The normal (or standard normal) reference distribution

The standard normal distribution is widely used in inferential statistics, owing to its

• applicability (many processes are intrinsically normally distributed, and the central limit theorem assures us that sample means, in specific, and many processes that involve summation or integration, in general, are normally distributed)
• simplicity (it has a relatively simple shape, owing to the particular choice of values for its two parameters, mu and sigma)

The standard normal distribution is described by an equation and is represented by its

Probability density function [pdf] (R/S-Plus term is Density),

# probability density function
x <- seq(-3, +3, by=.1)
pdf.norm <- dnorm(x, mean=0, sd=1)
plot(x, pdf.norm, type="l")

Cumulative density function [cdf] (R/S-Plus term is Probability), and

# cumulative density function
x <- seq(-3, +3, by=.1)
cdf.norm <-pnorm(x, mean=0, sd=1)
plot(x, cdf.norm, type="l")

Inverse cumulative density function [invcdf] (R/S-Plus term is Quantile).

# inverse cumulative density function
pr <- seq(0, 1, by=.01)
invcdf.norm <- qnorm(pr, mean=0, sd=1)
plot(pr, invcdf.norm, type="l")

Random numbers drawn from the normal distribution can be obtained as follows

# normally distributed random numbers
z <- rnorm(n=1000, mean=0, sd=1)
hist(z, nclass=40)
plot(z)

The Central Limit Theorem

The Central Limit Theorem provide the explanation for the wide applicability of the normal distribution in significance testing, because it guarantees us that (providing certain assumptions are met) that statistics (like the mean) obtained by integrating (as in summing up for computing the mean) will turn out to be normally distributed, no matter what the distribution of the underlying data is.

empirical demonstration of the Central Limit Theorem

Application of the normal distribution in significance testing

The application of the standard normal distribution for making inferences about the significance of test statistics can be illustrated using a few representative questions that might be asked in practice.  To answer these questions, we first need to know the xxx

• mean and variance of the sample mean

Then, questions of the following kind can be answered

• probability of occurrence of an observation with a particular value
• probability of occurrence of a sample mean with a particular value
• "confidence interval" for a sample mean with a particular value

Simple statistical inferences