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Many distributions in statistics can be regarded as limiting forms of other distributions. What this statement means can be best demonstrated through an example of how the binomial and normal distributions are related. When the number of discrete categories along the x‐axis grows larger and larger, the areas under the binomial distribution more and more resemble the probabilities computed under the normal curve. It is in this sense that for a large number of trials on the binomial, it begins to more closely approximate the normal distribution.

As an example, consider once again the binomial distribution for n = 5, p = 0.5, but this time with a normal density overlaying the binomial (Figure 2.5).

We can see that the normal curve “approximates” the binomial distribution, though perhaps not tremendously well for only five trials. If we increase the number of trials, however, to say, 20, the approximation is much improved. And when we increase the number of trials to 100, the binomial distribution looks virtually like a normal density. That is, we say that the normal distribution is the limiting form of the binomial distribution.

We can express this idea more formally. If the number of trials n in a binomial experiment is made large, the distribution of the number of successes x will tend to resemble a normal distribution. That is, the normal distribution is the limiting form of a binomial distribution as n → ∞ for a fixed p (and where q = 1 − p), where E(x_i) is the expectation of the random variable x_i (the meaning of “random variable” will be discussed shortly):

Figure 2.5 Binomial distributions approximated by normal densities for 5 (far left), 20 (middle), and 100 trials (far right).

Notice that in a z‐score calculation using the population mean μ, in the numerator, we are actually calculating the difference between the obtained score and the expectation, E(x_i). We can change this to a binomial function by replacing the expectation μ with the expectation from a binomial distribution, that is, np, where np is the mean of a binomial distribution. Similarly, we replace the standard deviation from a normal distribution with the standard deviation from the binomial distribution, . As n grows infinitely large, the normal and the binomial probabilities become identical for any standardized interval.³