Relationship between Binomial and Normal Distributions

Theorem 1: If x is a random variable with distribution B(n, p), then for sufficiently large n, the following random variable has a standard normal distribution:

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where
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ProofClick here for a proof of Theorem 1, which requires knowledge of calculus.

Corollary 1: Provided n is large enough, N(μ,σ) is a good approximation for B(n, p) where μ = np and σ2 = np (1 – p).

Observation: The normal distribution is generally considered to be a pretty good approximation for the binomial distribution when np ≥ 5 and n(1 – p) ≥ 5.  For values of p close to .5, the number 5 on the right side of these inequalities may be reduced somewhat, while for more extreme values of p (especially for p < .1 or p > .9) the value 5 may need to be increased.

Example 1: What is the normal distribution approximation for the binomial distribution where n = 20 and p = .25 (i.e. the binomial distribution displayed in Figure 1 of Binomial Distribution)?

As in Corollary 1, define the following parameters:

image528 image529 image530

Since np = 5 ≥ 5 and n(1 – p) = 15 ≥ 5, based on Corollary 1 we can conclude that B(20,.25) ~ N(5,1.94).

We now show the graph of both pdf’s to see visibly how close these distributions are:

Binomial normal distribution chart

Figure 1 – Binomial vs. normal distribution

8 Responses to Relationship between Binomial and Normal Distributions

  1. Florence says:

    Thank you so much. This is a good tutorial.

  2. Sonya says:

    Thank you for the clear explanations!
    I was wondering if there is a standard (peer reviewed?) reference for the observation that the normal distribution is a good approximation for the binomial distribution when n > 10 and .4 < p 30 and .1 < p < .9. That would be very helpful!

    • Charles says:

      Sonya,
      I was not able to find the reference to this, but I have now checked the statement against real data. When n > 10 and .4 < p < .6, the approximation is pretty good. However, for p near .1 or .9, the approximations weren't that good for n near 30. I have now changed the wording on the referenced webpage. Charles

  3. Gagan says:

    Thank you for the information!

  4. Kindly give the full information about Binomial and Normal Distribution.

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