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The different notations used by statisticians to describe the binomial distribution can be very confusing. In the summary below, ones represent (for example) successes or positives whilst zeros represent (for example) failures or negatives.

  • n (or k): the sample size, that is the number of observations - ones plus zeros - in your sample.

  • p (or ) and q (or ): The proportions of ones and zeros in a sample of n observations. The proportion of ones in the sample is equal to the mean (ΣY / n) of the binary variable (Y).

  • P, Q: The parametric proportions - that is the proportions with and without the characteristic in the population. (If the binomial model is correct, P is also the probability of obtaining a one.) Following the usual system you might expect the Greek equivalent of p (π) to be used for its parameter (the proportion of successes in the population). In order to avoid confusion with π (= 3.14159...) this is generally avoided.

    The proportion without the characteristic is also known as the proportion that are 'not P' or 'P compliment' (Pc). It is sometimes more convenient arithmetically, to treat Q as if it is P compliment (1 - P). Another form of notation is to write the proportion of the variable having characteristic 'A' (P[A]) or its compliment (P[Ac]). This notation, and ones related to it, are commonly used in multivariate analysis of categorical data.

  • P(ΣY = r) (or P(r)):  The probability of getting a specific frequency (r).

  • f (or fy)  The frequency of ones in a sample of (binary) variable Y - is equal to ΣY, or pn.

  • λ (or μf or, misleadingly, μy or μ):   The expected (mean) frequency of ones in samples of variable Y - is equal to Pn.

  • m (or, misleadingly, ):  The observed mean frequency of ones in N samples of variable Y - is equal to Σf / N.