# R Dataset / Package HistData / Macdonell

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On this Picostat.com statistics page, you will find information about the Macdonell data set which pertains to Macdonell's Data on Height and Finger Length of Criminals, used by Gosset (1908). The Macdonell data set is found in the HistData R package. You can load the Macdonell data set in R by issuing the following command at the console data("Macdonell"). This will load the data into a variable called Macdonell. If R says the Macdonell data set is not found, you can try installing the package by issuing this command install.packages("HistData") and then attempt to reload the data. If you need to download R, you can go to the R project website. You can download a CSV (comma separated values) version of the Macdonell R data set. The size of this file is about 19,647 bytes. ## Macdonell's Data on Height and Finger Length of Criminals, used by Gosset (1908)## DescriptionIn the second issue of W. S. Gosset (aka "Student") used these data in two classic papers in 1908, in which he derived various characteristics of the sampling distributions of the mean, standard deviation and Pearson's r. He said, "Before I had succeeded in solving my problem analytically, I had endeavoured to do so empirically." Among his experiments, he randomly shuffled the 3000 observations from Macdonell's table, and then grouped them into samples of size 4, 8, ..., calculating the sample means, standard deviations and correlations for each sample. ## Usagedata(Macdonell) data(MacdonellDF) ## Format
`height` lower class boundaries of height, in decimal ft. `finger` length of the left middle finger, in mm. `frequency` frequency of this combination of `height` and`finger`
`height` a numeric vector `finger` a numeric vector
## DetailsClass intervals for For convenience, the data frame ## SourceMacdonell, W. R. (1902).
On Criminal Anthropometry and the Identification of Criminals.
The data used here were obtained from: Hanley, J. (2008). Macdonell data used by Student. http://www.medicine.mcgill.ca/epidemiology/hanley/Student/ ## ReferencesHanley, J. and Julien, M. and Moodie, E. (2008).
Student's z, t, and s: What if Gosset had R?
Gosett, W. S. [Student] (1908).
Probable error of a mean.
Gosett, W. S. [Student] (1908).
Probable error of a correlation coefficient.
## Examplesdata(Macdonell)# display the frequency table xtabs(frequency ~ finger+round(height,3), data=Macdonell)## Some examples by james.hanley@mcgill.ca October 16, 2011 ## http://www.biostat.mcgill.ca/hanley/ ## See: http://www.biostat.mcgill.ca/hanley/Student/############################################### ## naive contour plots of height and finger ## ############################################### # make a 22 x 42 table attach(Macdonell) ht <- unique(height) fi <- unique(finger) fr <- t(matrix(frequency, nrow=42)) detach(Macdonell) dev.new(width=10, height=5) # make plot double wide op <- par(mfrow=c(1,2),mar=c(0.5,0.5,0.5,0.5),oma=c(2,2,0,0))dx <- 0.5/12 dy <- 0.5/12plot(ht,ht,xlim=c(min(ht)-dx,max(ht)+dx), ylim=c(min(fi)-dy,max(fi)+dy), xlab="", ylab="", type="n" )# unpack 3000 heights while looping though the frequencies heights <- c() for(i in 1:22) { for (j in 1:42) { f <- fr[i,j] if(f>0) heights <- c(heights,rep(ht[i],f)) if(f>0) text(ht[i], fi[j], toString(f), cex=0.4, col="grey40" ) } } text(4.65,13.5, "Finger length (cm)",adj=c(0,1), col="black") ; text(5.75,9.5, "Height (feet)", adj=c(0,1), col="black") ; text(6.1,11, "Observed bin\nfrequencies", adj=c(0.5,1), col="grey40",cex=0.85) ; # crude countour plot contour(ht, fi, fr, add=TRUE, drawlabels=FALSE, col="grey60") # smoother contour plot (Galton smoothed 2-D frequencies this way) # [Galton had experience with plotting isobars for meteorological data] # it was the smoothed plot that made him remember his 'conic sections' # and ask a mathematician to work out for him the iso-density # contours of a bivariate Gaussian distribution... dx <- 0.5/12; dy <- 0.05 ; # shifts caused by averagingplot(ht,ht,xlim=c(min(ht),max(ht)),ylim=c(min(fi),max(fi)), xlab="", ylab="", type="n" ) sm.fr <- matrix(rep(0,21*41),nrow <- 21) for(i in 1:21) { for (j in 1:41) { smooth.freq <- (1/4) * sum( fr[i:(i+1), j:(j+1)] ) sm.fr[i,j] <- smooth.freq if(smooth.freq > 0 ) text(ht[i]+dx, fi[j]+dy, sub("^0.", ".",toString(smooth.freq)), cex=0.4, col="grey40" ) } } contour(ht[1:21]+dx, fi[1:41]+dy, sm.fr, add=TRUE, drawlabels=FALSE, col="grey60") text(6.05,11, "Smoothed bin\nfrequencies", adj=c(0.5,1), col="grey40", cex=0.85) ; par(op) dev.new() # new default device####################################### ## bivariate kernel density estimate #######################################if(require(KernSmooth)) { MDest <- bkde2D(MacdonellDF, bandwidth=c(1/8, 1/8)) contour(x=MDest$x1, y=MDest$x2, z=MDest$fhat, xlab="Height (feet)", ylab="Finger length (cm)", col="red", lwd=2) with(MacdonellDF, points(jitter(height), jitter(finger), cex=0.5)) }############################################################# ## sunflower plot of height and finger with data ellipses ## #############################################################with(MacdonellDF, { sunflowerplot(height, finger, size=1/12, seg.col="green3", xlab="Height (feet)", ylab="Finger length (cm)") reg <- lm(finger ~ height) abline(reg, lwd=2) if(require(car)) { dataEllipse(height, finger, plot.points=FALSE, levels=c(.40, .68, .95)) } }) ############################################################ ## Sampling distributions of sample sd (s) and z=(ybar-mu)/s ############################################################# note that Gosset used a divisor of n (not n-1) to get the sd. # He also used Sheppard's correction for the 'binning' or grouping. # with concatenated height measurements...mu <- mean(heights) ; sigma <- sqrt( 3000 * var(heights)/2999 ) c(mu,sigma)# 750 samples of size n=4 (as Gosset did)# see Student's z, t, and s: What if Gosset had R? # [Hanley J, Julien M, and Moodie E. The American Statistician, February 2008] # see also the photographs from Student's notebook ('Original small sample data and notes") # under the link "Gosset' 750 samples of size n=4" # on website http://www.biostat.mcgill.ca/hanley/Student/ # and while there, look at the cover of the Notebook containing his yeast-cell counts # http://www.medicine.mcgill.ca/epidemiology/hanley/Student/750samplesOf4/Covers.JPG # (Biometrika 1907) and decide for yourself why Gosset, when forced to write under a # pen-name, might have taken the name he did!# PS: Can you figure out what the 750 pairs of numbers signify? # hint: look again at the numbers of rows and columns in Macdonell's (frequency) Table III. n <- 4 Nsamples <- 750y.bar.values <- s.over.sigma.values <- z.values <- c() for (samp in 1:Nsamples) { y <- sample(heights,n) y.bar <- mean(y) s <- sqrt( (n/(n-1))*var(y) ) z <- (y.bar-mu)/s y.bar.values <- c(y.bar.values,y.bar) s.over.sigma.values <- c(s.over.sigma.values,s/sigma) z.values <- c(z.values,z) } op <- par(mfrow=c(2,2),mar=c(2.5,2.5,2.5,2.5),oma=c(2,2,0,0)) # sampling distributions hist(heights,breaks=seq(4.5,6.5,1/12), main="Histogram of heights (N=3000)") hist(y.bar.values, main=paste("Histogram of y.bar (n=",n,")",sep=""))hist(s.over.sigma.values,breaks=seq(0,4,0.1), main=paste("Histogram of s/sigma (n=",n,")",sep="")); z=seq(-5,5,0.25)+0.125 hist(z.values,breaks=z-0.125, main="Histogram of z=(ybar-mu)/s") # theoretical lines(z, 750*0.25*sqrt(n-1)*dt(sqrt(n-1)*z,3), col="red", lwd=1) par(op)##################################################### ## Chisquare probability plot for bivariate normality #####################################################mu <- colMeans(MacdonellDF) sigma <- var(MacdonellDF) Dsq <- mahalanobis(MacdonellDF, mu, sigma)Q <- qchisq(1:3000/3000, 2) plot(Q, sort(Dsq), xlab="Chisquare (2) quantile", ylab="Squared distance") abline(a=0, b=1, col="red", lwd=2) -- Dataset imported from https://www.r-project.org. |

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dataset-25306.csv | 19.19 KB |