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Chemistry LibreTexts

6.3: Using R to Model Properties of a Normal Distribution

  • Page ID
    220901
  • Given a mean and a standard deviation, we can use R’s dnorm()function to plot the corresponding normal distribution

    dnorm(x, mean, sd)

    wheremeanis the value for \(\mu\),sdis the value for \(\sigma\), andxis a vector of values that spans the range of x-axis values we want to plot.

    # define the mean and the standard deviation

    mu = 12
    sigma = 2

    # create vector for values of x that span a sufficient range of

    # standard deviations on either side of the mean; here we use values

    # for x that are four standard deviations on either side of the mean

    x = seq(4, 20, 0.01)

    # use dnorm() to calculate probabilities for each x

    y = dnorm(x, mean = mu, sd = sigma)

    # plot normal distribution curve

    plot(x, y, type = "l", lwd = 2, col = "blue", ylab = "probability", xlab = "x")

    clipboard_e7aed94b615f26041ff81bacd0bae52e8.png
    Figure \(\PageIndex{1}\): Plot showing the normal distribution curve for a population with \(\mu = 12\) and \(\sigma = 2\).

    To annotate the normal distribution curve to show an area of interest to us, we use R’s polygon() function, as illustrated here for the normal distribution curve in Figure \(\PageIndex{1}\), showing the area that includes values between 8 and 15.

    # define the mean and the standard deviation

    mu = 12
    sigma = 2

    # create vector for values of x that span a sufficient range of

    # standard deviations on either side of the mean; here we use values

    # for x that are four standard deviations on either side of the mean

    x = seq(4, 20, 0.01)

    # use dnorm() to calculate probabilities for each x

    y = dnorm(x, mean = mu, sd = sigma)

    # plot normal distribution curve; the options xaxt = "i" and yaxt = "i"

    # force the axes to begin and end at the limits of the data

    plot(x, y, type = "l", lwd = 2, col = "ivory4", ylab = "probability", xlab = "x", xaxs = "i", yaxs = "i")

    # create vector for values of x between a lower limit of 8 and an upper limit of 15
    lowlim = 8

    uplim = 15

    dx = seq(lowlim, uplim, 0.01)

    # use polygon to fill in area; x and y are vectors of x,y coordinates

    # that define the shape that is then filled using the desired color

    polygon(x = c(lowlim, dx, uplim), y = c(0, dnorm(dx, mean = 12, sd = 2), 0), border = NA, col = "ivory4")

    clipboard_e23f252d30b5f2fc7c07e9ce75522637e.png
    Figure \(\PageIndex{2}\): Plot showing the normal distribution curve for a population with \(\mu = 12\) and \(\sigma = 2\), and highlighting probability of obtaining a result between 8 and 15.

    To find the probability of obtaining a value within the shaded are, we use R’spnorm()command

    pnorm(q, mean, sd, lower.tail)

    whereqis a limit of interest,meanis the value for \(\mu\),sdis the value for \(\sigma\), andlower.tailis a logical value that indicates whether we return the probability for values below the limit (lower.tail = TRUE) or for values above the limit (lower.tail = FALSE). For example, to find the probability of obtaining a result between 8 and 15, given \(\mu = 12\) and \(\sigma = 2\), we use the following lines of code.

    # find probability of obtaining a result greater than 15

    prob_greater15 = pnorm(15, mean = 12, sd = 2, lower.tail = FALSE)

    # find probability of obtaining a result less than 8

    prob_less8 = pnorm(8, mean = 12, sd = 2, lower.tail = TRUE)

    # find probability of obtaining a result between 8 and 15

    prob_between = 1 - prob_greater15 - prob_less8 # display results

    prob_greater15

    [1] 0.0668072

    prob_less8

    [1] 0.02275013

    prob_between

    [1] 0.9104427

    Thus, 91.04% of values fall between the limits of 8 and 15.

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