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

Dan Finlay

Last updated

Did this video help you?

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.

👀 You've read 1 of your 5 free revision notes this week
An illustration of students holding their exam resultsUnlock more revision notes. It's free!

By signing up you agree to our Terms and Privacy Policy.

Already have an account? Log in

Did this page help you?

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.

Download notes on Normal Approximation of Binomial