Using a continuity correction

In this normal approximations section we shall explore using a continuous correction. Knowledge of binomial and poisson distributions will be very beneficial here.
In rounding values you should be familiar with the idea that if x=5 to the nearest integer then X must be between 4.5 and 5.5. In a continuity correction we use this idea when approximating discrete distributions by a normal (or any continuous) distribution; this is called continuity correction and we shall look at this later in this article.
When approximating a discrete distribution, X, by a continuous distribution, Y, you have to consider how to treat the decimal values of Y between the discrete values of X.
It is important to note before we continue for a continuous variable such as the normal P(X ≤ 5) and P(X < 5) give different answer values. P(X ≤ 5) != P(X < 5); they are not equal because P(X < 5) = P(X ≤ 4) Below we shall look at some examples;

Applying continuity correction to discrete distributions

Example

In the following discrete distributions, we’re going to apply a continuity correction

• P(X ≤ 6)
• P(X > 10)
• P(2 ≤ X < 5)

We shall start with P(X ≤ 6). To get a better view of what is happening we shall use a diagram as shown below;
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Because any value of X such that X is between 6 and 6.5 would round down to 6, we write this as;
[IMAGE]
The continuous variable Y could be a decimal number for instance it could be 6.26 so when approximating X ≤ 6 you use Y < 6.5 Now we shall work with P(X > 10). We know that P(X > 10) = P(X ≥ 11). Drawing a diagram is helpful so we shall draw one below;
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Here because you want P(X > 10), 10 should not be included, 10.5 round to 11 so all you need Y ≥ 10.5, the answer is;
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And lastly P(2 ≤ X ≤ 5). The same approach applies as we have done and you may have noticed a general rule which we will explore lastly in this article. Again drawing a diagram should be helpful;
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We 1.5 and 4.5 for the approximation as shown in the diagram. So for Y ≥ 2 would round to Y ≥ 1.5 and for Y < 5 would round to Y < 4.5 so the answer becomes; [IMAGE] Below is a summary for the continuity correction; You may have noticed some simple rules while working through the examples above. Here are the rules you must take while applying a continuity correction;

• First write the probability using ≤ or ≥
• For P(X ≤ n) approximate by P(Y < n+0.5)
• For P(X > n) approximate by P(Y ≥ n-0.5)