The density returns a number that in itself does not translate directly into a probability. But it gives the height of a curve that, if drawn over the full range of possible numbers, has the area underneath it that adds up to 1.
Consider this. If I make the vector x of evenly spaced numbers from -7.5 to 7.5, 0.1 apart, and get the density of a normal variable with mean 0 and standard deviation 2.5 for each value of x.
x <- seq(from = -7.5, to = 7.55, by = 0.1)
y <- dnorm(x, 0, 2.5)
The approximate value of the area under the curve formed by those densities (which I have stored as y), multiplied by their distance apart (0.1) is nearly 1:
> sum(y * 0.1)
[1] 0.9974739
If you did this properly with calculus rather than approximating it with numbers, it would be exactly one.
Why is this useful? The cumulative area under parts of the curve can be used to estimate the probability of the variable coming anywhere in a particular range, even though as one of your sources points out, the chance of any precise number is technically zero for a continuous variable.
Consider this graphic. The area of the shaded space shows the chance of a variable from your normal distribution (mean zero, standard deviation 2.5) being between -7.5 and 4. This leads to many useful applications.
Made with:
library(ggplot2)
d <- data.frame(x, y)
ggplot(d, aes(x = x, y = y)) +
geom_line() +
geom_point() +
geom_ribbon(fill = "steelblue", aes(ymax = y), ymin = 0, alpha = 0.5, data = subset(d, x <= 4)) +
annotate("text", x= -4, y = 0.13, label = "Each point is an individual density\nestimate of dnorm(x, 0, 2.5)") +
annotate("text", x = -.3, y = 0.02, label = "Filled area under the curve shows the cumulative probability\nof getting a number as high as a given x, in this case 4") +
ggtitle("Density of a random normal variable with mean zero and standard deviation 2.5")
