Notation

We now generalise the brand loyalty example on the previous page. Consider an infinite categorical population that contains a proportion π of some category that we will call 'success'. We call the other values in the population 'failures'.

In the brand loyalty example, picking the Pizza Hut slice might be called a 'success' and a other slice would be a 'failure'. The probability of success is π = 0.5.

The labels 'success' and 'failure' provide terminology that can describe a wide range of data sets. For example,

Data set  'Success'   'Failure' 
Change in share price increase same or decrease
  Quality of items from production line   acceptable defective
  Bank balances   in credit overdrawn

When a random sample of n values is selected from such a population, we denote the number of successes by x and the proportion of successes by p  = x/n.

Distribution of a proportion from a simple random sample

The number of successes, x , has a 'standard' discrete distribution called a binomial distribution which has two parameters, n and π. In practical applications, n is a known constant, but π may be unknown. The sample proportion, p , has a distribution with the same shape, but is scaled by n .

With appropriate choice of the parameters n and π, the binomial distribution can describe the distribution of any proportion from a random sample.

Shape of the binomial distribution

The diagram below shows some possible shapes of the binomial distribution. The barchart has dual axes and therefore shows the distributions of both x and p.

Drag the sliders to adjust the two parameters of the binomial distribution. Observe that

The diagram can be used to obtain binomial probabilities by setting π and n to the appropriate values, then clicking on one of the bars in the barchart.

Brand loyalty

For example, to find the probability of 4 out of 5 subjects picking the Pizza Hut slice in the previous page if there was no brand awareness, set π = 0.5 and n = 5, then click on the bar for x = 4.

The probability is shown under the barchart. In the actual experiment performed by the grade 9 children, there were 16 subjects. Use the diagram to find the probability that all 16 picked the Pizza Hut slice if there was no tendency to pick the 'famous' brand.


The diagram below demonstrates that a binomial distribution does indeed describe sample-to-sample variability. The pink barchart at the bottom of the diagram shows the binomial distribution with parameters n = 16 and π = 0.5 that describes the distribution of the sample proportion of Pizza Hut slices picked as 'best' from n = 16 guesses.

Click Accumulate and take several samples. Observe that the distribution of p matches the theoretical binomial distribution. Repeat the exercise with different sample sizes.