Statistics: Normal Approximation to the Binomial

Normal Approximation to the Binomial

Some variables are continuous, that 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, one kind of discrete variable, can take only two values, often termed successes and failures. Examples include coin flips 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.

The outcomes of binomial trials have a frequency distribution, just as continuous variables do. The more binomial outcomes there are (for example, the more coins you flip simultaneously), the more closely the sampling distribution resembles a normal curve (see Figure 1 ).

Figure 1

As the number of trials increases, the binomial distribution approaches the normal distribution.

The mean of the normal approximation to the binomial is

and the standard deviation is

where n is the number of trials and π is the probability of success. The approximation will be more accurate the larger the n and the closer the proportion of successes in the population to .5.

Example 1: Assuming an equal chance of a new baby being a boy or a girl (that is, π = 0.5), what is the likelihood that 60 or more out of the next 100 births at a local hospital will be boys?

A z-score of 2 corresponds to a probability of .9772. As you can see in Figure 2 , there is a .9772 chance that there will be 60 percent or fewer boys, which means that the probability that there will be more than 60 percent boys is 1 − .9772 = .0228, or just over 2 percent. If the assumption that the chance of a new baby being a girl is the same as it being a boy is correct, the probability of obtaining 60 or fewer girls in the next 100 births is also .9772.