Binomial Distribution - Mode and Median

Mode and Median

Usually the mode of a binomial B(n, p) distribution is equal to, where is the floor function. However when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes: (n + 1)p and (n + 1)p − 1. When p is equal to 0 or 1, the mode will be 0 and n correspondingly. These cases can be summarized as follows:

$\text{mode} = \begin{cases} \lfloor (n+1)\,p\rfloor & \text{if }(n+1)p\text{ is 0 or a noninteger}, \\ (n+1)\,p\ \text{ and }\ (n+1)\,p - 1 &\text{if }(n+1)p\in\{1,\dots,n\}, \\ n & \text{if }(n+1)p = n + 1. \end{cases}$

In general, there is no single formula to find the median for a binomial distribution, and it may even be non-unique. However several special results have been established:

If np is an integer, then the mean, median, and mode coincide and equal np.
Any median m must lie within the interval ⌊np⌋ ≤ m ≤ ⌈np⌉.
A median m cannot lie too far away from the mean: |m − np| ≤ min{ ln 2, max{p, 1 − p} }.
The median is unique and equal to m = round(np) in cases when either p ≤ 1 − ln 2 or p ≥ ln 2 or |m − np| ≤ min{p, 1 − p} (except for the case when p = ½ and n is odd).
When p = 1/2 and n is odd, any number m in the interval ½(n − 1) ≤ m ≤ ½(n + 1) is a median of the binomial distribution. If p = 1/2 and n is even, then m = n/2 is the unique median.

Read more about this topic: Binomial Distribution

Famous quotes containing the word mode:

“If Thought is capable of being classed with Electricity, or Will with chemical affinity, as a mode of motion, it seems necessary to fall at once under the second law of thermodynamics as one of the energies which most easily degrades itself, and, if not carefully guarded, returns bodily to the cheaper form called Heat. Of all possible theories, this is likely to prove the most fatal to Professors of History.”
—Henry Brooks Adams (1838–1918)