Normal Approximation to the Binomial
Some variables are continuous—there is no limit to the number of times you could divide their intervals into still smaller ones, although you may round them off for convenience. Examples include age, height, and cholesterol level. Other variables are discrete, or made of whole units with no values between them. Some discrete variables are the number of children in a family, the sizes of televisions available for purchase, or the number of medals awarded at the Olympic Games.
A binomial variable can take only two values, often termed successes and failures. Examples include coin tosses that come up either heads or tails, manufactured parts that either continue working past a certain point or do not, and basketball tosses that either fall through the hoop or do not.
You discovered that the outcomes of binomial trials have a frequency distribution, just as continuous variables do. The more binomial trials there are (for example, the more coins you toss simultaneously), the more closely the sampling distribution resembles a normal curve (see Figure 1). You can take advantage of this fact and use the table of standard normal probabilities (Table 2 in "Statistics Tables") to estimate the likelihood of obtaining a given proportion of successes. You can do this by converting the test proportion to a z‐score and looking up its probability in the standard normal table.
Figure 1.As the number of trials increases, the binomial distribution approaches the normal distribution.