How does sampling affect the distribution of frequencies of individual types?

Question

Consider a population of size N in which individuals can be of x different types. Take a sample (with replacement) of size samplesize from this population and then compare the distribution of frequencies of individual type in the whole population and in the sample.

In the whole population the distribution of frequencies of individual types seems symmetric. In the sample, there are many individual types of low frequency and few individual type of high frequency (the distribution is right skewed). Why is it so? What kind of predictions can we make about how the distribution of frequencies will differ after sampling?

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Here a short R code to simulate this process. I also added one typical output

x = 75
N = 1000
samplesize = N/10
nb_replicates = 10 

par(mfrow=c(1,2))
for (replicate in 1:nb_replicates){
    pop = sample(1:x, N, replace=TRUE)
    s = sample(pop, samplesize, replace=TRUE)
    hist(table(pop)/N, main="pop")
    hist(table(s)/samplesize, main="sample")
    Sys.sleep(1)
}

Answer 1

In fact your plots are perfectly normal, except that it would be better to not divide by N and samplesize (and find a way to plot the number of missing types, especially for the sample histogram) .

The number of individuals of a specific type in a population of size $N$ follow a Binomial distribution $\mathcal{B} (N, p= \frac{1}{x}) $

For $n$ big and $x$ not too big, this binomial is nearly a Normal law, but if $n$ and $x$ are about the same, it's more like a Poisson law.

Here with the sample, $n=100$ and $x=75$, it's close to a Poisson law of parameter $\frac{100}{75}=\frac{4}{3}$

And what you plot is the density of these law