Basic Statistics

128 CATEGORICAL DATA: PROPORTIONS

:& (a; ,

0.2

0 0.2 0.4 0.6 0.8 LOP

(cJ R = .2,n = 10

wu

0 0.2 0.4 0.6 p

ii :ha.5t;;5h

u ,e U 0.2

0 0.2 0.4 0.6 0.8 1.0 p

(d) R = .5, n = 10

9 0.3

0 0.2 0.4 0.6 0.8 1.0 p

Figure 10.3 Frequency distributions of sample proportions.

is done somewhat more briefly than with the measurement data, since the problems
and the methods are analogous.

10.2 SAMPLES FROM CATEGORICAL DATA

The proportion of successes in the sample is given by the number of observed suc-
cesses divided by the sample size. The proportion of successes is called p. Here the
frequency disributions are illustrated in Figure 10.3. This figure illustrates that the
appearance of the frequency distributions for two different sample sizes (5 and 10)
and for T = .2 and rr = .5. For rr = .2 the distribution tends to be skewed to the
right. For rr = .5 the distribution does not look skewed.
The distribution is called the binomial distribution and is computed using a math-
ematical formula. The mathematical formula for the binomial distribution is beyond
the scope of this book.
The$rst statement is that the mean of all possible sample proportions p is equal to
T, the population proportion. In other words, some samples give sample proportions
that are higher than rr; some give sample proportions that are lower than 7r; on the
average, however, the sample proportions equal 7r. In symbols, pLp = T. In other
words, p is an unbiased statistic.
The second statement is that the variance of the distribution of all sample pro-
portions equals a: = ~(l - .)/.. For continuous measurements, we have shown
in Section 5.3 that the variance of the sample mean is equal to the variance of the
observations divided by the sample size n. It can be shown that the same statement
applies here.