Reputation: 25
Displaying area under the curve between two z values in standard normal distribution
I am trying to draw a sketch something like this:
where I can denote the area under curve left/right of the z score value in standard normal distribution I am trying to make a similar graph but this time want to display all areas in -2.25<Z<1.77 with two lines perhaps? I have searched so many places but cannot seem to find an answer and mosaic::xpnorm() seems to be the closest one but I don't believe it has an option for that.
library(mosaic)
xpnorm(1.77, mean=0, sd=1)
Upvotes: 0
Views: 2399
Answers (2)
Reputation: 101
I'm not sure if I've completely understood what you want. If you are looking for a way to shade the area under a normal curve between two z scores, then you could use a function I wrote, expanding a function from https://gist.github.com/jrnold/6799152.
You can use the function normal_prob_area_plot_p_value() to shade the area beyond your Z scores:
normal_prob_area_plot_p_val <- function(lb, ub,
mean = 0, sd = 1,
limits = c(mean - 3 * sd, mean + 3 * sd),
type = "lower_upper_bounds") {
if (type == "lower_upper_bounds") {
# X axis
x <- seq(limits[1],
limits[2],
length.out = 100)
# Define the lower and upper bound
xmin <- max(lb, limits[1])
xmax <- min(ub, limits[2])
# Define the lower tail area x axis
areax1 <- seq(limits[1],
xmin,
length.out = 100)
# Define the upper tail area x axis
areax2 <- seq(limits[2], xmax, length.out = 100)
# Calculate densities for the areas
area1 <- data.frame(x = areax1,
ymin = 0,
ymax = dnorm(areax1, mean, sd))
area2 <- data.frame(x = areax2,
ymin = 0,
ymax = dnorm(areax2, mean, sd))
return(ggplot()
# Normal pdf representation
+ geom_line(data.frame(x = x,
y = dnorm(x, mean, sd)),
mapping = aes(x = x, y = y))
# Lower tail area
+ geom_ribbon(data = area1,
mapping = aes(x = x,
ymin =
ymin,
ymax =
ymax),
fill = "dodgerblue", alpha=0.4)
# Upper tail area
+ geom_ribbon(data = area2,
mapping = aes(x = x,
ymin = ymin,
ymax = ymax),
fill = "dodgerblue", alpha=0.4)
# X axis
+ scale_x_continuous(limits = limits)
+ labs(y="Density", x="z-score") +
theme_bw())
}
# The same than before, but now
# only matters the lower tail
else if (type == "lower_bound") {
x <- seq(limits[1],
limits[2],
length.out = 100)
xmin <- max(lb, limits[1])
xmax <- min(ub, limits[2])
areax1 <- seq(limits[1],
xmin,
length.out = 100)
areax2 <- seq(limits[2],
xmax,
length.out = 100)
area1 <- data.frame(x = areax1,
ymin = 0,
ymax = dnorm(areax1, mean, sd))
area2 <- data.frame(x = areax2,
ymin = 0,
ymax = dnorm(areax2, mean, sd))
return(ggplot()
+ geom_line(data.frame(x = x,
y = dnorm(x, mean, sd)),
mapping = aes(x = x,
y = y))
+ geom_ribbon(data = area1,
mapping = aes(x = x,
ymin = ymin,
ymax = ymax),
fill = "dodgerblue", alpha=0.4)
+ scale_x_continuous(limits = limits)
+ labs(y="Density", x="z-score") +
theme_bw())
}
# Only matters the upper tail
else if (type == "upper_bound") {
x <- seq(limits[1],
limits[2],
length.out = 100)
xmin <- max(lb, limits[1])
xmax <- min(ub, limits[2])
areax1 <- seq(limits[1],
xmin,
length.out = 100)
areax2 <- seq(limits[2],
xmax,
length.out = 100)
area1 <- data.frame(x = areax1,
ymin = 0,
ymax = dnorm(areax1, mean, sd))
area2 <- data.frame(x = areax2,
ymin = 0,
ymax = dnorm(areax2, mean, sd))
return(ggplot()
+ geom_line(data.frame(x = x,
y = dnorm(x, mean, sd)),
mapping = aes(x = x,
y = y))
+ geom_ribbon(data = area2,
mapping = aes(x = x,
ymin = ymin,
ymax = ymax),
fill = "dodgerblue", alpha=0.4)
+ scale_x_continuous(limits = limits)
+ labs(y="Density", x="z-score") +
theme_bw())
}
# Shade the inner zone between the bounds
else if (type=="inner") {
# X axis
x <- seq(limits[1],
limits[2],
length.out = 100)
# Define the lower and upper bound
xmin <- lb
xmax <- ub
# Define the area to be shaded
areax <- seq(xmin,
xmax,
length.out = 100)
# Calculate densities for the areas
area <- data.frame(x = areax,
ymin = 0,
ymax = dnorm(areax, mean, sd))
return(ggplot()
# Normal pdf representation
+ geom_line(data.frame(x = x,
y = dnorm(x, mean, sd)),
mapping = aes(x = x, y = y))
# Area
+ geom_ribbon(data = area,
mapping = aes(x = x,
ymin = ymin,
ymax = ymax),
fill = "dodgerblue", alpha=0.4)
# X axis
+ scale_x_continuous(limits = limits)
+ labs(y="Density", x="z-score") +
theme_bw())
}
}
In your case, the usage would be:
normal_prob_area_plot_p_val(-2.25, 1.77)
And that's what you would get:
However, given this function, it's easy to shift the shaded area. If you actually want to shade the area between the z scores, then you can use:
normal_prob_area_plot_p_val(-2.25, 1.77, type="inner")
You would get:
Upvotes: 0
Reputation: 73272
We can use polygon() to color areas under a curve. polygon() needs a two-dimensional coordinates matrix of the points of the polygon line where it draws straight lines between these points. Actually their number is infinity, but the function will smooth sufficiently above a certain number, i.e. we want a sequence of length, say, 200. For the coordinates matrix we want the x values and their corresponding y=dnorm(x) values.
# given z values
z.rg <- c(-2.25, 1.77)
# define cuts and canonical z-scores
cuts <- sort(sort(c(z.rg, -3.5, 3.5))) # c(-3.5, 3.5) will be the xlim of the plot
x.sq <- seq(cuts[1], cuts[4], len=200)
alpha <- c(.001, .01, .05)
z <- c(qnorm(alpha/2), 0, abs(qnorm(alpha/2)))
# plot
plot(x.sq, dnorm(x.sq), type="l", xlab="z-score", ylab="density",
main="Standard Normal Distribution", xaxt="n")
z <- c(qnorm(alpha/2), 0, abs(qnorm(alpha/2)))
axis(1, z, round(z, 2))
abline(h=0)
# random middle part (optional)
polygon(c(cuts[2], cuts[2], x.sq[x.sq > cuts[2] & x.sq < cuts[3]], cuts[3], cuts[3]),
c(0, dnorm(cuts[2]), dnorm(x.sq[x.sq > cuts[2] & x.sq < cuts[3]]), dnorm(cuts[3]), 0),
col="lightgrey", border=1)
# left tail
polygon(c(cuts[3], cuts[3], x.sq[x.sq > cuts[3]], cuts[4], cuts[4]),
c(0, dnorm(cuts[3]), dnorm(x.sq[x.sq > cuts[3]]), dnorm(cuts[4]), 0),
col="#4da6ff", border=1)
# right tail
polygon(c(cuts[1], cuts[1], x.sq[x.sq < cuts[2]], cuts[2], cuts[2]),
c(0, dnorm(cuts[1]), dnorm(x.sq[x.sq < cuts[2]]), dnorm(cuts[2]), 0),
col="#4da6ff", border=4)
# labels
arrows(z.rg, rep(dnorm(z.rg), length(z.rg)), z.rg, rep(dnorm(0)*.666, length(z.rg)),
length=0, lty=2, col=4)
sapply(z.rg, function(x) text(x, dnorm(0)*.666 + .02, bquote(italic("z=")~.(x)), col=4))
Upvotes: 1
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