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The colored lines are 50% confidence intervals for the mean, μ. At the center of each interval is the sample mean, marked with a diamond. The blue intervals contain the population mean, and the red ones do not. In statistics, a confidence interval (CI) is a tool for estimating a parameter, such as the mean of a population. [1]
A bar chart with confidence intervals (shown as red lines) ... one standard error, or a particular confidence interval (e.g., a 95% interval). These quantities are ...
A confidence interval states there is a 100γ% confidence that the parameter of interest is within a lower and upper bound. A common misconception of confidence intervals is 100γ% of the data set fits within or above/below the bounds, this is referred to as a tolerance interval, which is discussed below.
Another is the R95, which is the radius of the circle where 95% of the values would fall, a 95% confidence interval. The concept of CEP also plays a role when measuring the accuracy of a position obtained by a navigation system, such as GPS or older systems such as LORAN and Loran-C .
A weaker three-sigma rule can be derived from Chebyshev's inequality, stating that even for non-normally distributed variables, at least 88.8% of cases should fall within properly calculated three-sigma intervals. For unimodal distributions, the probability of being within the interval is at least 95% by the Vysochanskij–Petunin inequality ...
The probability density function (PDF) for the Wilson score interval, plus PDF s at interval bounds. Tail areas are equal. Since the interval is derived by solving from the normal approximation to the binomial, the Wilson score interval ( , + ) has the property of being guaranteed to obtain the same result as the equivalent z-test or chi-squared test.
Confidence bands can be constructed around estimates of the empirical distribution function.Simple theory allows the construction of point-wise confidence intervals, but it is also possible to construct a simultaneous confidence band for the cumulative distribution function as a whole by inverting the Kolmogorov-Smirnov test, or by using non-parametric likelihood methods.
In particular, the quantile is 1.96; therefore a normal random variable will lie outside the interval in only 5% of cases. The following table gives the quantile z p {\textstyle z_{p}} such that X {\displaystyle X} will lie in the range μ ± z p σ {\textstyle \mu \pm z_{p}\sigma } with a specified probability p {\displaystyle p} .