Standardized Variables

Standardizing a raw score.

A (interval or ratio) variable on a raw score can be standardized to have mean 0 and standard deviation 1 by simply subtracting the mean and dividing by the standard deviation. This formula come in two flavors: one using the population mean and standard deviation (mu and sigma) and one using the sample statistics (x-bar and s). The subscripts are to remind you what variable you are using, as there is often both an X and Y wandering around.

\[ z = \frac{x-\mu_X}{\sigma_X}; \qquad Z = \frac{X-\bar X}{s_X} \]

#| standalone: true
#| viewerHeight: 300
library(shiny)

ui <- fluidPage(
inputPanel(
  numericInput("mn", label = "Mean of X:",value=0,width=130),
  
  numericInput("sd", label = "Standard Deviation of X:",value=1,
              min = 0, width=130),
  numericInput("X", label = "x:",value=0, width=130)
),
mainPanel(
h3(textOutput("z")))
)

server <- function (input,output) {
  output$z <- renderText({
   paste("z = ",round((input$X-input$mn)/input$sd,3))
})
}
shinyApp(ui=ui,server=server)

Often the next step is to look up the Z score on a normal calculator.

Going from a standard (z) score to a raw score.

Solving the above equations for X allows the z-score to be translated back into a raw score. Often, a new variable is needed, so lets change the variables from X to Y. Once again, there are two variants based on whether sample or population means and standard deviations are used:

\[ y = \sigma_Y z + \mu_Y\, ; \qquad Y = s_Y Z + \bar{Y}\ .\]

#| standalone: true
#| viewerHeight: 300
library(shiny)

ui1 <- fluidPage(
inputPanel(
  numericInput("mny", label = "Mean of Y:",value=0,width=130),
  
  numericInput("sdy", label = "Standard Deviation of Y:",value=1,
              min = 0, width=130),
  numericInput("ZZ", label = "z:",value=0, width=130)
),
mainPanel(h3(textOutput("Y")))
)

server1 <- function (input,output) {
  output$Y <- renderText({
 paste("Y = ",round(input$ZZ*input$sdy+input$mny,3))
})
}
shinyApp(ui=ui1,server=server1)

Note that these formulae are well worth memorizing, as they will come up over and over again.