Normal Distribution"

In NDP: Interactive Presentation for Working with Normal Distribution

knitr::opts_chunk$set(echo = FALSE)

Normal Distribution Characteristics

• The distribution is symmetric; its skewness measure is zero.
• The entire family of normal probability distributions is defined by its mean and its standard deviation.
• The mean can be any numerical value: negative, zero, or positive.
• Probabilities for the normal random variable are given by areas under the curve.
• The standard deviation determines the width of the curve: larger values result in wider, flatter curves.

Normal Distribution Problem

Scores on a recent national statistics exam were normally distributed with a mean of 80 and a standard deviation of 6.

cat(sprintf("\n" ))

• What is the probability that a randomly selected exam will have a score of at least 71?

cat(sprintf("\n" ))

• What percentage of exams will have scores between 89 and 92?

cat(sprintf("\n" ))

• If the top 2.5% of test scores receive merit awards, what is the lowest score eligible for an award?

Problem Formulation

inputPanel(

 numericInput("mean", label = "Enter the mean:",80),
 numericInput("sd", label = "Enter the standard deviation:",6),
 textInput("rv", label = "Enter what random variable indicates","Score"),
 textInput("cl",label = "Enter color for filling","red")
 )

inputPanel(
 numericInput("ss",label = "Sample Size for creating dataset",10000 ),
downloadButton("downloadData", "Download the dataset")
  )

renderPrint({
      cat(sprintf("Let X be the Random variable indicating %s",input$rv))
      expectednumber  = as.numeric(input$mean)
      standarddeviation = as.numeric(input$sd)
     limitingvaluen = as.numeric(input$limitingvaluen)
     limitingvaluenl = as.numeric(input$limitingvaluenl)
     limitingvaluenu = as.numeric(input$limitingvaluenu)
      cat(sprintf("\nThe mean of the distribution is %f",expectednumber))
      cat(sprintf("\nThe standard deviation of the distribution  is %f", standarddeviation))
    #  cat(sprintf("\nProbability(X %s %f)",input$signnormal,limitingvaluen))
      if(input$signnormal == "<=")
        probvalue = pnorm(q = limitingvaluen, mean = expectednumber,sd = standarddeviation)

      if(input$signnormal == ">=")
        probvalue = 1-pnorm(q = limitingvaluen,mean = expectednumber,sd = standarddeviation)

    #  cat(sprintf(" : %f",probvalue))
})
datasetInput <- reactive({
    rnorm2 = function(n,mean,sd){mean + sd * scale(rnorm(n))}

    dataset1= rnorm2(n= input$ss, mean = input$mean,sd =input$sd)
    dataset = data.frame(X = dataset1)

  })
output$downloadData <- downloadHandler(
    filename = function() {
      filetitle = paste(input$rv,"dataset")
      paste(filetitle, ".csv", sep = "")
    },
    content = function(file) {

      write.csv(datasetInput(), file, row.names = FALSE)
    }
  )

Computation and Visualization

inputPanel(
  selectInput("signnormal", label = "Select the sign",
                                     choices = c("<=",">="), selected = ">="),
   numericInput("limitingvaluen", label = "Enter the limitingvalue",71),
downloadButton("downloadPlot", "Download the Plot")
)
renderPrint({

      expectednumber  = as.numeric(input$mean)
      standarddeviation = as.numeric(input$sd)
     limitingvaluen = as.numeric(input$limitingvaluen)
     limitingvaluenl = as.numeric(input$limitingvaluenl)
     limitingvaluenu = as.numeric(input$limitingvaluenu)
     # cat(sprintf("The mean of the distribution is %f",expectednumber))
     # cat(sprintf("\nThe standard deviation of the distribution  is %f", standarddeviation))
      cat(sprintf("Probability(X %s %f)",input$signnormal,limitingvaluen))
      if(input$signnormal == "<=")
        probvalue = pnorm(q = limitingvaluen, mean = expectednumber,sd = standarddeviation)

      if(input$signnormal == ">=")
        probvalue = 1-pnorm(q = limitingvaluen,mean = expectednumber,sd = standarddeviation)

      cat(sprintf(" : %f",probvalue))
})

renderPlot({
  mean = as.numeric(input$mean)
      sd = as.numeric(input$sd)
      limitingvaluen = as.numeric(input$limitingvaluen)
      x <- seq(-3.5,3.5,length=100)*sd + mean
      y <- dnorm(x,mean,sd)

      if(input$signnormal == "<=")
    {  title = paste("Normal Distribution followed by " ,  input$rv)
        plot(x, y, type="l",xlab = input$rv, ylab = "Probability Value",main = title,col = "black")
      polygon(c( x[x<=limitingvaluen], limitingvaluen ),  c(y[x<=limitingvaluen],0 ), col= input$cl)
     }
if(input$signnormal == ">=")
    {
       title = paste("Normal Distribution followed by  " ,  input$rv)
       plot(x, y, type="l",xlab = input$rv, ylab = " Probability Value",main = title,col = "black")
        polygon(c( x[x>=limitingvaluen], limitingvaluen ),  c(y[x>=limitingvaluen],0 ), col=input$cl)
      }
})


output$downloadPlot <- downloadHandler(
    filename = function() {
      paste("NormalDistributionplot", ".png", sep = "")
    },
    content = function(file) {
      png(file)

      mean = as.numeric(input$mean)
      sd = as.numeric(input$sd)
      limitingvaluen = as.numeric(input$limitingvaluen)
      x <- seq(-3.5,3.5,length=100)*sd + mean
      y <- dnorm(x,mean,sd)

      if(input$signnormal == "<=")
    {   title = paste("Normal Distribution followed by " ,  input$rv)
        plot(x, y, type="l",xlab = input$rv, ylab = "Probability Value",main = title,col = "black")
      polygon(c( x[x<=limitingvaluen], limitingvaluen ),  c(y[x<=limitingvaluen],0 ), col = input$cl)
     }
if(input$signnormal == ">=")
    {  title = paste("Normal Distribution followed by " ,  input$rv)
       plot(x, y, type="l",xlab = input$rv , ylab = " Probability Value",main = title,col = "black")
        polygon(c( x[x>=limitingvaluen], limitingvaluen ),  c(y[x>=limitingvaluen],0 ), col = input$cl)
      }
      dev.off()
    })

Visualization between limits

inputPanel(
numericInput("limitingvaluenl", label = "Enter the lower limit",89),
numericInput("limitingvaluenu", label = "Enter the upper limit",92),
downloadButton("downloadPlot2", "Download the Plot")
)
renderPrint({
  expectednumber  = as.numeric(input$mean)
  standarddeviation = as.numeric(input$sd)
  limitingvaluenl = as.numeric(input$limitingvaluenl)
  limitingvaluenu = as.numeric(input$limitingvaluenu)
  cat(sprintf("Probability(%f <= X <= %f)",limitingvaluenl,limitingvaluenu))
probvaluebetween = pnorm(q = limitingvaluenu, mean = expectednumber,sd = standarddeviation)-pnorm(q = limitingvaluenl, mean = expectednumber,sd = standarddeviation)
      cat(sprintf(" : %f",probvaluebetween))
})
renderPlot({
      title = paste("Normal Distribution followed by  " ,  input$rv)
      mean = as.numeric(input$mean)
      sd = as.numeric(input$sd)
      limitingvaluenl = as.numeric(input$limitingvaluenl)
      limitingvaluenu = as.numeric(input$limitingvaluenu)
      x <- seq(-3.5,3.5,length=100)*sd + mean
      y <- dnorm(x,mean,sd)
      plot(x, y, type="l",xlab = input$rv, ylab = " Probability Value",main = title,col = "black")
      polygon(c(limitingvaluenl,x[x<=limitingvaluenu & x>=limitingvaluenl], limitingvaluenu ),  c(0,y[x<=limitingvaluenu & x>=limitingvaluenl],0 ), col= input$cl)

})

output$downloadPlot2 <- downloadHandler(
    filename = function() {
      paste("NormalDistributionplot", ".png", sep = "")
    },
    content = function(file) {
      png(file)

       title = paste("Normal Distribution followed by  " ,  input$rv)
      mean = as.numeric(input$mean)
      sd = as.numeric(input$sd)
      limitingvaluenl = as.numeric(input$limitingvaluenl)
      limitingvaluenu = as.numeric(input$limitingvaluenu)
      x <- seq(-3.5,3.5,length=100)*sd + mean
      y <- dnorm(x,mean,sd)
      plot(x, y, type="l",xlab = input$rv, ylab = " Probability Value",main = title,col = "black")
      polygon(c(limitingvaluenl,x[x<=limitingvaluenu & x>=limitingvaluenl], limitingvaluenu ),  c(0,y[x<=limitingvaluenu & x>=limitingvaluenl],0 ), col= input$cl)
      dev.off()
    })

Area Covered

inputPanel(
  numericInput("auc",label= "Enter probability coverage from the left",0.975),
downloadButton("downloadPlot3", "Download the Plot")
)
renderPrint({
  expectednumber  = as.numeric(input$mean)
  standarddeviation = as.numeric(input$sd)

  cat(sprintf("limiting value of %s",input$rv))
X = qnorm(p = input$auc, mean = expectednumber,sd = standarddeviation)
cat(sprintf(" : %f",X))
})
renderPlot({
    mean = as.numeric(input$mean)
      sd = as.numeric(input$sd)

limitingvaluenauc = qnorm(p = input$auc, mean = mean ,sd = sd)
  title = paste("Normal Distribution followed by " ,  input$rv)
   x <- seq(-3.5,3.5,length=100)*sd + mean
      y <- dnorm(x,mean,sd)
        plot(x, y, type="l",xlab = input$rv, ylab = "Probability Value",main = title,col = "black")
      polygon(c( x[x<=limitingvaluenauc], limitingvaluenauc ),  c(y[x<=limitingvaluenauc],0 ), col = input$cl)
})
output$downloadPlot3 <- downloadHandler(
    filename = function() {
      paste("NormalDistributionplot", ".png", sep = "")
    },
    content = function(file) {
      png(file)
       mean = as.numeric(input$mean)
      sd = as.numeric(input$sd)

limitingvaluenauc = qnorm(p = input$auc, mean = mean ,sd = sd)
  title = paste("Normal Distribution followed by " ,  input$rv)
   x <- seq(-3.5,3.5,length=100)*sd + mean
      y <- dnorm(x,mean,sd)
        plot(x, y, type="l",xlab = input$rv, ylab = "Probability Value",main = title,col = "black")
      polygon(c( x[x<=limitingvaluenauc], limitingvaluenauc ),  c(y[x<=limitingvaluenauc],0 ), col = input$cl)

      dev.off()
    })

Number of standard deviations

inputPanel(
 sliderInput("sscore",label = "Enter Z",min = 1, max = 6,step = 1,value = 1),
 downloadButton("downloadPlot4", "Download the Plot")

)
renderPrint({
  expectednumber  = as.numeric(input$mean)
  standarddeviation = as.numeric(input$sd)
  limitingvaluenl = expectednumber - (input$sscore * standarddeviation)
  limitingvaluenu = expectednumber + (input$sscore * standarddeviation)
  cat(sprintf("Probability(%f <= X <= %f)",limitingvaluenl,limitingvaluenu))
probvaluebetween = pnorm(q = limitingvaluenu, mean = expectednumber,sd = standarddeviation)-pnorm(q = limitingvaluenl, mean = expectednumber,sd = standarddeviation)
      cat(sprintf(" : %f",probvaluebetween))
})
renderPlot({
      title = paste("Normal Distribution followed by  " ,  input$rv)
      mean = as.numeric(input$mean)
      sd = as.numeric(input$sd)
      limitingvaluenl = mean - (input$sscore * sd)
      limitingvaluenu = mean + (input$sscore *sd)
      x <- seq(-6.5,6.5,length=600)*sd + mean
      y <- dnorm(x,mean,sd)
      plot(x, y, type="l",xlab = input$rv, ylab = " Probability Value",main = title,col = "black")
      polygon(c(limitingvaluenl,x[x<=limitingvaluenu & x>=limitingvaluenl], limitingvaluenu ),  c(0,y[x<=limitingvaluenu & x>=limitingvaluenl],0 ), col= input$cl)

})
output$downloadPlot4 <- downloadHandler(
    filename = function() {
      paste("NormalDistributionplot", ".png", sep = "")
    },
    content = function(file) {
      png(file)
       title = paste("Normal Distribution followed by  " ,  input$rv)
      mean = as.numeric(input$mean)
      sd = as.numeric(input$sd)
      limitingvaluenl = mean - (input$sscore * sd)
      limitingvaluenu = mean + (input$sscore *sd)
      x <- seq(-6.5,6.5,length=600)*sd + mean
      y <- dnorm(x,mean,sd)
      plot(x, y, type="l",xlab = input$rv, ylab = " Probability Value",main = title,col = "black")
      polygon(c(limitingvaluenl,x[x<=limitingvaluenu & x>=limitingvaluenl], limitingvaluenu ),  c(0,y[x<=limitingvaluenu & x>=limitingvaluenl],0 ), col= input$cl)

      dev.off()
    })

Not satisfied with spoon feeding!

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Then contact

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NDP documentation built on May 2, 2019, 1:03 p.m.