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Normal Approximation of Binomial (Edexcel A Level Maths): Revision Note

Dan Finlay

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Normal Approximation of Binomial

When can I use a normal distribution to approximate a binomial distribution?

  • A binomial distribution begin mathsize 16px style X tilde straight B left parenthesis n comma p right parenthesis end style can be approximated by a normal distribution begin mathsize 16px style X subscript N tilde straight N left parenthesis mu comma sigma squared right parenthesis end style  provided

    • n is large

    • p is close to 0.5

  • The mean and variance of a binomial distribution can be calculated by:

    • mu equals n p

    • sigma squared equals n p left parenthesis 1 minus p right parenthesis

4-4-2-normal-approximation-of-binomial-diagram-1

Why do we use approximations?

  • These days calculators can calculate binomial probabilities so approximations are no longer necessary

  • However it is easier to work with a normal distribution

    • You can calculate the probability of a range of values quickly

    • You can use the inverse normal distribution function (most calculators don't have an inverse binomial distribution function)

What are continuity corrections?

  • The binomial distribution is discrete and the normal distribution is continuous

  • A continuity correction takes this into account when using a normal approximation

  • The probability being found will need to be changed from a discrete variable, X,   to a continuous variable, XN

    • For example, X = 4 for binomial can be thought of as begin mathsize 16px style 3.5 less or equal than X subscript N less than 4.5 end style for normal as every number within this interval rounds to 4

    • Remember that for a normal distribution the probability of a single value is zero so begin mathsize 16px style straight P left parenthesis 3.5 less or equal than X subscript N less than 4.5 right parenthesis equals straight P left parenthesis 3.5 less than X subscript N less than 4.5 right parenthesis end style

How do I apply continuity corrections?

  • Think about what is largest/smallest integer that can be included in the inequality for the discrete distribution and then find its upper/lower bound

  • begin mathsize 16px style P left parenthesis X equals k right parenthesis almost equal to P left parenthesis k space minus 0.5 less than X subscript N less than k plus 0.5 right parenthesis end style

  • size 16px P size 16px left parenthesis size 16px X size 16px less or equal than size 16px k size 16px right parenthesis size 16px almost equal to size 16px P size 16px left parenthesis size 16px X subscript size 16px N size 16px less than size 16px k size 16px plus size 16px 0 size 16px. size 16px 5 size 16px right parenthesis

    • You add 0.5 as you want to include k in the inequality

  • size 16px P size 16px left parenthesis size 16px X size 16px less than size 16px k size 16px right parenthesis size 16px almost equal to size 16px P size 16px left parenthesis size 16px X subscript size 16px N size 16px less than size 16px k size 16px minus size 16px 0 size 16px. size 16px 5 size 16px right parenthesis

    • You subtract 0.5 as you don't want to include k in the inequality

  • size 16px P size 16px left parenthesis size 16px X size 16px greater or equal than size 16px k size 16px right parenthesis size 16px almost equal to size 16px P size 16px left parenthesis size 16px X subscript size 16px N size 16px greater than size 16px k size 16px minus size 16px 0 size 16px. size 16px 5 size 16px right parenthesis

    • You subtract 0.5 as you want to include k in the inequality

  • size 16px P size 16px left parenthesis size 16px X size 16px greater than size 16px k size 16px right parenthesis size 16px almost equal to size 16px P size 16px left parenthesis size 16px X subscript size 16px N size 16px greater than size 16px k size 16px plus size 16px 0 size 16px. size 16px 5 size 16px right parenthesis

    • You add 0.5 as you don't want to include k  in the inequality

  • For a closed inequality such as begin mathsize 16px style straight P left parenthesis a less than X less or equal than b right parenthesis end style

    • Think about each inequality separately and use above

    • begin mathsize 16px style P left parenthesis X greater than a right parenthesis almost equal to P left parenthesis X subscript N greater than a plus 0.5 right parenthesis end style

    • size 16px P size 16px left parenthesis size 16px X size 16px less or equal than size 16px b size 16px right parenthesis size 16px almost equal to size 16px P size 16px left parenthesis size 16px X subscript size 16px N size 16px less than size 16px b size 16px plus size 16px 0 size 16px. size 16px 5 size 16px right parenthesis

    • Combine to give

    • begin mathsize 16px style straight P left parenthesis a plus 0.5 less than X subscript N less than b plus 0.5 right parenthesis end style

How do I approximate a probability?

  • STEP 1: Find the mean and variance of the approximating distribution

    • mu equals n p

    • begin mathsize 16px style sigma squared equals n p left parenthesis 1 minus p right parenthesis end style

  • STEP 2: Apply continuity corrections to the inequality

  • STEP 3: Find the probability of the new corrected inequality

    • Use the "Normal Cumulative Distribution" function on your calculator

  • The probability will not be exact as it is an approximate but provided n is large and p is close to 0.5 then it will be a close approximation

Worked Example

The random variable X tilde straight B left parenthesis 1250 comma 0.4 right parenthesis.

Use a suitable approximating distribution to approximate straight P left parenthesis 485 less or equal than X less or equal than 530 right parenthesis.

4-4-2-normal-approximation-of-binomial-we-solution

Examiner Tips and Tricks

  • In the exam, only use a normal approximation if the question tells you to. Otherwise use the binomial distribution.

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    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.