Properties of a sample proportion

Sample proportions vary from sample to sample. Our model (of sampling from an underlying population) therefore means that they have probability distributions.

The sample proportion in a particular category from a random sample of size n has a distribution that ...

Count and proportion of successes

Although the sample proportion in a category, p , is a good summary statistic, the raw count of sample values in the category, x  = np, contains equivalent information and is often easier to use.

p and x  = np have distributions with the same shape (other than the scaling constant n).

Telepathy experiment

Consider an experiment which investigates whether one subject can telepathically pass shape information to another subject. A deck of cards containing equal numbers of cards with circles, squares and crosses is shuffled. One subject selects cards at random and attempts to 'send' the shape on the card to the other subject who is seated behind a screen; this second subject writes the shape imagined for the card.

If the second subject is just guessing, then the probability of giving a correct answer for each card is a third. The diagram below simulates such an experiment where the second subject guesses each card. Values are therefore sampled from an infinite categorical population of one third 'correct' and two thirds 'wrong' values.

Click the checkbox Accumulate then take a few samples to build up the distribution of the proportion of 'correct' guesses in samples of size n = 20. The distribution is discrete since only a few distinct values are possible. The distribution of the sample proportion should be centred on π = 0.333.

Use the pop-up menu to change the sample size and repeat sampling to verify that the sample proportion has lower spread when the sample size is higher.

Finally, use the pop-up menu to change the scale on the sampling distribution of the proportions to show the the distribution of the sample number of correct guesses. Observe that the shape of the distribution is unchanged.