In statistical inference, specifically predictive inference, a prediction interval is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are often used in regression analysis. A simple example is given by a six-sided dice with face values ranging from 1 to 6. The confidence interval for the estimated expected value of the face value will be around 3.5 and will become narrower with a larger sample size. However, the prediction interval for the next roll will approximately range from 1 to 6, even with any number of samples seen so far. Prediction intervals are used in both frequentist statistics and Bayesian statistics: a prediction interval bears the same relationship to a future observation that a frequentist confidence interval or Bayesian credible interval bears to an unobservable population parameter: prediction intervals predict the distribution of individual future points, whereas confidence intervals and credible intervals of parameters predict the distribution of estimates of the true population mean or other quantity of interest that cannot be observed.
Introduction If one makes the parametric assumption that the underlying distribution is a normal distribution, and has a sample set {X1, ..., Xn}, then confidence intervals and credible intervals may be used to estimate the population mean μ and population standard deviation σ of the underlying population, while prediction intervals may be used to estimate the value of the next sample variable, Xn+1. Alternatively, in Bayesian terms, a prediction interval can be described as a credible interval for the variable itself, rather than for a parameter of the distribution thereof. The concept of prediction intervals need not be restricted to inference about a single future sample value but can be extended to more complicated cases. For example, in the context of river flooding where analyses are often based on annual values of the largest flow within the year, there may be interest in making inferences about the largest flood likely to be experienced within the next 50 years. Since prediction intervals are only concerned with past and future observations, rather than unobservable population parameters, they are advocated as a better method than confidence intervals by some statisticians, such as Seymour Geisser, following the focus on observables by Bruno de Finetti.
Normal distribution Given a sample from a normal distribution, whose parameters are unknown, it is possible to give prediction intervals in the frequentist sense, i.e., an interval [a, b] based on statistics of the sample such that on repeated experiments, Xn+1 falls in the interval the desired percentage of the time; one may call these "predictive confidence intervals". A general technique of frequentist prediction intervals is to find and compute a pivotal quantity of the observables X1, ..., Xn, Xn+1 – meaning a function of observables and parameters whose probability distribution does not depend on the parameters – that can be inverted to give a probability of the future observation Xn+1 falling in some interval computed in terms of the observed values so far, X 1 , … , X n . {\displaystyle X_{1},\dots ,X_{n}.} Such a pivotal quantity, depending only on observables, is called an ancillary statistic. The usual method of constructing pivotal quantities is to take the difference of two variables that depend on location, so that location cancels out, and then take the ratio of two variables that depend on scale, so that scale cancels out. The most familiar pivotal quantity is the Student's t-statistic, which can be derived by this method and is used in the sequel.
Known mean, known variance
A prediction interval [ℓ,u] for a future observation X in a normal distribution N(μ,σ2) with known mean and variance may be calculated from
γ = P ( ℓ < X < u ) = P ( ℓ − μ σ < X − μ σ < u − μ σ ) = P ( ℓ − μ σ < Z < u − μ σ ) , {\displaystyle \gamma =P(\ell <X<u)=P\left({\frac {\ell -\mu }{\sigma }}<{\frac {X-\mu }{\sigma }}<{\frac {u-\mu }{\sigma }}\right)=P\left({\frac {\ell -\mu }{\sigma }}<Z<{\frac {u-\mu }{\sigma }}\right),}
where Z = X − μ σ {\displaystyle Z={\frac {X-\mu }{\sigma }}} , the standard score of X, is distributed as standard normal. Hence
ℓ − μ σ = − z , u − μ σ = z , {\displaystyle {\frac {\ell -\mu }{\sigma }}=-z,\quad {\frac {u-\mu }{\sigma }}=z,}
or
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