How to approximate the Binomial distribution to the Normal Distribution

X~Binomial (n,p) with n= fixed number of independent trials and p= probability of a trail succeeding.

X ~Binomial can be approximated to Y~ Normal  if:

1)      The number of trials, n, is large

2)      Probability of succeeding in a trial , p, is close to 0.5

 

Step 1: Apply continuity correction

If given P(X=>n) then P(Y=>n-0.5)

                P(X>n) then P(Y>n+0.5)

                P(X<=n) then P(Y<n+0.5)

                P(X<n) then P(Y<n-0.5)

 

Step 2: Find mean and variance of Y~Normal

For X, we know the mean= np and variance = np(1-p)

So Y~Normal(np, np(1-p)

 

Step 3: Standardise Y~Normal to the standard normal

Values given in statistical tables for the normal distribution are based on the standard normal Z~Normal(0,1)

Z= (Y-np)/ sqrt(np(1-p))= m

 

Step 4:

Using the table find the desired probability relating to value, m, of Z and this approximates to probability relating to n of X

                P(Z<=m) = probablity corresponding to m in the table

                P(Z=>m) = 1- P(Z<=m)

                P(Z<-m)= P(Z=>m) = 1- P(Z<=m)

                P(Z>-m)= 1-P(Z<-m)= 1-(1- P(Z<m)= P(Z<m)