For any given vs , the p value of any given point x is , Where

-- See R. Weber's Statistics Note (Chap 6.2 & 7.1)

I made some wrong comment on the pdf Null Ritual (Gigerenzer, Krauss, & Vitouch, 2004) Where three types of significance level (rather than p value) were discussed. I had written the comment to note that the chapter had ignored the role of in definition of p value. In almost every textbook, the two-tail p vs single-tail p are differentiated. Usually, the two-tail p is defined by like .

Here I demonstrate a three-tail p value case on R platform.

z=(-1000:1000)*0.02;

f=0.5 * dchisq(abs(z),df=5);

h=dchisq(10,df=5)*.5;

plot(z,f,type="h",col=c("black","grey")[1+(f>h)]);

lines(c(-20,20),c(h,h));

## is * binomial(-1 vs 1) ##

Do you agree the region nearby zero under the "V" curve (which is below the horizontal line) should be the 3rd tail? I think so, if only includes all other possible distributions in the same shape.

You'll also agree there will be two asymmetrical tails if includes just two asymmetrical curves, for example, and () while is the standardized normal distribution.

However, the 3rd tail is still a freak even if there were two peaks in the population density.In psychometrics, judgment is rarely based just one single sample. With a realistic sample size, the density of calculated statistics like mean and variance would easily average into a single peak.