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As the level of confidence decreases, the size of the corresponding interval will decrease. In most practical research, the standard deviation for the population of interest is not known. 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. Definition of Standard DeviationStandard deviation measures how widely spread data points are. If a high proportion of data points lie near the mean value, then the standard deviation is small.
If a high proportion of data points lie far from the mean value, then the standard deviation is large. An experiment that yields data with a high standard deviation is said to have low precision.
The equations provide precision measures for a limited number of repetitive measurements, i.e.

The equation at the end is the true standard deviation for any number of repeat measurements. The standard deviation, s, is a statistical measure of the precision for a series of repetitive measurements. The advantage of using s to quote uncertainty in a result is that it has the same units as the experimental data.
The relative standard deviation (RSD) is useful for comparing the uncertainty between different measurements of varying absolute magnitude. Confidence limits are another statistical measure of the precision for a series of repetitive measurements. You would say that with some confidence, for example 95%, the true value is between the confidence limits. The t term is taken from a table for the number of degrees of freedom and the degree of confidence desired. The quantitative measures of precision described above are the most common for reporting analytical results. You might encounter other measures of precisions, and several other quantities are listed here for completeness. The advantage of working with variance is that variances from independent sources of variation may be summed to obtain a total variance for a measurement.

All of the equations above are intended to obtain the precision of a relatively small numbers of repeated measurements.
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. 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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