Sampling error in the context of Errors and residuals

In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by ${\displaystyle \alpha }$, is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true; and the p-value of a result, ${\displaystyle p}$, is the probability of obtaining a result at least as extreme, given that the null hypothesis is true. The result is said to be statistically significant, by the standards of the study, when ${\displaystyle p\leq \alpha }$. The significance level for a study is chosen before data collection, and is typically set to 5% or much lower—depending on the field of study.

In any experiment or observation that involves drawing a sample from a population, there is always the possibility that an observed effect would have occurred due to sampling error alone. But if the p-value of an observed effect is less than (or equal to) the significance level, an investigator may conclude that the effect reflects the characteristics of the whole population, thereby rejecting the null hypothesis.

The margin of error is a statistic expressing the amount of random sampling error in the results of a survey. The larger the margin of error, the less confidence one should have that a poll result would reflect the result of a simultaneous census of the entire population. The margin of error will be positive whenever a population is incompletely sampled and the outcome measure has positive variance, which is to say, whenever the measure varies.

The term margin of error is often used in non-survey contexts to indicate observational error in reporting measured quantities.

In statistics, the score (or informant) is the gradient of the log-likelihood function with respect to the parameter vector. Evaluated at a particular value of the parameter vector, the score indicates the steepness of the log-likelihood function and thereby the sensitivity to infinitesimal changes to the parameter values. If the log-likelihood function is continuous over the parameter space, the score will vanish at a local maximum or minimum; this fact is used in maximum likelihood estimation to find the parameter values that maximize the likelihood function.

Since the score is a function of the observations, which are subject to sampling error, it lends itself to a test statistic known as score test in which the parameter is held at a particular value. Further, the ratio of two likelihood functions evaluated at two distinct parameter values can be understood as a definite integral of the score function.