geom_density y-axis goes above 1

Question

I think this might partly be an R question and partly a statistics question, so please excuse me if there is a better place for it (if so, please let me know where).

Let's say I have a dataset my_measurements like this:

> glimpse(my_measurements)
Observations: 200
Variables: 2
$ sample_id <int> 18, 22, 30, 59, 74, 126, 133, 137, 147, 186, 189, 195, 203, 248, 294, 303, 320, 324, 353, 3...
$ value     <dbl> 0.9565217, 1.0000000, 0.7500000, 0.7142857, 1.0000000, 0.8571429, 1.0000000, 1.0000000, 0.8...

Where each sample_id has a corresponding measurement of something which gives a value between 0 and 1 (so e.g. they might be proportions of something).

The full dput() output for it is:

my_measurements <- tibble::tibble(
  sample_id = c(
    18L, 22L, 30L, 59L, 74L, 126L, 133L, 137L, 147L, 186L, 189L,
    195L, 203L, 248L, 294L, 303L, 320L, 324L, 353L, 375L, 384L, 385L,
    395L, 400L, 401L, 411L, 459L, 468L, 479L, 482L, 497L, 502L, 528L,
    556L, 576L, 601L, 640L, 657L, 659L, 674L, 687L, 688L, 709L, 711L,
    716L, 737L, 744L, 771L, 784L, 791L, 793L, 794L, 813L, 845L, 854L,
    864L, 866L, 887L, 891L, 899L, 915L, 917L, 919L, 934L, 948L, 969L,
    975L, 980L, 998L, 1006L, 1011L, 1015L, 1021L, 1036L, 1047L, 1056L,
    1062L, 1073L, 1074L, 1082L, 1087L, 1101L, 1102L, 1108L, 1113L,
    1119L, 1130L, 1160L, 1175L, 1176L, 1179L, 1187L, 1188L, 1206L,
    1224L, 1227L, 1411L, 1412L, 1431L, 1472L, 1481L, 1485L, 1488L,
    1491L, 1501L, 1519L, 1531L, 1534L, 1537L, 1559L, 1579L, 1592L,
    1603L, 1608L, 1629L, 1643L, 1684L, 1721L, 1726L, 1736L, 1744L,
    1756L, 1778L, 1800L, 1807L, 1813L, 1829L, 1839L, 1901L, 1905L,
    1926L, 1975L, 1980L, 2004L, 2006L, 2019L, 2062L, 2069L, 2079L,
    2087L, 2091L, 2116L, 2123L, 2141L, 2147L, 2159L, 2160L, 2163L,
    2168L, 2173L, 2191L, 2194L, 2208L, 2214L, 2231L, 2244L, 2246L,
    2253L, 2273L, 2290L, 2291L, 2302L, 2318L, 2326L, 2353L, 2371L,
    2372L, 2388L, 2412L, 2415L, 2423L, 2443L, 2451L, 2452L, 2468L,
    2470L, 2472L, 2481L, 2485L, 2502L, 2503L, 2504L, 2521L, 2572L,
    2601L, 2621L, 2625L, 2635L, 2643L, 2644L, 2674L, 2698L, 2710L,
    2723L, 2742L, 2757L, 2794L, 2824L, 2835L, 2837L
  ),
  value = c(
    0.956521739130435, 1, 0.75, 0.714285714285714, 1, 0.857142857142857, 1, 1,
    0.869565217391304, 0, 0.892857142857143, 0.9, 1, 0.892857142857143, 1, 1, 0,
    0.883333333333333, 1, 0.976190476190476, 0.973684210526316, 0.914285714285714,
    1, 0.6, 0.6, 1, 0.931818181818182, 1, 0.882352941176471, 0.75, 1, 1, 1,
    0.826086956521739, 1, 0.8, 0.75, 1, 0.931034482758621, 1, 1,
    0.980769230769231, 1, 0.875, 1, 0.985294117647059, 1, 1, 0.5,
    0.826086956521739, 0.833333333333333, 0.75, 0.631578947368421, 1, 0.875, 1, 1,
    0.904761904761905, 1, 1, 0.666666666666667, 0.96551724137931, 1,
    0.636363636363636, 1, 0.681818181818182, 0.78125, 0.285714285714286,
    0.833333333333333, 0.928571428571429, 0.991735537190083, 1, 0.5,
    0.833333333333333, 0.666666666666667, 0.8, 0.666666666666667,
    0.710526315789474, 0.787878787878788, 1, 1, 0.888888888888889, 1, 1,
    0.703703703703704, 1, 1, 0.875, 0.686274509803922, 0.714285714285714, 1, 1, 1,
    1, 1, 1, 0.805309734513274, 0.774193548387097, 1, 1, 1, 0.62962962962963, 1,
    0.782608695652174, 1, 1, 0.5, 0.666666666666667, 1, 1, 0.5, 0.5,
    0.555555555555556, 0.666666666666667, 0.5, 0.5, 0.697674418604651,
    0.593220338983051, 1, 0.6, 1, 1, 0.615384615384615, 0.673913043478261, 0.5, 1,
    1, 0, 1, 1, 0.555555555555556, 0.366666666666667, 0.333333333333333, 1, 1, 1,
    0.888888888888889, 1, 1, 1, 1, 1, 1, 0.6, 0.26530612244898, 1, 0.3, 1, 1, 0.5,
    1, 1, 1, 0.888888888888889, 0.666666666666667, 1, 1, 0.866666666666667,
    0.193548387096774, 1, 1, 0.181818181818182, 1, 1, 0.947368421052632, 1, 1, 1,
    0.851851851851852, 1, 1, 0.0769230769230769, 0.125, 0.1875, 1,
    0.230769230769231, 0.111111111111111, 1, 1, 0.444444444444444, 1, 0.5,
    0.153846153846154, 0.3, 0, 0.0714285714285714, 0.166666666666667, 1,
    0.166666666666667, 1, 0.181818181818182, 0.0714285714285714,
    0.142857142857143, 1, 0, 0, 0.888888888888889, 0, 0, 0
  ),
)

I was able to use ggplot()'s geom_histogram() to make a histogram of the distribution of values which shows me that a lot of them are close to 1:

ggplot(data = my_measurements) +
    geom_histogram(mapping = aes(x = value))

[

Then, I tried to plot the same data with geom_density():

library(ggplot2)
ggplot(data = my_measurements) +
    geom_density(mapping = aes(x = value))

What I am confused by is why does the y-axis ("density") go above 1? I had the (probably erroneous) understanding that the total area under this curve should be 1. If not, (a) how do I intepret this plot, and (b) in case I want the area under the curve to be 1, how do I do it?

Answer 1

I think the confusion is about discrete and continuous variables. For discrete variables all probability mass functions are in [0, 1]. For continuous variables with density, the area under the curve in 1. If a certain point has density value larger than 1, that does not imply that the particular point has probability greater than 1. Probability for that point is still zero. Value of density is combined with the range on the x-axis to calculate area under the curve. Hence, area and density value are different. Your plot is all good.

Answer 2

You have to use ..scaled.. (density estimate, scaled to maximum of 1) within geom_density, by default it uses ..density...

library(ggplot2)
# In aes by default first argument is x and second argument is y
ggplot(my_measurements, aes(value, ..scaled..)) +
    geom_density()

---

All code to reproduce result:

library(ggplot2)
p1 <- ggplot(my_measurements, aes(value, ..density..)) +
    geom_density() +
    ggtitle("Density")
p2 <- ggplot(my_measurements, aes(value, ..count..)) +
    geom_density() +
    ggtitle("Count")
p3 <- ggplot(my_measurements, aes(value, ..scaled..)) +
    geom_density() +
    ggtitle("Scaled")
egg::ggarrange(p1, p2, p3, ncol = 3)