General Instructions

This demonstration allows you to explore the binomial distribution. Here you can specify the number of trials (N) and the propotion of successes (p). Note that the proporton of successes is sometimes referred to with the Greek letter p; here it will be referred to as p.

The starting values for N and p are 10 and 0.5. The graph therefore shows the probability of from 0 to 10 successes out of 10 trials. The probability of 0 successes is so small that it just barely shows up on the graph. The most likely number of successes is 5, which has a probability of about 0.25. Naturally, the sum of the probabilities is 1.0.

The demonstration also shows the mean and the standard deviation of the binomial distribution.

Step by Step Instructions

Show Questions

Notice that the distribution is symmetric when p = 0.5.

Set p to 0.8. Is the distribution still symmetric, or is its tail longer in one direction or the other? If the tail is longer on the right side, it is called positive skew; if it is longer to the left, it is called negative skew. Now set p to 0.2 and note the results.

Now consider how the value of p affects how spread out the distribution is. Using the standard deviation as the measure of spread, examine the spread for p= 0.1, 0.3, 0.5, 0.7, and 0.9. Which value has the largest spread? Which value(s) have the smallest spread?

How does N affect the mean of the binomial distribution? Setting p back to 0.5, find the mean for N = 10 and N = 20. It makes sense that the mean number of successes is greater when you have more trials.

You learned earlier using N = 10 that the distribution is symmetric only when p = 0.5; otherwise it is skewed. How does N affect the skew? Compare the skew for N = 10, p = 0.8 with the skew for N = 50, p = 0.8. Both distributions are negatively skewed, but notice which one only has a very small skew.

Summary

The binomial distribution is symmetric when p = 0.5. It has a negative skew when p is larger than 0.5 and a positive skew when p is smaller than 0.5. The larger the value of N, the less the skew for a given value of p (other than 0.5 where there is no skew for any sample size).

The mean of the binomial distribution is equal to Np. Therefore, the larger the number of trials the larger the expected number of successes.

The spread of the binomial distribution is greatest when p = 0.5. The farther p is from 0.5, the smaller the spread.