Can kernel density be more than 1?

[Guest]

Dear Stata experts,

when drawing a histogram and kernel density I see density bigger than 1. I was googling it and didn't find an answer. Is it all good or it is something wrong with my data?

[Hemanshu Kumar]

The area under a density curve is a probability. So that area can never be less than 0 or greater than 1. But it is entirely possible for a density itself to be greater than 1, or -- at least for singleton points -- even be negative.

Take a look at the density function of the normal distribution with standard deviation 0.1:

twoway function normalden(x,0,0.1), range(-0.3 0.3)

[Nick Cox]

An easy example, extending Hemanshu Kumar's helpful reply, is a uniform distribution over the interval [0, 0.5] for which the density must be constant at 2 for the probability to integrate to 1.

[Nick Cox]

David Finney's paper on dimensions of statistics in Applied Statistics (1977) remains as relevant as when written. You may have access via https://www.jstor.org/stable/pdf/2346969.pdf
Finney emphasises that probability density has dimensions of the reciprocal of the variable in question, and units correspondingly.

[Guest]

Thank you Hemanshu and Nick for your helpful reply!