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1 Introduction2 Graphing Distributions3 Summarizing Distributions4 Describing Bivariate Data5 Probability6 Research Design7 Normal Distribution8 Advanced Graphs9 Sampling Distributions10 Estimation11 Logic of Hypothesis Testing12 Tests of Means13 Power14 Regression15 Analysis of Variance16 Transformations17 Chi Square18 Distribution Free Tests19 Effect Size
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5.8 Binomial Demonstration

This section is an interactive demonstration. The live simulation runs on the original site: open the demonstration at onlinestatbook.com.

Learning Objectives

  1. State the relationship beween p and skew.
  2. Compute the mean and standard deviation of a binomial distribution
  3. State how N affects the shape of a binomial distribution.

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.

Illustrated Instructions
The screenshot below shows the binomial demonstration with its default data. You can adjust the number of trials (N) as well as the proportion of successes (p).
Screenshot of the binomial distribution calculator with N: 10 and p: 0.5. The plot of Probability against number of successes 0 to 10 is a symmetric spike pattern peaking at 0.246 at 5, with 0.205 at 4 and 6, 0.117 at 3 and 7, 0.044 at 2 and 8, 0.010 at 1 and 9, and near 0 at the ends. Below: Mean = N x p = 5.00, Sd = sqrt(N x p x (1-p)) = 1.58.
The example below shows a distribution of 20 trials with a probability of success of.7. Notice that the mean and standard deviation for the distribution are also shown.
Screenshot of the same calculator after changing to N: 20 and p: .7. The distribution has shifted right and turned slightly asymmetric, peaking at 0.192 at 14 with 0.179 at 15, 0.166 at 13, 0.130 at 16, 0.114 at 12 and a tail thinning out below 8 and above 19. Below: Mean = N x p = 14.00, Sd = sqrt(N x p x (1-p)) = 2.05.

The graph will adjust to the entered number of trials (N) and probability of success (P).

Adapted from Online Statistics Education: A Multimedia Course of Study (onlinestatbook.com), Project Leader: David M. Lane, Rice University. Developed with NSF support. The original work is in the public domain; it is cited here at the authors' request. Changes were made: reformatted as an accessible XYZ web edition with native MathML. License: Public-Domain.