Skewness Practice with Quantile-Quantile Plots
Skewness Determination Exercise.
In this exercise, the computer will generate 3 datasets: A, B and C. These will be randomly assigned to a positively skewed, negatively skewed, and symmetric distribution type. The data (sorted in order) are plotted on the Y axis and the quantiles of a standard normal (qnorm) distribution are plotted on the X axis. A normal distribution should appear as a straight line; positively skewed distributions will yield a ‘U’ shaped curve, and negatively skewed distributions a ‘C’ shaped curve. (An ‘S’ or ‘Z’ shaped curve indicates kurtosis, not skewness). Note: SPSS plots the normal quantiles on the Y axis and the data on the X: Which means that leptokurtic and platykurtic distributions will curve in the opposite direction from these Q-Q plots.
You can redraw from the same distributions by changing the sample size. (Bigger sample sizes are easier.)
#| standalone: true
#| viewerHeight: 500
library(lattice)
library(shiny)
distlist <-list(
skewNeg = list("beta(8,2)"=function(n) rbeta(n,8,2),
"normal with neg outliers"=function (n) {
ifelse(runif(n)<.05,rnorm(n,-3),rnorm(n))
},
"hypergeometric(975,25,100)" =
function (n) rhyper(n,975,25,100)),
skewPos = list("gamma(3)"=function(n) rgamma(n,3),
"normal with positive outliers"=function(n) {
ifelse(runif(n)<.05,rnorm(n,3),rnorm(n))
},
"lognormal"=function (n) rlnorm(n,0,.3)),
sym = list("normal"=rnorm, "uniform"=runif,
"t(5 d.f.)"=function (n) rt(n,5)))
longnames <- c("Negatively Skewed"="skewNeg",
"Positively Skewed"="skewPos",
"Symmetric"="sym")
## Initial draw, so that we have some starting values.
key <-
{
## Randomly permute the types.
key <- sample(names(distlist),length(distlist))
## Label from A -- C (or whatever)
names(key) <- sapply(1L:length(key),
function (i)
intToUtf8(utf8ToInt("A")-1L+i))
key
}
kdist <-
{
# draw random distribution for each plot
sapply(key, function (r)
sample(names(distlist[[r]]),1L))
}
ui <- fluidPage(
inputPanel(
selectInput("nn", label = "Sample Size:",
choices = c(50, 100, 500, 1000), selected = 500)),
mainPanel(
plotOutput("QQplots")),
h4("Which is which?"),
p("Identify the skewness of each distribution."),
do.call(inputPanel,
lapply(names(key), function (k)
selectInput(k, label=k,
choices=c(Unknown="unknown",
longnames),
selected="unknown"))),
h4("Answers:\n"),
tableOutput("answers"))
server <- function (input,output) {
output$QQplots <- renderPlot({
## Draw random data
print(key)
kdat <- lapply(names(key), function (k) {
x <-do.call(distlist[[key[k]]][[kdist[k]]],
list(input$nn))
scale(x,(min(x)+max(x))/2,(max(x)-min(x)))*100+50
})
names(kdat) <- names(key)
kdat <-
data.frame(dat=do.call(c,kdat),
group=rep(names(key),
each=input$nn))
qqmath(~dat|group, data = kdat,
layout=c(3,1),horizontal=FALSE,
panel=function(x,...) {
panel.qqmathline(x,...)
panel.qqmath(x,...)
})
})
output$answers <- renderTable({
answer <- sapply(names(key),
function (k) {
if (input[[k]]=="unknown") {
"Make your selection.\n"
} else {
paste(ifelse(input[[k]]==key[k],
"Correct:", "Incorrect:"),
"Distribution was",kdist[k],
"(",
names(longnames)[grep(key[k],longnames)],
")\n")
}})
names(answer) <- names(key)
as.data.frame(answer)
}, colnames=FALSE,rownames=TRUE)
}
shinyApp(ui=ui,server=server)To try again with different distributions, reload the page. If you are having trouble, try increasing the sample size: sometimes a small sample won’t display the characteristics of the distribution strongly.
Here are the other exercises in this series:
- Skewness Practice:
- Kurtosis Practice: