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title: "Normal approximation to the binomial method of estimating a probability p"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575356/Normal%20approximation%20to%20the%20binomial%20method%20of%20estimating%20a%20probability%20p"
format: markdown
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<span style="color: #000000">A Binomial(</span><span style="color: #000000">*p, n*</span><span style="color: #000000">) has a mean and standard deviation given by:</span>

 

![image](media://4b00ab9f-d3c1-4b57-b058-65a02af68761)

> Macro (mathblock)

![image](media://8a76e6f0-cf80-4009-9808-735d9b36a23d)


> Macro (mathblock)

From [Central Limit Theorem](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574953/), as *n* gets large the number of observed successes *s* will tend to:

 

![image](media://f82af368-ae0e-455f-b779-f32fbf5406f9)

 

> Macro (mathblock)

 

[Equation 2](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575355) for the binomial method can then be rewritten and p can be approximated by a Normal when n is large, as follows:

 

![image](media://2f1e007a-eadf-4611-b90f-ddd9097ec2ef)

 

> Macro (mathblock)

                                                                        (1)

 

which can be rearranged to:

 

![image](media://6e2ad27b-aec0-4d2e-9f9c-e81726d44529)

 

> Macro (mathblock)

                                                                        (2)

 

and which results in the following equation:

 

![image](media://d85178f2-842d-42ce-a40d-22c7a608d840)

 

> Macro (mathblock)

                                                                                  (3)

 

![image](media://2dc37463-24d9-4b41-b243-39dec01f24aa)

<span style="color: #000000">**Figure 1**</span><span style="color: #000000">: Example of Equation 3 estimate of p where </span><span style="color: #000000">*s*</span><span style="color: #000000"> = 5, </span><span style="color: #000000">*n*</span><span style="color: #000000"> = 10</span>

 

 

![image](media://731ac7e1-6d1c-4f78-b62a-af0657bcffd7)

<span style="color: #000000">**Figure 2**</span><span style="color: #000000">: Example of Equation 3 estimate of p where </span><span style="color: #000000">*s*</span><span style="color: #000000"> = 1, </span><span style="color: #000000">*n*</span><span style="color: #000000"> = 10</span>

 

<span style="color: #000000">Equation 3 works nicely in the plot above for small n (10) because the number of successes was half of n, and so the uncertainty distribution is symmetric about 0.5, which nicely matches the properties of a Normal distribution. However, if one had observed just 1 success from 10 trials, it would look quite different, as shown in Figure 2: now the Normal approximation of Equation 3 is completely inaccurate, assigning considerable confidence to negative values, and fails to reflect the asymmetric nature of the uncertainty distribution.</span>

 

 

 

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