Calculator: \(t\)-distribution
Summary
A calculator to work out cdfs for the \(t\)-distribution.
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#| viewerHeight: 640
library(shiny)
library(bslib)
library(ggplot2)
ui <- page_fluid(
title = "t-distribution calculator",
layout_columns(
col_widths = c(4, 8),
# Left column - Inputs
card(
card_header("Parameters"),
card_body(
numericInput("df", "Degrees of freedom (v):", value = 5, min = 1, step = 1),
hr(),
radioButtons("prob_type", "Probability to Calculate:",
choices = list("P(X ≤ x)" = "less",
"P(X ≥ x)" = "greater",
"P(x ≤ X ≤ y)" = "between"),
selected = "less"),
conditionalPanel(
condition = "input.prob_type == 'less'",
sliderInput("x_less", "x value:", min = -6, max = 6, value = 0, step = 0.1)
),
conditionalPanel(
condition = "input.prob_type == 'greater'",
sliderInput("x_greater", "x value:", min = -6, max = 6, value = 0, step = 0.1)
),
conditionalPanel(
condition = "input.prob_type == 'between'",
sliderInput("x_lower", "Lower bound (x):", min = -6, max = 6, value = -2, step = 0.1),
sliderInput("x_upper", "Upper bound (y):", min = -6, max = 6, value = 2, step = 0.1)
)
)
),
# Right column - Plot
card(
card_header("t-distribution plot"),
card_body(
uiOutput("plot_title"),
plotOutput("distPlot", height = "300px")
)
)
),
# Bottom row - Results
card(
card_header("Results"),
card_body(
textOutput("explanation")
)
)
)
server <- function(input, output, session) {
# Ensure that x_upper is always greater than or equal to x_lower
observe({
if (input$x_upper < input$x_lower) {
updateSliderInput(session, "x_upper", value = input$x_lower)
}
})
# Display the plot title with distribution parameters
output$plot_title <- renderUI({
title <- sprintf("t-distribution(df = %d, μ = 0)", input$df)
tags$h4(title, style = "text-align: center; margin-bottom: 15px;")
})
# Calculate the probability based on user selection
probability <- reactive({
if (input$prob_type == "less") {
prob <- pt(input$x_less, df = input$df)
explanation <- sprintf("P(X ≤ %.1f) = %.6f or %.4f%%",
input$x_less, prob, prob * 100)
return(list(prob = prob, explanation = explanation, type = "less", x = input$x_less))
} else if (input$prob_type == "greater") {
prob <- 1 - pt(input$x_greater, df = input$df)
explanation <- sprintf("P(X ≥ %.1f) = %.6f or %.4f%%",
input$x_greater, prob, prob * 100)
return(list(prob = prob, explanation = explanation, type = "greater", x = input$x_greater))
} else if (input$prob_type == "between") {
if (input$x_lower == input$x_upper) {
# For continuous distributions, P(X = a) = 0
prob <- 0
} else {
upper_prob <- pt(input$x_upper, df = input$df)
lower_prob <- pt(input$x_lower, df = input$df)
prob <- upper_prob - lower_prob
}
explanation <- sprintf("P(%.1f ≤ X ≤ %.1f) = %.6f or %.4f%%",
input$x_lower, input$x_upper, prob, prob * 100)
return(list(prob = prob, explanation = explanation, type = "between",
lower = input$x_lower, upper = input$x_upper))
}
})
# Display an explanation of the calculation
output$explanation <- renderText({
res <- probability()
return(res$explanation)
})
# Generate the t-distribution plot
output$distPlot <- renderPlot({
# Create data frame for plotting
df <- input$df
x_values <- seq(-6, 6, length.out = 500)
density_values <- dt(x_values, df = df)
plot_df <- data.frame(x = x_values, density = density_values)
# Create base plot
p <- ggplot(plot_df, aes(x = x, y = density)) +
geom_line(size = 1, color = "darkgray") +
labs(x = "X", y = "probability density function") +
theme_minimal() +
theme(panel.grid.minor = element_blank()) +
geom_vline(xintercept = 0, linetype = "dashed", color = "gray", alpha = 0.7)
# Add shaded area based on selected probability type
res <- probability()
if (res$type == "less") {
# Create data for the filled area
fill_x <- seq(-6, res$x, length.out = 200)
fill_y <- dt(fill_x, df = df)
fill_df <- data.frame(x = fill_x, density = fill_y)
p <- p + geom_area(data = fill_df, aes(x = x, y = density),
fill = "#3F6BB6", alpha = 0.6)
} else if (res$type == "greater") {
# Create data for the filled area
fill_x <- seq(res$x, 6, length.out = 200)
fill_y <- dt(fill_x, df = df)
fill_df <- data.frame(x = fill_x, density = fill_y)
p <- p + geom_area(data = fill_df, aes(x = x, y = density),
fill = "#3F6BB6", alpha = 0.6)
} else if (res$type == "between") {
# Create data for the filled area
fill_x <- seq(res$lower, res$upper, length.out = 200)
fill_y <- dt(fill_x, df = df)
fill_df <- data.frame(x = fill_x, density = fill_y)
p <- p + geom_area(data = fill_df, aes(x = x, y = density),
fill = "#3F6BB6", alpha = 0.6)
}
# Add normal distribution comparison if degrees of freedom is high enough
if (df >= 30) {
norm_x <- seq(-6, 6, length.out = 500)
norm_y <- dnorm(norm_x)
norm_df <- data.frame(x = norm_x, density = norm_y)
p <- p + geom_line(data = norm_df, aes(x = x, y = density),
linetype = "dotted", color = "#db4315", size = 0.8, alpha = 0.7)
}
return(p)
})
}
shinyApp(ui = ui, server = server)Further reading
[This interactive element appears in Overview: Probability distributions. Please click this link to go to the guide.]
Version history
v1.0: initial version created 04/24 by tdhc and Michelle Arnetta as part of a University of St Andrews VIP project.