# What does the y axis in a kernel density plot mean? [duplicate]

Possible Duplicate:
Probability distribution value exceeding 1 is OK?

I thought the area under the curve of a density function represents the probability of getting an x value between a range of x values, but then how can the y-axis be greater than 1 when I make the bandwidth small? See this R plot:

range <- seq(2,6,.01)
n <- 1000
d <- sample(range,n, replace=TRUE)
d <- c(d,rep(0,100))
d <- c(d,rep(1,50))
df <- data.frame(counts=d)
plot(dens)


Also, the probability of getting $P(x<2)=\frac{150}{1000}=.15$, how can I see this in the plot?

• Consider a uniform density on $(0,0.1)$. What's the height of the density in that range? Density is not probability. Commented Jan 20, 2013 at 6:17
• Density is "unit probability". Commented Jun 11, 2014 at 11:10

But remember area is not just height: width is also important. So if you have a spike at 0, if the width is very small (say 0.1) then the height can be quite a bit higher than 1 (up to 10, if the spike is perfectly rectangular, since $0.1\times10 = 1$) without violating any rules of probability. The height of the spike is large, but the area under the spike is still quite small.
I missed your second question initially, but $P(x<2)=\frac{150}{1000}=.15$ means that the area under the curve to the left of 2 (i.e., the area of the two spikes at 0 and 1, more or less) is .15.
• Thanks, that makes sense. In R I can do library(pracma);trapz(dens$x[dens$x < 2], dens$y[dens$x < 2]) and I get .139. Close enough Commented Jan 20, 2013 at 2:23