli <- -999 for (i in 1:numlam) { l[i] <- (i-par2-1)/100 if (l[i] != 0) { if (par1 == "Full Box-Cox transform") x1 <- (x^l[i] - 1) / l[i] if (par1 == "Simple Box-Cox transform") x1 <- x^l[i] } else { x1 <- log(x) } c[i] <- cor(qnorm(ppoints(x), mean=0, sd=1),sort(x1)) if (mx < c[i]) { mx <- c[i] mxli <- l[i] x1.best <- x1 } } print(c) print(mx) print(mxli) print(x1.best) if (mxli != 0) { if (par1 == "Full Box-Cox transform") x1 <- (x^mxli - 1) / mxli if (par1 == "Simple Box-Cox transform") x1 <- x^mxli } else { x1 <- log(x) } mypT <- powerTransform(x) summary(mypT) bitmap(file="test1.png") plot(l,c,main="Box-Cox Normality Plot", xlab="Lambda",ylab="correlation") mtext(paste("Optimal Lambda =",mxli)) grid() dev.off() bitmap(file="test2.png") hist(x,main="Histogram of Original Data",xlab="X",ylab="frequency") grid() dev.off() bitmap(file="test3.png") hist(x1,main="Histogram of Transformed Data", xlab="X",ylab="frequency") grid() dev.off() bitmap(file="test4.png") qqPlot(x) grid() mtext("Original Data") dev.off() bitmap(file="test5.png") qqPlot(x1) grid() mtext("Transformed Data") dev.off() load(file="createtable") a<-table.start() a<-table.row.start(a) a<-table.element(a,"Box-Cox Normality Plot",2,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,"# observations x",header=TRUE) a<-table.element(a,n) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,"maximum correlation",header=TRUE) a<-table.element(a,mx) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,"optimal lambda",header=TRUE) a<-table.element(a,mxli) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,"transformation formula",header=TRUE) if (par1 == "Full Box-Cox transform") { a<-table.element(a,"for all lambda <> 0 : T(Y) = (Y^lambda - 1) / lambda") } else { a<-table.element(a,"for all lambda <> 0 : T(Y) = Y^lambda") } a<-table.row.end(a) if(mx<0) { a<-table.row.start(a) a<-table.element(a,"Warning: maximum correlation is negative! The Box-Cox transformation must not be used.",2) a<-table.row.end(a) } a<-table.end(a) table.save(a,file="mytable.tab") if(par5=="Yes") { a<-table.start() a<-table.row.start(a) a<-table.element(a,"Obs.",header=T) a<-table.element(a,"Original",header=T) a<-table.element(a,"Transformed",header=T) a<-table.row.end(a) for (i in 1:n) { a<-table.row.start(a) a<-table.element(a,i) a<-table.element(a,x[i]) a<-table.element(a,x1.best[i]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file="mytable1.tab") } a<-table.start() a<-table.row.start(a) a<-table.element(a,"Maximum Likelihood Estimation of Lambda",1,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,paste("<pre>",RC.texteval("summary(mypT)"),"</pre>",sep="")) a<-table.row.end(a) a<-table.end(a) table.save(a,file="mytable3.tab")

Cite this software as:

Wessa P., (2021), Box-Cox Normality Plot (v1.1.13) in Free Statistics Software (v1.2.1), Office for Research Development and Education, URL http://www.wessa.net/rwasp_boxcoxnorm.wasp/

The R code is based on :

NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, 2006-10-03.

Box, G. E. P. and Cox, D. R. (1964) An analysis of transformations. Journal of the Royal Statisistical Society, Series B. 26 211-46.

Cook, R. D. and Weisberg, S. (1999) Applied Regression Including Computing and Graphics. Wiley

Fox, J. and Weisberg, S. (2011) An R Companion to Applied Regression, Second Edition, Sage.

Velilla, S. (1993) A note on the multivariate Box-Cox transformation to normality. Statistics and Probability Letters, 17, 259-263.

Weisberg, S. (2014) Applied Linear Regression, Fourth Edition, Wiley.

Yeo, I. and Johnson, R. (2000) A new family of power transformations to improve normality or symmetry. Biometrika, 87, 954-959.

To cite Wessa.net in publications use:
Wessa, P. (2024), Free Statistics Software, Office for Research Development and Education,
version 1.2.1, URL https://www.wessa.net/

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Software Version : 1.2.1
Algorithms & Software : Patrick Wessa, PhD
Server : supernova.wessa.net

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