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:

graph

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:

graph

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: 

graph

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:

graph

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.

Interactivity

In order to participate in this interactivity, you will need to make sure that you have already downloaded a free version of the Mathematica player:

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Once you've downloaded Mathematica, you then need to download this demo (by clicking on Download Live Version). Once, you've downloaded and opened the demo in Mathematica, you can participate in the interactivity:

  1. First, use the sliders (or the plus signs +) to set n = 5 and p = 0.2. Notice that the binomial distribution is skewed to the right.
  2. Then, as you move the sample size slider to the right in order to increase n, notice that the distribution moves from being skewed to the right to approaching symmetry.
  3. Now, set p = 0.5. Then, as you move the sample size slider in either direction, notice that regardless of the value of n, the binomial distribution is symmetric.
  4. Then, do whatever you want with the sliders until you thing you fully understand the effect of n and p on the shape of the binomial distribution. Smile