# Inference vs. Implication

**Difference Between Inference And Implication**

Inference

In the science of statistics, inference is the process of using information from observed phenomena to derive conclusions about the underlying probability distribution of the observations (see distribution, statistics). Suppose a coin is known to have some unknown probability p of coming up heads. (In most cases, p will not be ½ and the coin will be biased.) Assume that in a coin-tossing experiment, 70 heads were observed out of 100 tosses. Two typical problems of inference are (1) to decide whether or not the coin is biased and (2) to estimate the value of p. The first of these questions is an example of hypothesis testing: the null hypothesis that the coin is unbiased is being tested. The second question is a problem in estimation; it does not seek a simple yes or no answer but rather an estimated value of a parameter of interest. In estimation, some measure of the precision of the estimate is also sought. This may take the form of the variance of the estimate in (hypothetical) repeated sampling. Alternatively, instead of giving a point estimate and the variance, there are ways of giving a confidence interval, with ends computed from the data, that will include the true value in some specified fraction of hypothetical repetitions.

Implication

Implication, a term in logic usually denoting “logical” (or “strict”) implication, although it sometimes denotes the weaker relation of “material” implication. Likewise, a group of statements logically implies B if, and only if, it is not possible for every statement in the group to be true while B is false. The statement “Mount McKinley is taller than Mount Logan, which is taller than Mount Rainier” logically implies the statement “Mount McKinley is taller than Mount Rainier.” It is not possible for the first of those statements to be true while the second is false.

Logical implication differs from material implication. To say that a statement, A, materially implies another statement, B, is to say merely this: it is not the case that A is true and B false. Any pair of true statements materially imply each other because, given that both are true, it is not the case that the first is true and the second false. Nevertheless, not every pair of true statements logically imply each other, for in many cases either of the two could be true even if the other were false.