Elementary Statistics – Normal Approximation for Certain Sampling Distributions

By the Central Limit Theorem (for “large” n), certain statistics have an approximate normal distribution. Thus, the normal distribution (a distribution we have studied in detail) can be used to approximate certain probabilities.

Using the normal approximation:

Step 1:

Is it a binomial setting or not? Consider the variable being measured. Does it only take two values? Or does it take a range of values? If it takes a range of values, then it is not the binomial setting. If it only takes two values (and the other binomial characteristics are present), then it is the binomial setting.

Step 2:

Is it appropriate to use the normal approximation? That is, will the approximation be good? Note that it is always physically possible to use the approximation, but it will not always provide an accurate approximation—this is an important distinction for a statistician to make.

Setting	Check
Binomial	np 10 and n(1 - p) 10
Not binomial	Depends on the original distribution (quick check: n 30)

Step 3:

For what statistic is the probability being calculated? Is it a total or an average?

Binomial Setting	Not Binomial Setting
Asking about a sample proportion or percentage? Then use and standardize with the appropriate mean and standard error:	Asking about a sample mean or average? Then use and standardize with the appropriate mean and standard error:
Asking about a sample count or number? Then use X and standardize with the appropriate mean and standard error:	Asking about a sample total or sum? Then use and standardize with the appropriate mean and standard error:

Step 4:

Draw the appropriate normal curve picture and standardize to obtain the probability of interest (or use reverse look-up to determine the value of interest).