Differences in Distribution

Simulating Data

• y is simulated normal distribution with a mean of 0 and sd 1.
• y0 is a highly skewed distribution you might see if teachers do not grade to a curve. The mean is also 0 and the sd also 1.

library(tidyverse)
library(fGarch)
library(psych)
library(gridExtra)
y = rnorm(1000)
yo = fGarch::rsnorm(1000,xi=-6)
d <- data.frame(y=y,yo=yo)


hist_top <- ggplot(d,aes(y))+geom_histogram()
hist_right <- ggplot(d, aes(yo))+geom_histogram()
gridExtra::grid.arrange(hist_top,  hist_right, ncol=2)

Distributions Qualities

d %>%
summarise(across(.fns = list(mean = mean, sd = sd, skew = skew), .names = "{col}_{fn}")) %>%
  mutate_all(.funs = ~round(., 2)) %>%
   rownames_to_column %>% 
   gather(var, value, -rowname) %>% 
   spread(rowname, value) %>%
  tidyr::separate(var, into = c("Variable", "Statistic")) %>%
  setNames(c("Variable","Statistic","Value")) %>%
  knitr::kable()

Variable	Statistic	Value
y	mean	-0.03
y	sd	1.04
y	skew	0.02
yo	mean	0.03
yo	sd	0.98
yo	skew	-0.94

KS Test

ks.test(y, yo)

## 
##  Two-sample Kolmogorov-Smirnov test
## 
## data:  y and yo
## D = 0.115, p-value = 3.61e-06
## alternative hypothesis: two-sided

d %>% 
  pivot_longer(cols = everything()) %>%
  ggplot(aes(value, group = name, color = name)) +
  stat_ecdf()

Relative Polarity

Note that the Skewed Variable has significantly more values in the upper tale.

r <- reldist::reldist(y=yo, yo=y,
        bar=TRUE,     
        ci=TRUE,
        smooth=0.1,                              
        yolabs=seq(-1, 3, by=0.5),                 
        ylim=c(0, 3.0),cex=0.8,                   
        ylab="Relative Density",                 
        xlab="Proportion of the Normal Variable")

tribble(
  ~ Polarity, ~ Estimate,  ~ `CI-95%`, ~ `CI+95%`,
  "Median Polarity", r$rp[2],r$rp[1],r$rp[3],
  "Lower Tail Polarity", r$rpl[2],r$rpl[1],r$rpl[3],
  "Upper Tail Polarity", r$rpu[2],r$rpu[1],r$rpu[3]
) %>%
  knitr::kable(digits = 2)

Polarity	Estimate	CI-95%	CI+95%
Median Polarity	0.79	0.75	0.82
Lower Tail Polarity	-0.33	-0.40	-0.26
Upper Tail Polarity	0.81	0.72	0.91