Calculator: Chi-squared distribution
Summary
A calculator to work out cdfs for the \(\chi^2\) distribution.
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library(shiny)
library(bslib)
library(ggplot2)
ui <- page_fluid(
title = "Chi-squared distribution calculator",
layout_columns(
col_widths = c(4, 8),
# Left column - Inputs
card(
card_header("Parameters"),
card_body(
numericInput("df", "Degrees of freedom (k):", value = 3, 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 = 0, max = 20, value = 5, step = 0.1)
),
conditionalPanel(
condition = "input.prob_type == 'greater'",
sliderInput("x_greater", "x value:", min = 0, max = 20, value = 5, step = 0.1)
),
conditionalPanel(
condition = "input.prob_type == 'between'",
sliderInput("x_lower", "Lower bound (x):", min = 0, max = 20, value = 2, step = 0.1),
sliderInput("x_upper", "Upper bound (y):", min = 0, max = 20, value = 7, step = 0.1)
)
)
),
# Right column - Plot
card(
card_header("Chi-squared 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) {
# When degrees of freedom change, adjust the range of sliders
observe({
# For chi-squared distribution, a reasonable upper limit for the x-axis depends on df
# Higher df means larger values make more sense
df <- input$df
# Use a heuristic to determine a reasonable upper bound
# This captures critical values at the 0.995 quantile
max_x <- min(qchisq(0.995, df = df), 50)
updateSliderInput(session, "x_less", max = max_x)
updateSliderInput(session, "x_greater", max = max_x)
updateSliderInput(session, "x_lower", max = max_x)
updateSliderInput(session, "x_upper", max = max_x)
})
# Ensure that x_upper is always greater than or equal to x_lower
observe({
if (input$x_upper < input$x_lower) {
updateSliderInput(session, "x_upper", value = input$x_lower)
}
})
# Display the plot title with distribution parameters
output$plot_title <- renderUI({
title <- sprintf("Chi-Squared(k = %d)", input$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 <- pchisq(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$prob_type == "greater") {
prob <- 1 - pchisq(input$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 <- pchisq(input$x_upper, df = input$df)
lower_prob <- pchisq(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 chi-squared distribution plot
output$distPlot <- renderPlot({
# Get parameters
df_val <- input$df
# Determine a reasonable max for x-axis based on df
max_x <- min(qchisq(0.995, df = df_val), 50)
# Create data frame for plotting - avoid x=0 when df=1 as density is infinite there
x_min <- if(df_val == 1) 0.01 else 0
x_values <- seq(x_min, max_x, length.out = 500)
density_values <- dchisq(x_values, df = df_val)
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()) +
xlim(0, max_x)
# Add shaded area based on selected probability type
res <- probability()
if (res$type == "less") {
# Create data for the filled area
fill_x <- seq(x_min, res$x, length.out = 200)
fill_y <- dchisq(fill_x, df = df_val)
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, max_x, length.out = 200)
fill_y <- dchisq(fill_x, df = df_val)
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 <- dchisq(fill_x, df = df_val)
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)
}
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.