You may be confusing population quantiles with the sample
quantiles that estimate them. Your population quantiles are
appropriately represented in your figures.
Population quantiles. If random variable $X \sim \mathsf{Norm}(\mu = 100, \sigma = 15),$
then quantiles $.01, .05, .25, .50, .95, .99$ of the distribution can be found in R by using the quantile function qnorm. (The quantile function is sometimes called the 'inverse CDF` function.)
q = round(qnorm(c(.01,.05,.25,.50,.75,.95,.99), 100, 15),3); q
[1] 65.105 75.327 89.883 100.000 110.117 124.673 134.895
These quantiles (at vertical lines) can be displayed along with the density function of $\mathsf{Norm}(100, 15)$ as shown in the graph below.
curve(dnorm(x, 100, 15), 50, 150, col="blue", lwd=2, ylab="PDF",
main="Density of NORM(100, 15) with Various Quantiles")
abline(h=0, col="green2"); abline(v=0, col="green2")
abline(v=q, col="red", lty="dotted", lwd=2)
The total area (representing probability) under the density curve is $1.$
Areas to the left of the three left-most vertical lines are $.01,.05,$ and $.25,$ respectively.
Sample quantiles. If I have a sufficiently large sample from
this distribution, then I can find the quantiles of the sample.
For example, the 50th sample percentile (quantile .5) is the sample median. These sample quantiles estimate the corresponding
population percentiles. Generally speaking, larger samples give better estimates. I will use $n = 1000$ in my example.
set.seed(2020) # for reproducibility
x = round(rnorm(1000, 100, 15), 3)
Here are some summary statistics of the sample, including the
sample first quartile (quantile .25), the sample median,
and the sample third quartile (quantile .75). The boxplot uses
the quartiles [upper and lower edges of the box]and the median [center line inside box], so we show it also.
summary(x)
Min. 1st Qu. Median Mean 3rd Qu. Max.
52.51 89.30 99.14 99.60 109.58 155.54
boxplot(x, col="skyblue2", horizontal=T,
main="n=1000; Boxplot of Sample from NORM(100,15)")
Without extra arguments, the R procedure quantile shows the maximum and minimum values in the sample and the three quantiles
shown in the summary.
quantile(x)
0% 25% 50% 75% 100%
52.50800 89.30475 99.13750 109.57850 155.54300
In order to get our full list of quantiles, we need to specify them individually.
samp.q = quantile(x, c(.01,.05,.25,.50,.75,.95,.99)); samp.q
1% 5% 25% 50% 75% 95% 99%
63.76255 74.46450 89.30475 99.13750 109.57850 126.38775 136.60263
In particular, notice that population quantile .05 (which is
$75.327$ from earlier) is estimated by the sample quantile .05 (which is $74.465$ just above).
Finally, we show a histogram of the $n=1000$ observations along
with the population density curve. Now the vertical dotted lines
show the positions of our chosen sample quantiles.
hist(x, prob=T, col="skyblue2", main="Histogram of Sample")
curve(dnorm(x, 100, 15), add=T, col="blue", lwd=2)
abline(v=samp.q, col="purple", lty="dotted", lwd=2)
Numbers of observations at or to the left of the three left-most vertical
lines are $10, 50,$ and $250,$ respectively, out of $1000.$
Note: All of the above is about quantiles for a normal distribution because your question deals only with normal distributions. But @Nick Cox makes a good point that quantiles are used similarly for other distributions. For example, here is a plot of an exponential distribution that has rate $\lambda = 0.1$ (hence mean $\mu = 10),$ with vertical lines at the same quantiles used above for the normal distribution.
q = round(qexp(c(.01,.05,.25,.50,.75,.95,.99), 0.1),3); q
[1] 0.101 0.513 2.877 6.931 13.863 29.957 46.052
curve(dexp(x, 0.1), 0, 60, col="blue", lwd=2, ylab="PDF", n=10001,
main="Density of EXP(mean=10) with Various Quantiles")
abline(h=0, col="green2"); abline(v=0, col="green2")
abline(v=q, col="red", lty="dotted", lwd=2)
