Idea

I want to simulate the central limit theorem.
With the inputs required are: population mean, population standard deviation, the users can see the effects of the number of simulations and the sample size on the distribution of the sample mean.

Formula to generate the sample means

nsim = 10
nsam = 4
pop_mean = 0
pop_sd = 1
matrix_data <- matrix(rnorm(nsim * nsam, mean=pop_mean, sd=pop_sd), nrow=nsim, ncol=nsam)
sample_means <- apply(matrix_data, 1, mean)
matrix_data

##             [,1]       [,2]        [,3]         [,4]
##  [1,] -1.7689285  0.9738033 -2.02231692 -0.135686038
##  [2,] -0.9769807 -0.7428236 -0.81423799 -0.724810822
##  [3,]  0.5753947 -1.1352441  1.48914436 -0.548698817
##  [4,]  0.7128147 -0.4303148  2.16688616 -0.471817880
##  [5,]  1.4595616 -0.1700032  0.01436222 -1.547878805
##  [6,] -0.8428305  0.4616640  1.55886877 -0.472174267
##  [7,]  0.1061522  1.3819628  1.74074909  0.007046604
##  [8,] -1.4868904 -0.3785397  0.11730153  1.102928537
##  [9,]  0.8631169 -0.3459268  0.33262690  0.220310140
## [10,]  1.3356739  1.6246608 -0.88304187  0.485259470

sample_means

##  [1] -0.73828204 -0.81471328  0.09514904  0.49439205 -0.06098954  0.17638202
##  [7]  0.80897768 -0.16130000  0.26753179  0.64063808

UI

The left part contains the presented plots, as well as the instructions to use the application. They are separated in different tab panels

mainPanel(
    h3('Central Limit Theorem Visualization'),
    tabsetPanel(type='tabs',
                tabPanel('Column plot', br(), plotOutput('output_column')),
                tabPanel('Line plot', br(), plotOutput('output_line')),
                tabPanel('Instruction', br(), div('Instruction: Use the sliders to modify parameters, which are use to randomly generate sample mean, which centers at the population means. You can see the effects of the number of simulations, as well as the size of sample on the theorem. With this, you can easily see what the Central Limit Theorem is.'))
    )
)

The right part contains the input sliders

sidebarPanel(
    h3('Simulation parameters'),
    sliderInput('nsim_slider', 'How many simulations?', 100, 10000, 100),
    sliderInput('nsam_slider', 'How samples to simulate?', 10, 100, 10),
    sliderInput('mean_slider', 'What is the population mean?', 0, 1000, 0),
    sliderInput('sd_slider', 'What is the population standard deviation?', 1, 100, 1)
)

Server

A reactive function to generate the data and calculate its sample means whenever the inputs change

create_random_data <- reactive({
    nsim <- input$nsim_slider
    nsam <- input$nsam_slider
    pop_mean <- input$mean_slider
    pop_sd <- input$sd_slider
    matrix_data <- matrix(rnorm(nsim * nsam, mean=pop_mean, sd=pop_sd), nrow=nsim, ncol=nsam)
    sample_means <- apply(matrix_data, 1, mean)
    data.frame(PMF=sample_means)
})

Code to present plots into the UI

output$output_column <- renderPlot({
    my_data <- create_random_data()
    ggplot(my_data, aes(x=PMF)) + geom_histogram(fill='lightblue', bins=100, aes(y=..density..)) +
        geom_vline(xintercept=input$mean_slider, color='red') +
        stat_function(fun=dnorm, args=list(mean=mean(my_data$PMF), sd=sd(my_data$PMF)), color='yellow')
})

output$output_line <- renderPlot({
    my_data <- create_random_data()
    ggplot(my_data, aes(x=PMF)) + geom_density(adjust=0.1, color='orange') +
        geom_vline(xintercept=input$mean_slider, color='red') +
        stat_function(fun=dnorm, args=list(mean=mean(my_data$PMF), sd=sd(my_data$PMF)), color='green')
})