Kurtosis (incomplete)

Kurtosis refers to how influenced a distribution is by its tails.

\(kurtosis=\frac{(N*(N+1)*m4 - 3*m2^2*(w-1))}{((N-1)*(N-2)*(N-3)*s1^4)}\)

\(kurtosis_{SE} = sqrt( 4*(w^2-1) * sdskew^2 / ((w-3)*(w+5)))\)

spssSkewKurtosis=function(x) {
  w=length(x)
  m1=mean(x)
  m2=sum((x-m1)^2)
  m3=sum((x-m1)^3)
  m4=sum((x-m1)^4)
  s1=sd(x)
  skew=w*m3/(w-1)/(w-2)/s1^3
  sdskew=sqrt( 6*w*(w-1) / ((w-2)*(w+1)*(w+3)) )
  kurtosis=(w*(w+1)*m4 - 3*m2^2*(w-1)) / ((w-1)*(w-2)*(w-3)*s1^4)
  sdkurtosis=sqrt( 4*(w^2-1) * sdskew^2 / ((w-3)*(w+5)) )
  
  ## z-scores added by reading-psych
  zskew = skew/sdskew
  zkurtosis = kurtosis/sdkurtosis

  mat=matrix(c(skew,kurtosis, sdskew,sdkurtosis, zskew, zkurtosis), 2,
        dimnames=list(c("skew","kurtosis"), c("estimate","se","zScore")))
  return(mat)
}

Is Platykurtic vs. leptokurtic data more sensitive to false positives!

or overly clustered around the mean (leptokurtik)

# 
# library(ggplot2)
# # https://stackoverflow.com/a/12429538
# norm_x<-seq(-4,4,0.01)
# norm_y<-dnorm(-4,4,0.0)/2
# 
# norm_data_frame<-data.frame(x=norm_x,y=dnorm(norm_x,0,1))
# 
# 
# shade_2.3 <- rbind(
#   c(-8,0), 
#   subset(norm_data_frame, x > -8), 
#   c(norm_data_frame[nrow(norm_data_frame), "X"], 0))
# 
# shade_13.6 <- rbind(
#   c(-2,0), 
#   subset(norm_data_frame, x > -2), 
#   c(norm_data_frame[nrow(norm_data_frame), "X"], 0))
# 
# shade_34.1 <- rbind(
#   c(-1,0), 
#   subset(norm_data_frame, x > -1), 
#   c(norm_data_frame[nrow(norm_data_frame), "X"], 0))
# 
# shade_50 <- rbind(
#   c(0,0), 
#   subset(norm_data_frame, x > 0), 
#   c(norm_data_frame[nrow(norm_data_frame), "X"], 0))
# 
# shade_84.1 <- rbind(
#   c(1,0), 
#   subset(norm_data_frame, x > 1), 
#   c(norm_data_frame[nrow(norm_data_frame), "X"], 0))
# 
# 
# shade_97.7 <- rbind(
#   c(2,0), 
#   subset(norm_data_frame, x > 2), 
#   c(norm_data_frame[nrow(norm_data_frame), "X"], 0))
# 
# 
# p<-qplot(
#   x=norm_data_frame$x,
#   y=norm_data_frame$y,
#   geom="line"
# )
# 
#  p +
#    geom_polygon(
#      data = shade_2.3,
#      aes(
#        x,
#        y,
#        fill="2.3"
#       )
#     ) +
#    geom_polygon(
#      data = shade_13.6,
#      aes(
#        x,
#        y,
#        fill="13.6"
#       )
#     ) +
#    geom_polygon(
#      data = shade_34.1,
#      aes(
#        x,
#        y,
#        fill="34.1"
#       )
#     ) +
#    geom_polygon(
#      data = shade_50,
#      aes(
#        x,
#        y,
#        fill="50"
#       )
#     ) +
#    geom_polygon(
#      data = shade_84.1,
#      aes(
#        x,
#        y,
#        fill="84.1"
#       )
#     ) +
#    geom_polygon(
#      data = shade_97.7, 
#      aes(
#        x, 
#        y,
#        fill="97.7"
#       )
#     ) +
#    xlim(c(-4,4)) +
#    
#    annotate("text", x=-2.3, y=0.01, label= "13.6%") + 
#    annotate("text", x=-1.4, y=0.01, label= "34.1%") + 
#    annotate("text", x=-0.3, y=0.01, label= "50%") + 
#    annotate("text", x=0.5, y=0.01, label= "84.1%") + 
#    annotate("text", x=1.5, y=0.01, label= "97.7%") + 
#    annotate("text", x=2.3, y=0.01, label= "100%") +
#    xlab("Z-score") +
#    ylab("Frequency") +
#    theme(legend.position="none")

or underly clustered around the mean (platykurtik)