June 26, 2012

For the presentation of norm values, often stanines are used (**stan**dard **nine**). These values mark a person’s relativ position in comparison to the sample or to norm values.

According to Wikipedia:

The underlying basis for obtaining stanines is that a normal distribution is divided into nine intervals, each of which has a width of 0.5 standard deviations excluding the first and last, which are just the remainder (the tails of the distribution). The mean lies at the centre of the fifth interval.

For illustration purposes, I wanted to plot the regions of the stanine values in the standard normal distribution – here’s the result:

*First: Calculate the stanine boundaries and draw the normal curve:*

# First: Calculate stanine breaks (on a z scale)

stan.z <- c(-3, seq(-1.75, +1.75, length.out=8), 3)

# Second: get cumulative probabilities for these z values

stan.PR <- pnorm(stan.z)

# define a color ramp from blue to red (... or anything else ...)

c_ramp <- colorRamp(c("darkblue", "red"), space="Lab")

# draw the normal curve, without axes; reduce margins on left, top, and right

par(mar=c(2,0,0,0))

curve(dnorm(x,0,1), xlim=c(-3,3), ylim=c(-0.03, .45), xlab="", ylab="", axes=FALSE)

stan.z <- c(-3, seq(-1.75, +1.75, length.out=8), 3)

# Second: get cumulative probabilities for these z values

stan.PR <- pnorm(stan.z)

# define a color ramp from blue to red (... or anything else ...)

c_ramp <- colorRamp(c("darkblue", "red"), space="Lab")

# draw the normal curve, without axes; reduce margins on left, top, and right

par(mar=c(2,0,0,0))

curve(dnorm(x,0,1), xlim=c(-3,3), ylim=c(-0.03, .45), xlab="", ylab="", axes=FALSE)

*Next: Calculate the shaded regions and plot a polygon for each region:*

# Calculate polygons for each stanine region

# S.x = x values of polygon boundary points, S.y = y values

for (i in 1:(length(stan.z)-1)) {

S.x <- c(stan.z[i], seq(stan.z[i], stan.z[i+1], 0.01), stan.z[i+1])

S.y <- c(0, dnorm(seq(stan.z[i], stan.z[i+1], 0.01)), 0)

polygon(S.x,S.y, col=rgb(c_ramp(i/9), max=255))

}

# S.x = x values of polygon boundary points, S.y = y values

for (i in 1:(length(stan.z)-1)) {

S.x <- c(stan.z[i], seq(stan.z[i], stan.z[i+1], 0.01), stan.z[i+1])

S.y <- c(0, dnorm(seq(stan.z[i], stan.z[i+1], 0.01)), 0)

polygon(S.x,S.y, col=rgb(c_ramp(i/9), max=255))

}

*And finally: add some legends to the plot:*

# print stanine values in white

# font = 2 prints numbers in boldface

text(seq(-2,2, by=.5), 0.015, label=1:9, col="white", font=2)

# print cumulative probabilities in black below the curve

text(seq(-1.75,1.75, by=.5), -0.015, label=paste(round(stan.PR[-c(1, 10)], 2)*100, "%", sep=""), col="black", adj=.5, cex=.8)

text(0, -0.035, label="Percentage of sample <= this value", adj=0.5, cex=.8)

# font = 2 prints numbers in boldface

text(seq(-2,2, by=.5), 0.015, label=1:9, col="white", font=2)

# print cumulative probabilities in black below the curve

text(seq(-1.75,1.75, by=.5), -0.015, label=paste(round(stan.PR[-c(1, 10)], 2)*100, "%", sep=""), col="black", adj=.5, cex=.8)

text(0, -0.035, label="Percentage of sample <= this value", adj=0.5, cex=.8)

And finally, here’s a short script for shading only one region (e.g., the lower 2.5%):

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