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Normal Approximation to the Binomial Distribution (Year 13)

Updated: Mar 22, 2023

Under certain circumstances we can approximate a binomial distribution (Discrete) by a normal distribution (Continuous).

For n large and p close to 0.5, we can approximate as follows

So μ = np and σ² = np(1-p) for the binomial are used as parameters in the normal.

As the binomial distribution is a discrete distribution, but the normal is a continuous distribution, the approximation can be refined by using a continuity correction.

In general,

Normal distribution curve superimposed on binomial distribution

Consider P(X ≦ 2) on the discrete scale i.e. the area of the blocks for X = 0, 1, 2. However 2 extends up to 2.5 on the continuous scale, so P(X ≦ 2) ≈ P(X ≦ 2.5)

Similarly, P(X = 3) ≈ P(2.5 ≦ X ≦ 3.5)


Recall also that for the normal distribution there is no distinction between < and ≤, > and ≥

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