The Normal Distribution

Introduction

• In this chapter we study the most important type of density curve: the normal curve.
• The normal curve is a symmetric "bell-shaped" curve whose exact form we will describe next.
• A distribution represented by a normal curve is called a normal distribution.

Example: serum cholesterol

The relationship between the concentration of cholesterol in the blood and the occurrence of heart disease has been the subject of much research. As part of a government health survey, researchers measured serum cholesterol levels for a large sample of Americans, including children. The distribution for children between $12$ and $14$ years of age can be fairly well approximated by a normal curve with mean $\mu=155$ mg/dl and standard deviation $\sigma=27$ mg/dl. The following figure shows a histogram based on a sample of $431$ children between $12$ and $14$ years old, with the normal curve superimposed.

The Normal Curves

• There are many normal curves; each particular normal curve is characterized by its mean and standard deviation.
• If a random variable $Y$ follows a normal distribution with mean $\mu$ and standard deviation $\sigma$, then it is common to write $Y\sim N(\mu, \sigma)$.
• The probability density function (pdf) of $Y\sim N(\mu, \sigma)$ is $$f(y)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(y-\mu)^2}{2\sigma^2}},$$ which expresses the height of the normal curve as a function of the position along the horizontal axis. The quantities $e$ and $\pi$ that appear in the formula are constants, with $e$ approximately equal to $2.71$ and $\pi$ approximately equal to 3.14.

• The figure below shows a graph of a normal curve. The shape of the curve is like a symmetric bell, centered at $y = \mu$.
• The direction of curvature is downward (like an inverted bowl) in the central portion of the curve, and upward in the tail portions.
• In principle the curve extends to $+\infty$ and $-\infty$, never actually reaching the $y$-axis; however, the height of the curve is very small for $y$ values more than three standard deviations from the mean.
• The area under the curve is exactly equal to $1$.

Normal curves with different means and SDs

• The location of the normal curve along the $y$-axis is governed by $\mu$ since the curve is centered at $y=\mu$;
• The width and the height of the curve (i.e., whether tall and thin or short and wide) are governed by $\sigma$.

Areas under a Normal Curve

• The standard normal distribution, represented by $Z$, is the normal distribution having a mean of $0$ and a standard deviation of $1$. That is, $Z\sim N(0, 1)$.
• If $X$ is a random variable from a normal distribution with mean $\mu$ and standard deviation $\sigma$, its Z-score (standardization) may be calculated from $X$ by subtracting $\mu$ and dividing by the standard deviation $\sigma$: $$Z=\frac{Y-\mu}{\sigma}.$$

• Z table gives areas under the standard normal curve, with distances along the horizontal axis measured in the Z scale.
• Each area tabled in the body of Z table is the area under the standard normal curve below a specified value of $z$, tabled in the margins.
• If we want to find the area above a given value of $z$, we subtract the tabulated area from $1$.
• How to find the area between two $z$ values?

The empirical rule for normal distribution

If the variable $Y$ follows a normal distribution, then

• about $68\%$ of the $y$'s are within $\pm1$ SD of the mean.
• about $95\%$ of the $y$'s are within $\pm2$ SDs of the mean.
• about $99.7\%$ of the $y$'s are within $\pm3$ SDs of the mean.

Determining areas for a normal curve

By taking advantage of the standardized scale, we can use $Z$ table to answer detailed questions about any normal population when the population mean and standard deviation are specified.

A professor's exam scores are approximately distributed normally with mean $80$ and standard deviation $5$.

• What is the probability that a student scores an $82$ or less? $0.65542$
• What is the probability that a student scores a $90$ or more? $0.02275$
• What is the probability that a student scores between $74$ and $82$? $0.54035$

Inverse reading of Z table

We often need to determine corresponding $z$-values when we want to determine a percentile of a normal distribution. For example, suppose we want to find the $70$th percentile of a standard normal distribution. We need to look in Z table for an area of $0.7000$. The closest value is an area of $0.6985$, corresponding to a $z$ value of $0.52$.

• What is the first quartile of the exam score distribution? $76.65$
• What is the $70$th percentile of the exam score distribution? $82.6$

Assessing Normality

Many statistical procedures are based on having data from a normal population. In this section we consider ways to assess whether it is reasonable to use a normal curve model for a set of data and, if not, how we might proceed.

Normal quantile plots

A normal quantile plot is a special statistical graph that is used to assess normality. We present this statistical tool with an example using the heights (in inches) of a sample of $11$ women, sorted from smallest to largest:

$$61, 62.5, 63, 64, 64.5, 65, 66.5, 67, 68, 68.5, 70.5$$

Based on these data, does it make sense to use a normal curve to model the distribution of women's heights?

• sort the data from smallest to largest.
• calculate the adjusted percentiles $100(i-1/2)/n$
• find the corresponding Z scores.
• calculate the theoretical quantiles $\mu+Z\times\sigma$.
• plot the sample quantiles against the theoretical quantiles in a scatterplot.

• In this case our plot appears fairly linear, suggesting that the observed values generally agree with the theoretical values and the normal model provides a reasonable approximation to the data.
• If the data do not agree with the normal model, then the plot will show strong nonlinear patterns such as curvature or S shapes.

Skewness in normal quantile plots

• Histogram and normal quantile plot of a distribution that is skewed to the left

• Histogram and normal quantile plot of a distribution that is skewed to the right

• Histogram and normal quantile plot of a distribution that has long tails

library(ggplot2)

# Create the normal quantile plot using ggplot2
g <- ggplot(data.frame(y = c(61, 62.5, 63, 64, 
                             64.5, 65, 66.5, 67, 
                             68, 68.5, 70.5)), 
            aes(sample = y)) +
    stat_qq() +
    stat_qq_line() +
    labs(x = "Theoretical Quantiles", y = "Sample Quantiles") +
    ggtitle("Normal Quantile Plot") +
    theme_bw() +
    theme(text = element_text(size = 20))
options(repr.plot.width=6, repr.plot.height=5)
g

Transformations for nonnormal data

• Sometimes a histogram or normal quantile plot shows that our data are nonnormal, but a transformation of the data gives us a symmetric, bell-shaped curve.
• In such a situation, we may wish to transform the data and continue our analysis in the new (transformed) scale.

• In general, if the distribution is skewed to the right then one of the following transformations should be considered: $$\sqrt{Y}, \log Y, 1/\sqrt{Y}, 1/Y.$$
• These transformations will pull in the long right-hand tail and push out the short left-hand tail, making the distribution more nearly symmetric. Each of these is more drastic than the one before. Thus, a square root transformation will change a mildly skewed distribution into a symmetric distribution, but a log transformation may be needed if the distribution is more heavily skewed, and so on.
• If the distribution of a variable $Y$ is skewed to the left, then raising $Y$ to a power greater than $1$ can be helpful.