using Statistics

Base.@kwdef mutable struct SampleAvg

    t::Int = 0

    avg::Float64 = 0.0

end

function (s::SampleAvg)(x)

    s.t += 1

    s.avg += (x - s.avg) / s.t

    s.avg

end

function run_once(b)

    rms = Float64[]

    distribution = randn(b)

    expectation = mean(distribution)

    sample_avg = SampleAvg()

    for i in 1:2*b

        avg = sample_avg(distribution[rand(1:b)])

        push!(rms, abs(avg - expectation))

    end

    rms

end

using Plots

begin

    n_runs = 1000

    p = plot(legend=:topright)

​

    for b in [2, 10, 100, 1000]

        rms = mean(run_once(b) for _ in 1:n_runs)

        xs = (1:2*b) ./ b

        plot!(p, xs, rms, label="b=$b")

    end

​

    p

end