Editing Quartile (section)

== Outliers ==
There are methods by which to check for [[outliers]] in the discipline of statistics and statistical analysis. Outliers could be a result from a shift in the location (mean) or in the scale (variability) of the process of interest.<ref>{{Cite journal|last=Walfish|first=Steven|date=November 2006|title=A Review of Statistical Outlier Method|url=http://www.statisticaloutsourcingservices.com/|journal=Pharmaceutical Technology}}</ref> Outliers could also be evidence of a sample population that has a non-normal distribution or of a contaminated population data set. Consequently, as is the basic idea of [[descriptive statistics]], when encountering an [[outlier]], we have to explain this value by further analysis of the cause or origin of the outlier.  In cases of extreme observations, which are not an infrequent occurrence, the typical values must be analyzed. The [[Interquartile Range]] (IQR), defined as the difference between the upper and lower quartiles (<math display="inline">Q_3 - Q_1 </math>), may be used to characterize the data when there may be extremities that skew the data; the [[interquartile range]] is a relatively [[robust statistic]] (also sometimes called "resistance") compared to the [[Range (statistics)|range]] and [[standard deviation]]. There is also a mathematical method to check for outliers and determining "fences", upper and lower limits from which to check for outliers.

After determining the first (lower) and third (upper) quartiles (<math display="inline">Q_1</math> and <math display="inline">Q_3</math> respectively) and the interquartile range (<math display="inline">\textrm{IQR} = Q_3 - Q_1  </math>) as outlined above, then fences are calculated using the following formula:

: <math>\text{Lower fence} = Q_1 - (1.5 \times \mathrm{IQR})    </math>
: <math>\text{Upper fence} = Q_3 + (1.5 \times \mathrm{IQR})</math>[[File:Boxplot outliers example.jpg|thumb|Boxplot Diagram with Outliers]]

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 is a boundary of a fence. It is common for the lower and upper fences along with the outliers to be represented by a [[Box plot|boxplot]]. For the boxplot shown on the right, only the vertical heights correspond to the visualized data set while horizontal width of the box is irrelevant. Outliers located outside the fences in a boxplot can be marked as any choice of symbol, such as an "x" or "o". The fences are sometimes also referred to as "whiskers" while the entire plot visual is called a "box-and-whisker" plot.

When spotting an outlier in the data set by calculating the interquartile ranges and boxplot features, it might be easy to mistakenly view it as evidence that the population is non-normal or that the sample is contaminated. However, this method should not take place of a [[hypothesis test]] for determining normality of the population. The significance of the outliers varies depending on the sample size. If the sample is small, then it is more probable to get interquartile ranges that are unrepresentatively small, leading to narrower fences. Therefore, it would be more likely to find data that are marked as outliers.<ref>{{Cite journal|last=Dawson|first=Robert|date=July 1, 2011|title=How Significant is a Boxplot Outlier?|journal=Journal of Statistics Education|volume=19|issue=2|doi=10.1080/10691898.2011.11889610|doi-access=free}}</ref>