# Effect of n and p on Shape

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Other than briefly looking at the picture of the histogram at the top of the cumulative binomial probability table in the back of your book, we haven't spent much time thinking about what a binomial distribution actually looks like. Well, let's do that now!  The bottom-line take-home message is going to be that the shape of the binomial distribution is directly related, and not surprisingly, to two things:

1. n, the number of independent trials
2. p, the probability of success

For small p and small n, the binomial distribution is what we call skewed right. That is, the bulk of the probability falls in the smaller numbers 0, 1, 2,..., and the distribution tails off to the right. For example, here's a picture of the binomial distribution when n = 15 and p = 0.2:

For large p and small n, the binomial distribution is what we call skewed left. That is, the bulk of the probability falls in the larger numbers n, n−1, n−2,...  and the distribution tails off to the left. For example, here's a picture of the binomial distribution when n = 15 and p = 0.8:

For p = 0.5 and large and small n, the binomial distribution is what we call symmetric. That is, the distribution is without skewness.  For example, here's a picture of the binomial distribution when n = 15 and p = 0.5:

For small p and large n, the binomial distribution approaches symmetry. For example, if p = 0.2 and n is small, we'd expect the binomial distribution to be skewed to the right. For large n, however, the distribution is nearly symmetric. For example, here's a picture of the binomial distribution when n = 40 and p = 0.2:

You might find it educational to play around yourself with various values of the n and p parameters to see their effect on the shape of the binomial distribution.

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