• Check the conditions for applying a normal distribution to approximate a binomial distribution

Continuity Correction

continuity correction

In probability theory, a continuity correction is an adjustment that is made when a discrete distribution is approximated by a continuous distribution. We do this by adjusting the discrete whole numbers in a binomial distribution so that any individual value, [latex]X[/latex], is represented in the normal distribution by the interval form: [latex]X±0.5[/latex].

Question: Why do we need to do continuity correction?

Consider the following:

Recall, the random variable used in a binomial distribution is discrete. The binomial distribution can be used to compute probabilities of events in a binomial experiment by finding exact binomial probabilities using the binomial formula for each value of  and adding the results.

For example, given [latex]1000[/latex] trials of a binomial experiment, to compute the probability of [latex]700[/latex] or more successes, we would need to compute the following probabilities: [latex]P(X \ge 700) = P(700) + P(701) + ... + P(1000)[/latex]. We see that a large number of trials of a binomial experiment can make this formula difficult to use, and it is time consuming to compute it by hand.

You may ask: “Why not use technology to find the binomial probabilities?” In practice, yes, technology is used to calculate these probabilities. But, there are cases where we need to use a normal approximation.

We have an alternative means for approximating binomial probabilities, provided that certain conditions are met: For a fixed [latex]p[/latex], as the number of trials, [latex]n[/latex], in a binomial experiment increases, the probability distribution of the random variable [latex]X[/latex] becomes more symmetric and bell shaped. As a rule of thumb, if [latex]np \ge 10[/latex] and [latex]n(1-p) \ge 10[/latex], then the probability distribution will become approximately symmetric and bell shaped.

Once we verify that the conditions are met, we can then use the continuity of correction to make the adjustment to the discrete random variable and use the normal distribution to approximate the binomial probabilities.