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The normal distribution is NOT assumed nor required in the calculation of control limits. Thus making the IndX/mR chart a very robust tool. Thus making the IndX/mR chart a very robust tool. This is demonstrated by Wheeler using real-world data [ 4 ] , [ 5 ] and for a number of highly non-normal probability distributions.
Minkowski's bound may be used to derive a lower bound for the discriminant of a field K given n, r 1 and r 2. Since an integral ideal has norm at least one, we have 1 ≤ M K , so that | D | ≥ ( π 4 ) r 2 n n n ! ≥ ( π 4 ) n / 2 n n n !
The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above.
The lower fence is the "lower limit" and the upper fence is the "upper limit" of data, and any data lying outside these defined bounds can be considered an outlier. The fences provide a guideline by which to define an outlier, which may be defined in other ways. The fences define a "range" outside which an outlier exists; a way to picture this ...
At about the same time, Makarov, [6] and independently, Rüschendorf [7] solved the problem, originally posed by Kolmogorov, of how to find the upper and lower bounds for the probability distribution of a sum of random variables whose marginal distributions, but not their joint distribution, are known.
In the most straightforward method, the boundary of the lower whisker is the minimum value of the data set, and the boundary of the upper whisker is the maximum value of the data set. Because of this variability, it is appropriate to describe the convention that is being used for the whiskers and outliers in the caption of the box-plot.
In this context, a percentage of an existing product set is evaluated to ensure that a percentage of the population is included within tolerance limits. When creating tolerance intervals, the bounds can be written in terms of an upper and lower tolerance limit, utilizing the sample mean, , and the sample standard deviation, s.
The five-number summary is a set of descriptive statistics that provides information about a dataset. It consists of the five most important sample percentiles: the sample minimum (smallest observation) the lower quartile or first quartile; the median (the middle value) the upper quartile or third quartile; the sample maximum (largest observation)