Why does the normal approximation to the binomial distribution break for small intervals




When the sample size is large and the random variables are Bernoulli variables we can use the normal approximation to the binomial distribution for calculation of p-values.However,for small intervals in the distribution,the approximation fails.
Example:If we wanted to calculate the probability of observing 69,70 or 71 smokers amongst a sample of 400 ,the normal approximation prob is 0.04 whereas the binomial is 0.07.
I am trying to understand why this happens.Is it because when the interval is small the distribution becomes more discrete than continuous??


The easiest way to understand this is by visualizing the distributions. Below is an overlapped diagram of the binomial and the corresponding approximated normal curves:

Here, the number of bars needs to be large for the normal curve to be a good approximation.

When we use the approximated values by the normal curve we use the value of y for a specific x as given by the blue line. We can clearly see the that the actual value is the top end of the bar. These approximated normal values rarely exactly match the actual values as shown by the red dots in the above plot (only half of the red dots are marked) .

On the other hand as we increase the data points the approximation gets better with each iteration as follows:

P.S : 1. If you want to look at the mathematical proof for the above it can be found here:
2, Some Examples explained: