density() — Density (KDE)

A smooth curve showing where data is crowded.

plot(density(x))

Key idea

• A density plot is a smoothed version of a histogram (a bar chart of how often values occur), drawn as a flowing curve.
• The total area under the curve always equals 1, because it covers all the data.
• The area over a range gives the proportion of values in that range. (e.g. 0.30 of data between 5 and 10)
• Taller parts of the curve show where values are most concentrated. (e.g. peak near 7)
• How smooth the curve is depends on the bandwidth — a smoothing dial, like bin width for a histogram.

Definition

$$\hat{f}(x)=\frac{1}{nh}\sum _{i=1}^{n}K\left(\frac{x-x_i}{h}\right)$$

$x_i$

each data point

$n$

how many points

$h$

the bandwidth (smoothing width)

$K$

the kernel — a little bump at each point

$\hat{f}(x)$

the estimated density at x

In plain words

Like a histogram — a bar chart of how often values occur — with the bumps sanded down into one smooth hill.

Where you'd use it

• You have 1000 incomes and do not want to fuss with histogram bin widths. A density curve draws one smooth hill that peaks near the typical income, say $40k, with a long tail stretching toward high earners.
• To compare two towns’ ages, overlay two density curves on one chart; if one peaks at 25 and the other at 50, you instantly see one town skews younger.
• A density plot of exam scores that rises to a single smooth peak around 72 but has a small second bump near 40 reveals a struggling subgroup hidden in the data.

Frequently asked questions

What controls the smoothness?

The bandwidth — too wide hides peaks, too narrow adds spikes.