Statistics
Chapter 6: The Normal Distribution
2026-10-08
Introduction
Figure 6-1 Histograms and normal model for the distribution of heights of adult women
Introduction
Introduction
Introduction
Shapes of Normal Distributions
The shape and position of a normal distribution curve depend on two parameters: the mean and the standard deviation (Figure 6-3).
x <- seq(-8, 10, by = 0.01)
curves <- data.frame(x, mu0_sd1 = dnorm(x, 0, 1), mu0_sd2 = dnorm(x, 0, 2),
mu2_sd2 = dnorm(x, 2, 2))
ggplot(curves, aes(x)) +
geom_line(aes(y = mu0_sd1)) +
geom_line(aes(y = mu0_sd2), linetype = "dashed") +
geom_line(aes(y = mu2_sd2), linetype = "dotted") +
labs(title = "Figure 6-3: shapes of normal distributions", x = "x", y = "Density")
The solid line has \mu = 0, \sigma = 1, the dashed line \mu = 0, \sigma = 2, and the dotted line \mu = 2, \sigma = 2. Solid and dashed have the same mean; dashed and dotted have the same standard deviation.
Areas Under a Normal Distribution Curve
Figure 6-4 — the areas in item 8 follow the empirical rule of Section 3-2.
Symmetric and Skewed Distributions
Figure 6-5 Normal and skewed distributions
x <- seq(0, 10, by = 0.1)
shapes <- data.frame(
shape = rep(c("(a) Normal", "(b) Negatively skewed", "(c) Positively skewed"),
each = length(x)),
x = c(x, x, x),
density = c(dnorm(x, 5, 1.5), dgamma(10 - x, shape = 3), dgamma(x, shape = 3))
)
ggplot(shapes, aes(x, density)) +
geom_line() +
facet_wrap(~ shape) +
labs(y = "Density")
The Standard Normal Distribution
Figure 6-6 Standard normal distribution
Example 6-1
Example 6-2
Example 6-3
Example 6-6
Example 6-7
Solution
Example 6-7
Example 6-11
Example 6-12
Normal Quantile Plots in R
Another method used to check normality is to draw a normal quantile plot. Quantiles (sometimes called fractiles) are values that separate the data set into approximately equal groups. A normal quantile plot graphs the data values against the z values of the corresponding quantiles. If the points do not lie in an approximately straight line, normality can be rejected.
Normal Quantile Plots in R
Other methods include normal probability graph paper, the chi-square goodness-of-fit test (Chapter 11), the Kolmogorov-Smirnov test and the Lilliefors test.
A 10-Point Quiz Illustration
[1] 6[1] 1.581139[1] 1.581139The histogram of the sample means appears approximately normal, the mean of the sample means equals the population mean, and the standard deviation of the sample means equals \sigma/\sqrt{n}.
The Central Limit Theorem in R
set.seed(42)
# rate = 0.5 gives a skewed population
simulation <- data.frame(
size = rep(c("(a) n = 2", "(b) n = 10", "(c) n = 30"), each = 2000),
sample_mean = c(replicate(2000, mean(rexp(2, rate = 0.5))),
replicate(2000, mean(rexp(10, rate = 0.5))),
replicate(2000, mean(rexp(30, rate = 0.5))))
)
ggplot(simulation, aes(sample_mean)) +
geom_histogram(bins = 40, color = "white") +
facet_wrap(~ size, scales = "free") +
labs(title = "Sample means approach normality as n increases",
x = "Sample mean", y = "Frequency")
The Conditions np \geq 5 and nq \geq 5 (cont.)
binomial_shapes <- data.frame(
label = rep(c("n = 10, p = 0.3 (np = 3)", "n = 10, p = 0.5 (np = 5, nq = 5)"),
each = 11),
x = c(0:10, 0:10),
probability = c(dbinom(0:10, 10, 0.3), dbinom(0:10, 10, 0.5))
)
ggplot(binomial_shapes, aes(x, probability)) +
geom_col() +
facet_wrap(~ label) +
labs(title = "Figure 6-39: binomial distribution and a normal distribution",
x = "X", y = "P(X)")
