z-binomial

Binomial Approximation of the Normal Distribution

The logic and computational details of binomial probabilities
are described in Chapters 5 and 6 of Concepts and Applications.

This page will calculate the binomial z-ratio for situations of the general "k out of n" type, according to the formula

z =

(k—

)±.5

where:

- = np	[the mean of the binomial sampling distribution]
- = sqrt[npq]	[the standard deviation of the binomial sampling distribution]

	n =	the number of opportunities for event x to occur;
	k =	the observed or stipulated number of occurences of event x;
	p =	the probability that event x will occur on any particular occasion; and
	q =	the complementary probability that event x will not occur on any particular occasion.

For example: In tossing a coin 20 times, what is the probability of ending up with as many as 16 heads among the 20 tosses (i.e., 16 or more heads out of 20 tosses)? In this case

	n = 20	[the number of opportunities for a head to occur]
	k = 16	[the stipulated number of heads]
	p = .5	[the probability that a head will occur on any particular toss]
	q = .5	[the probability that a head will not occur on any particular toss]

Alternatively: In tossing a coin 20 times, what is the probability of ending up with as few as 8 heads among the 20 tosses (i.e., 8 or fewer heads out of 20 tosses)? In this case, n = 20, k = 8, p = .5, and q = .5.

To perform a calculation of this type, enter the appropriate values for n, k, and p (the value of q will be calculated and entered automatically). Then click the "Calculate" button. To enter a new set of values for n, k, and p, click the "Reset" button. The value entered for p can be either a decimal fraction such as .25 or a common fraction such as 1/4. Whenever possible, it is better to enter the common fraction rather than a rounded decimal fraction: 1/3 rather than .3333; 1/6 rather than .1667; and so forth.

The one-tailed and two-tailed probabilities associated with the resulting value of z will be calculated and displayed in the designated text cells. Note that this analysis is valid only if np and nq are both equal to or greater than 5. If this requirement is not met, use the Exact Binomial Probability Calculator if n is 170 or less; if n>170, use the Poisson Approximation of Binomial Probabilities.

n	k	p	q

binomial mean	Probability
binomial sd	One-Tail	Two-Tail
binomial z
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