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

Normal approximation of a binomial distribution

When can a normal distribution be used to approximate a binomial distribution?

• A binomial distribution $X~\mathrm{B}(n,p)$ can be approximated by a normal distribution $X_N~\mathrm{N}(\mu ,{\sigma }^2)$  provided

  • n is large

  • p is close to 0.5

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

  • $\mu =np$

  • ${\sigma }^2=np(1-p)$

Why do we use approximating distributions?

• 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, X_N

  • For example, X = 4 for binomial can be thought of as $3.5\le X_N<4.5$ 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 $\mathrm{P}(3.5\le X_N<4.5)=\mathrm{P}(3.5<X_N<4.5)$

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

• $P(X=k)\approx P(k-0.5<X_N<k+0.5)$

• $P(X\le k)\approx P(X_N<k+0.5)$

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

• $P(X<k)\approx P(X_N<k-0.5)$

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

• $P(X\ge k)\approx P(X_N>k-0.5)$

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

• $P(X>k)\approx P(X_N>k+0.5)$

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

• For a closed inequality such as $\mathrm{P}(a<X\le b)$

  • Think about each inequality separately and use above

  • $P(X>a)\approx P(X_N>a+0.5)$

  • $P(X\le b)\approx P(X_N<b+0.5)$

  • Combine to give

  • $\mathrm{P}(a+0.5<X_N<b+0.5)$

How do I approximate a binomial probability with a normal probability?

• You may be asked to calculate a probability from a binomial distribution using a suitable approximating distribution (i.e. a normal distribution)

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

  • $\mu =np$

  • ${\sigma }^2=np(1-p)$

• 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