For a univariate random variable $$ x $$, a confidence interval is a range $$ [a,b] $$, $$ a<b $$, so that $$ x $$ lies between $$ a $$ and $$ b $$ with given probability.

If $$ x $$ has a continuous probability distribution $$ P $$, then $$ [a,b] $$ is a 95% confidence interval if $$ \int_a^b P(x) = 0.95 $$.

The figure on the right shows the 75% confidence interval (in orange) for a normally distributed variable with mean 4 and standard deviation 1. Commonly used reliability values are 99% (referred to as A-basis allowables) and 90% reliability values (referred to as B-basis allowables). A confidence interval of 95% is specified in both MIL-HDBK-5 and MIL-HDBK-17 for calculating A and B basis values.

We expect 90% (for a B-basis value) of the strength values from the population to be greater than this value, or falling on the right side of the 10th percentile line.

Neither the service provider nor the domain owner maintain any relationship with the advertisers. For example, a 95% confidence interval is a range in which $$ x $$ falls 95% of the time (or with probability 0.95). If $$ x $$ has a symmetric distribution, then the confidence intervals are usually centred around the mean. This means that only 1% and 10% of the material is likely to fall below these strength values.

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Confidence intervals provide intuitive summaries of the statistics of the variable $$ x $$. However, non-centred confidence intervals are possible and are better described by their lower and upper quantiles or levels.

The top plot shows 100 samples from $$ x $$, the middle plot shows the probability density function of $$ x $$ (with 75% of the probability in orange) and the bottom plot shows the corresponding cumulative density function (the height of the highlighted portion is 0.75). For example, a 50% confidence interval would usually lie between the 25% and 75% quantiles, but could in theory also lie between the 10% and 60% quantiles, although this would be rare in practice. UncertML allows you the flexibility to specify non-symmetric confidence intervals however in practice we would expect the main usage to be for symmetric intervals and this will be addressed in the API.

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