Suppose you toss a fair coin times. What is the probability of getting at most heads? We can answer this question directly using binomial probability. This corresponds to adding the areas of the through rectangles of the binomial distribution. Alternatively, we can approximate it using a normal distribution with mean and standard deviation .
Below the normal curve and binomial distribution are shown for tossing the coin 30 times. Notice that the area under the curve and the area included by the rectangles are slightly off. In order to get a good approximation, we need to include the right half of the last rectangle, by using an upper bound of when we compute the cumulative normal distribution.