6.3: Using R to Model Properties of a Normal Distribution

Last updated
Save as PDF

Page ID: 220901

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

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")

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")

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.