In statistics, a confidence interval (CI) is a type of interval estimate, computed from the statistics of the observed data, that might contain the true value of an unknown population parameter. The interval has an associated confidence level, or coverage that, loosely speaking, quantifies the level of confidence that the deterministic parameter is captured by the interval. More strictly speaking, the confidence level represents the frequency (i.e. the proportion) of possible confidence intervals that contain the true value of the unknown population parameter. In other words, if confidence intervals are constructed using a given confidence level from an infinite number of independent sample statistics, the proportion of those intervals that contain the true value of the parameter will be equal to the confidence level.^[1][2][3]

Confidence intervals consist of a range of potential values of the unknown population parameter. However, the interval computed from a particular sample does not necessarily include the true value of the parameter. Based on the (usually taken) assumption that observed data are random samples from a true population, the confidence interval obtained from the data is also random.

The confidence level is designated prior to examining the data. Most commonly, the 95% confidence level is used.^[4] However, other confidence levels can be used, for example, 90% and 99%.

Factors affecting the width of the confidence interval include the size of the sample, the confidence level, and the variability in the sample. A larger sample will tend to produce a better estimate of the population parameter, when all other factors are equal. A higher confidence level will tend to produce a broader confidence interval.

Confidence intervals were introduced to statistics by Jerzy Neyman in a paper published in 1937.^[3]

Interval estimation can be contrasted with point estimation. A point estimate is a single value given as the estimate of a population parameter that is of interest, for example, the mean of some quantity. An interval estimate specifies instead a range within which the parameter is estimated to lie. Confidence intervals are commonly reported in tables or graphs along with point estimates of the same parameters, to show the reliability of the estimates.

For example, a confidence interval can be used to describe how reliable survey results are. In a poll of election–voting intentions, the result might be that 40% of respondents intend to vote for a certain party. A 99% confidence interval for the proportion in the whole population having the same intention on the survey might be 30% to 50%. From the same data one may calculate a 90% confidence interval, which in this case might be 37% to 43%. A major factor determining the length of a confidence interval is the size of the sample used in the estimation procedure, for example, the number of people taking part in a survey.

Various interpretations of a confidence interval can be given (taking the 90% confidence interval as an example in the following).

In each of the above, the following applies: If the true value of the parameter lies outside the 90% confidence interval, then a sampling event has occurred (namely, obtaining a point estimate of the parameter at least this far from the true parameter value) which had a probability of 10% (or less) of happening by chance.

Confidence intervals and levels are frequently misunderstood, and published studies have shown that even professional scientists often misinterpret them.^[6][7][8][9]

The principle behind confidence intervals was formulated to provide an answer to the question raised in statistical inference of how to deal with the uncertainty inherent in results derived from data that are themselves only a randomly selected subset of a population. There are other answers, notably that provided by Bayesian inference in the form of credible intervals. Confidence intervals correspond to a chosen rule for determining the confidence bounds, where this rule is essentially determined before any data are obtained, or before an experiment is done. The rule is defined such that over all possible datasets that might be obtained, there is a high probability ("high" is specifically quantified) that the interval determined by the rule will include the true value of the quantity under consideration. The Bayesian approach appears to offer intervals that can, subject to acceptance of an interpretation of "probability" as Bayesian probability, be interpreted as meaning that the specific interval calculated from a given dataset has a particular probability of including the true value, conditional on the data and other information available. The confidence interval approach does not allow this since in this formulation and at this same stage, both the bounds of the interval and the true values are fixed values, and there is no randomness involved. On the other hand, the Bayesian approach is only as valid as the prior probability used in the computation, whereas the confidence interval does not depend on assumptions about the prior probability.

The questions concerning how an interval expressing uncertainty in an estimate might be formulated, and of how such intervals might be interpreted, are not strictly mathematical problems and are philosophically problematic.^[13] Mathematics can take over once the basic principles of an approach to 'inference' have been established, but it has only a limited role in saying why one approach should be preferred to another: For example, a confidence level of 95% is often used in the biological sciences, but this is a matter of convention or arbitration. In the physical sciences, a much higher level may be used.^[14]

Confidence intervals are closely related to statistical significance testing. For example, if for some estimated parameter θ one wants to test the null hypothesis that θ = 0 against the alternative that θ ≠ 0, then this test can be performed by determining whether the confidence interval for θ contains 0.

More generally, given the availability of a hypothesis testing procedure that can test the null hypothesis θ = θ0 against the alternative that θ ≠ θ0 for any value of θ0, then a confidence interval with confidence level γ = 1 − α can be defined as containing any number θ0 for which the corresponding null hypothesis is not rejected at significance level α.^[15]

If the estimates of two parameters (for example, the mean values of a variable in two independent groups) have confidence intervals that do not overlap, then the difference between the two values is more significant than that indicated by the individual values of α.^[16] So, this "test" is too conservative and can lead to a result that is more significant than the individual values of α would indicate. If two confidence intervals overlap, the two means still may be significantly different.^[17][18][19] Accordingly, and consistent with the Mantel-Haenszel Chi-squared test, is a proposed fix whereby one reduces the error bounds for the two means by multiplying them by the square root of ½ (0.707107) before making the comparison.^[20]

While the formulations of the notions of confidence intervals and of statistical hypothesis testing are distinct, they are in some senses related and to some extent complementary. While not all confidence intervals are constructed in this way, one general purpose approach to constructing confidence intervals is to define a 100(1 − α)% confidence interval to consist of all those values θ0 for which a test of the hypothesis θ = θ0 is not rejected at a significance level of 100α%. Such an approach may not always be available since it presupposes the practical availability of an appropriate significance test. Naturally, any assumptions required for the significance test would carry over to the confidence intervals.

It may be convenient to make the general correspondence that parameter values within a confidence interval are equivalent to those values that would not be rejected by a hypothesis test, but this would be dangerous. In many instances the confidence intervals that are quoted are only approximately valid, perhaps derived from "plus or minus twice the standard error," and the implications of this for the supposedly corresponding hypothesis tests are usually unknown.

It is worth noting that the confidence interval for a parameter is not the same as the acceptance region of a test for this parameter, as is sometimes thought. The confidence interval is part of the parameter space, whereas the acceptance region is part of the sample space. For the same reason, the confidence level is not the same as the complementary probability of the level of significance.

Confidence regions generalize the confidence interval concept to deal with multiple quantities. Such regions can indicate not only the extent of likely sampling errors but can also reveal whether (for example) it is the case that if the estimate for one quantity is unreliable, then the other is also likely to be unreliable.

A confidence band is used in statistical analysis to represent the uncertainty in an estimate of a curve or function based on limited or noisy data. Similarly, a prediction band is used to represent the uncertainty about the value of a new data point on the curve, but subject to noise. Confidence and prediction bands are often used as part of the graphical presentation of results of a regression analysis.

Confidence bands are closely related to confidence intervals, which represent the uncertainty in an estimate of a single numerical value. "As confidence intervals, by construction, only refer to a single point, they are narrower (at this point) than a confidence band which is supposed to hold simultaneously at many points."^[21]

This example assumes that the samples are drawn from a Normal distribution. The basic procedure for calculating a confidence interval for a population mean is as follows:

Let X be a random sample from a probability distribution with statistical parameters θ, which is a quantity to be estimated, and φ, representing quantities that are not of immediate interest. A confidence interval for the parameter θ, with confidence level or confidence coefficient γ, is an interval with random endpoints (u(X), v(X)), determined by the pair of random variables u(X) and v(X), with the property:

The quantities φ in which there is no immediate interest are called nuisance parameters, as statistical theory still needs to find some way to deal with them. The number γ, with typical values close to but not greater than 1, is sometimes given in the form 1 − α (or as a percentage 100%·(1 − α)), where α is a small non-negative number, close to 0.

Here Prθ,φ indicates the probability distribution of X characterised by (θ, φ). An important part of this specification is that the random interval (u(X), v(X)) covers the unknown value θ with a high probability no matter what the true value of θ actually is.

Note that here Prθ,φ need not refer to an explicitly given parameterized family of distributions, although it often does. Just as the random variable X notionally corresponds to other possible realizations of x from the same population or from the same version of reality, the parameters (θ, φ) indicate that we need to consider other versions of reality in which the distribution of X might have different characteristics.

In a specific situation, when x is the outcome of the sample X, the interval (u(x), v(x)) is also referred to as a confidence interval for θ. Note that it is no longer possible to say that the (observed) interval (u(x), v(x)) has probability γ to contain the parameter θ. This observed interval is just one realization of all possible intervals for which the probability statement holds.

In many applications, confidence intervals that have exactly the required confidence level are hard to construct. But practically useful intervals can still be found: the rule for constructing the interval may be accepted as providing a confidence interval at level γ if

to an acceptable level of approximation. Alternatively, some authors^[24] simply require that

which is useful if the probabilities are only partially identified, or imprecise.

When applying standard statistical procedures, there will often be standard ways of constructing confidence intervals. These will have been devised so as to meet certain desirable properties, which will hold given that the assumptions on which the procedure rely are true. These desirable properties may be described as: validity, optimality, and invariance. Of these "validity" is most important, followed closely by "optimality". "Invariance" may be considered as a property of the method of derivation of a confidence interval rather than of the rule for constructing the interval. In non-standard applications, the same desirable properties would be sought.

For non-standard applications, there are several routes that might be taken to derive a rule for the construction of confidence intervals. Established rules for standard procedures might be justified or explained via several of these routes. Typically a rule for constructing confidence intervals is closely tied to a particular way of finding a point estimate of the quantity being considered.

A machine fills cups with a liquid, and is supposed to be adjusted so that the content of the cups is 250 g of liquid. As the machine cannot fill every cup with exactly 250.0 g, the content added to individual cups shows some variation, and is considered a random variable X. This variation is assumed to be normally distributed around the desired average of 250 g, with a standard deviation, σ, of 2.5 g. To determine if the machine is adequately calibrated, a sample of n = 25 cups of liquid is chosen at random and the cups are weighed. The resulting measured masses of liquid are X1, ..., X25, a random sample from X.

To get an impression of the expectation μ, it is sufficient to give an estimate. The appropriate estimator is the sample mean:

The sample shows actual weights x1, ..., x25, with mean:

If we take another sample of 25 cups, we could easily expect to find mean values like 250.4 or 251.1 grams. A sample mean value of 280 grams however would be extremely rare if the mean content of the cups is in fact close to 250 grams. There is a whole interval around the observed value 250.2 grams of the sample mean within which, if the whole population mean actually takes a value in this range, the observed data would not be considered particularly unusual. Such an interval is called a confidence interval for the parameter μ. How do we calculate such an interval? The endpoints of the interval have to be calculated from the sample, so they are statistics, functions of the sample X1, ..., X25 and hence random variables themselves.

In our case we may determine the endpoints by considering that the sample mean X from a normally distributed sample is also normally distributed, with the same expectation μ, but with a standard error of:

By standardizing, we get a random variable:

dependent on the parameter μ to be estimated, but with a standard normal distribution independent of the parameter μ. Hence it is possible to find numbers −z and z, independent of μ, between which Z lies with probability 1 − α, a measure of how confident we want to be.

We take 1 − α = 0.95, for example. So we have:

The number z follows from the cumulative distribution function, in this case the cumulative normal distribution function:

and we get:

In other words, the lower endpoint of the 95% confidence interval is:

and the upper endpoint of the 95% confidence interval is:

With the values in this example, the confidence interval is:

This might be interpreted as: with probability 0.95 we will find a confidence interval in which the value of parameter μ will be between the stochastic endpoints

and

This does not mean there is 0.95 probability that the value of parameter μ is in the interval obtained by using the currently computed value of the sample mean,

Instead, every time the measurements are repeated, there will be another value for the mean X of the sample. In 95% of the cases μ will be between the endpoints calculated from this mean, but in 5% of the cases it will not be. The actual confidence interval is calculated by entering the measured masses in the formula. Our 0.95 confidence interval becomes:

In other words, the 95% confidence interval is between the lower endpoint 249.22 g and the upper endpoint 251.18 g.

As the desired value 250 of μ is within the resulted confidence interval, there is no reason to believe the machine is wrongly calibrated.

The calculated interval has fixed endpoints, where μ might be in between (or not). Thus this event has probability either 0 or 1. One cannot say: "with probability (1 − α) the parameter μ lies in the confidence interval." One only knows that by repetition in 100(1 − α)% of the cases, μ will be in the calculated interval. In 100α% of the cases however it does not. And unfortunately one does not know in which of the cases this happens. That is (instead of using the term "probability") why one can say: "with confidence level 100(1 − α) %, μ lies in the confidence interval."

The maximum error is calculated to be 0.98 since it is the difference between the value that we are confident of with upper or lower endpoint.

The figure on the right shows 50 realizations of a confidence interval for a given population mean μ. If we randomly choose one realization, the probability is 95% we end up having chosen an interval that contains the parameter; however, we may be unlucky and have picked the wrong one. We will never know; we are stuck with our interval.

Suppose {X1, ..., X**n} is an independent sample from a normally distributed population with unknown (parameters) mean μ and variance σ2. Let

Where X is the sample mean, and S2 is the sample variance. Then

has a Student's t-distribution with n − 1 degrees of freedom.^[26] Note that the distribution of T does not depend on the values of the unobservable parameters μ and σ2; i.e., it is a pivotal quantity. Suppose we wanted to calculate a 95% confidence interval for μ. Then, denoting c as the 97.5th percentile of this distribution,

Consequently,

and we have a theoretical (stochastic) 95% confidence interval for μ.

After observing the sample we find values x for X and s for S, from which we compute the confidence interval

an interval with fixed numbers as endpoints, of which we can no longer say there is a certain probability it contains the parameter μ; either μ is in this interval or isn't.

Confidence intervals are one method of interval estimation, and the most widely used in frequentist statistics. An analogous concept in Bayesian statistics is credible intervals, while an alternative frequentist method is that of prediction intervals which, rather than estimating parameters, estimate the outcome of future samples. For other approaches to expressing uncertainty using intervals, see interval estimation.

A prediction interval for a random variable is defined similarly to a confidence interval for a statistical parameter. Consider an additional random variable Y which may or may not be statistically dependent on the random sample X. Then (u(X), v(X)) provides a prediction interval for the as-yet-to-be observed value y of Y if

Here Prθ,φ indicates the joint probability distribution of the random variables (X, Y), where this distribution depends on the statistical parameters (θ, φ).

A Bayesian interval estimate is called a credible interval. Using much of the same notation as above, the definition of a credible interval for the unknown true value of θ is, for a given γ,^[27]

Here Θ is used to emphasize that the unknown value of θ is being treated as a random variable. The definitions of the two types of intervals may be compared as follows.

Note that the treatment of the nuisance parameters above is often omitted from discussions comparing confidence and credible intervals but it is markedly different between the two cases.

In some simple standard cases, the intervals produced as confidence and credible intervals from the same data set can be identical. They are very different if informative prior information is included in the Bayesian analysis, and may be very different for some parts of the space of possible data even if the Bayesian prior is relatively uninformative.

There is disagreement about which of these methods produces the most useful results: the mathematics of the computations are rarely in question–confidence intervals being based on sampling distributions, credible intervals being based on Bayes' theorem–but the application of these methods, the utility and interpretation of the produced statistics, is debated.

An approximate confidence interval for a population mean can be constructed for random variables that are not normally distributed in the population, relying on the central limit theorem, if the sample sizes and counts are big enough. The formulae are identical to the case above (where the sample mean is actually normally distributed about the population mean). The approximation will be quite good with only a few dozen observations in the sample if the probability distribution of the random variable is not too different from the normal distribution (e.g. its cumulative distribution function does not have any discontinuities and its skewness is moderate).

One type of sample mean is the mean of an indicator variable, which takes on the value 1 for true and the value 0 for false. The mean of such a variable is equal to the proportion that has the variable equal to one (both in the population and in any sample). This is a useful property of indicator variables, especially for hypothesis testing. To apply the central limit theorem, one must use a large enough sample. A rough rule of thumb is that one should see at least 5 cases in which the indicator is 1 and at least 5 in which it is 0. Confidence intervals constructed using the above formulae may include negative numbers or numbers greater than 1, but proportions obviously cannot be negative or exceed 1. Additionally, sample proportions can only take on a finite number of values, so the central limit theorem and the normal distribution are not the best tools for building a confidence interval. See "Binomial proportion confidence interval" for better methods which are specific to this case.

Since confidence interval theory was proposed, a number of counter-examples to the theory have been developed to show how the interpretation of confidence intervals can be problematic, at least if one interprets them naïvely.

Welch ^[28] presented an example which clearly shows the difference between the theory of confidence intervals and other theories of interval estimation (including Fisher's fiducial intervals and objective Bayesian intervals). Robinson ^[29] called this example "[p]ossibly the best known counterexample for Neyman's version of confidence interval theory." To Welch, it showed the superiority of confidence interval theory; to critics of the theory, it shows a deficiency. Here we present a simplified version.

A fiducial or objective Bayesian argument can be used to derive the interval estimate

This counter-example is used to argue against naïve interpretations of confidence intervals. If a confidence procedure is asserted to have properties beyond that of the nominal coverage (such as relation to precision, or a relationship with Bayesian inference), those properties must be proved; they do not follow from the fact that a procedure is a confidence procedure.

This behavior is consistent with the relationship between the confidence procedure and significance testing: as F becomes so small that the group means are much closer together than we would expect by chance, a significance test might indicate rejection for most or all values of ω2. Hence the interval will be very narrow or even empty (or, by a convention suggested by Steiger, containing only 0). However, this does not indicate that the estimate of ω2 is very precise. In a sense, it indicates the opposite: that the trustworthiness of the results themselves may be in doubt. This is contrary to the common interpretation of confidence intervals that they reveal the precision of the estimate.