# Shading regions of the normal: The Stanine scale

June 26, 2012
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For the presentation of norm values, often stanines are used (standard 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:

?View Code RSPLUS

 ```# 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)```

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

?View Code RSPLUS

 ```# 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)) }```

And finally: add some legends to the plot:

?View Code RSPLUS

 ```# 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)```

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

?View Code RSPLUS

 ```# draw the normal curve curve(dnorm(x,0,1), xlim=c(-3,3), main="Normal density")   # define shaded region from.z <- -3 to.z <- qnorm(.025)   S.x <- c(from.z, seq(from.z, to.z, 0.01), to.z) S.y <- c(0, dnorm(seq(from.z, to.z, 0.01)), 0) polygon(S.x,S.y, col="red")```

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