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* The second quartile (''Q''<sub>2</sub>) is the [[median]] of a data set; thus 50% of the data lies below this point.
* The third quartile (''Q''<sub>3</sub>) is the 75th percentile where lowest 75% data is below this point. It is known as the ''upper'' quartile, as 75% of the data lies below this point.<ref name=":0">{{Cite book |author=Dekking, Michel <!--1946– --> |url=https://archive.org/details/modernintroducti0000unse_h6a1 |title=A modern introduction to probability and statistics: understanding why and how |date=2005 |publisher=Springer |isbn=978-1-85233-896-1 |location=London |pages=[https://archive.org/details/modernintroducti0000unse_h6a1/page/236/ 236-238] |oclc=262680588 |url-access=limited}}</ref>
Along with the minimum and maximum of the data (which are also quartiles), the three quartiles described above provide a [[five-number summary]] of the data. This summary is important in statistics because it provides information about both the [[Mean (Statistics)|center]] and the [[Statistical dispersion|spread]] of the data. Knowing the lower and upper quartile provides information on how big the spread is and if the dataset is [[Skewness|skewed]] toward one side. Since quartiles divide the number of data points evenly, the [[Range (statistics)|range]] is generally not the same between adjacent quartiles (i.e. usually (''Q''<sub>3</sub> - ''Q''<sub>2</sub>) ≠ (''Q''<sub>2</sub> - ''Q''<sub>1</sub>)). [[Interquartile range]] (IQR) is defined as the difference between the 75th and 25th percentiles or ''Q''<sub>3</sub> - ''Q''<sub>
== Definitions ==
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==== Method 1 ====
# Use the [[median]] to divide the ordered data set into two
#* If there are an odd number of data points in the original ordered data set, '''do not include''' the median (the central value in the ordered list) in either half.
#* If there are an even number of data points in the original ordered data set, split this data set exactly in half.
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==== Method 2 ====
# Use the
#* If there are an odd number of data points in the original ordered data set, '''include''' the median (the central value in the ordered list) in both halves.
#* If there are an even number of data points in the original ordered data set, split this data set exactly in half.
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==== Method 3 ====
# Use the median to divide the ordered data set into two halves. The median becomes the second quartiles.
# If there are even numbers of data points, then Method 3 starts off the same as Method 1 or Method 2 above and you can choose to include or not include the median as a new datapoint. If you choose to include the median as the new datapoint, then proceed to the step 2 or 3 below because you now have an odd number of datapoints. If you do not choose the median as the new data point, then continue the Method 1 or 2 where you have started.▼
## If there are odd numbers of data points, then go to the next step.
▲## If there are even numbers of data points, then the Method 3 starts off the same as the Method 1 or the Method 2 above and you can choose to include or not include the median as a new datapoint. If you choose to include the median as the new datapoint, then proceed to the step 2 or 3 below because you now have an odd number of datapoints. If you do not choose the median as the new data point, then continue the Method 1 or 2 where you have started.
# If there are (4''n''+1) data points, then the lower quartile is 25% of the ''n''th data value plus 75% of the (''n''+1)th data value; the upper quartile is 75% of the (3''n''+1)th data point plus 25% of the (3''n''+2)th data point.
# If there are (4''n''+3) data points, then the lower quartile is 75% of the (''n''+1)th data value plus 25% of the (''n''+2)th data value; the upper quartile is 25% of the (3''n''+2)th data point plus 75% of the (3''n''+3)th data point.
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==== Example 1 ====
Ordered Data Set (of an odd number of data points): 6, 7, 15, 36, 39, '''40''', 41, 42, 43, 47, 49.
The bold number (40) is the median splitting the data set into two halves with equal number of data points.
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==== Example 2 ====
Ordered Data Set (of an even number of data points): 7, 15, '''36, 39''', 40, 41.
The bold numbers (36, 39) are used to calculate the median as their average. As there are an even number of data points, the first three methods all give the same results. (The Method 3 is executed such that the median is not chosen as a new data point and the Method 1 started.)
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<math>F_X(x) = P(X \leq x)</math>.<ref name=":0" />
The [[Cumulative distribution function|CDF]] gives the probability that the random variable <math>X</math> is less than or equal to the value <math>x</math>. Therefore, the first quartile is the value of <math>x</math> when <math>F_X(x) = 0.25</math>, the second quartile is <math>x</math> when <math>F_X(x) = 0.5</math>, and the third quartile is <math>x</math> when <math>F_X(x) = 0.75</math>.<ref>{{Cite web|url=https://math.bme.hu/~nandori/Virtual_lab/stat/dist/CDF.pdf|title=6. Distribution and Quantile Functions|website=math.bme.hu}}</ref> The values of <math>x</math> can be found with the [[quantile function]] <math>Q(p)</math> where <math>p = 0.25</math> for the first quartile, <math>p = 0.5</math> for the second quartile, and <math>p = 0.75</math> for the third quartile. The quantile function is the inverse of the cumulative distribution function if the cumulative distribution function is [[Monotonic function|monotonically increasing]] because the [[Bijection|one-to-one correspondence]] between the input and output of the cumulative distribution function holds.
== 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.
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
: <math>\text{Upper fence} = Q_3 + (1.5
When spotting an outlier in the data set by calculating the interquartile ranges and boxplot features, it might be
== Computer software for quartiles ==
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=== Excel
The Excel function ''QUARTILE(array, quart)'' provides the desired quartile value for a given array of data, using Method 3 from above. In the ''
▲The Excel function ''QUARTILE(array, quart)'' provides the desired quartile value for a given array of data, using Method 3 from above. In the ''Quartile'' function, array is the dataset of numbers that is being analyzed and quart is any of the following 5 values depending on which quartile is being calculated. <ref>{{Cite web|url=https://exceljet.net/excel-functions/excel-quartile-function|title=How to use the Excel QUARTILE function {{!}} Exceljet|website=exceljet.net|access-date=December 11, 2019}}</ref>
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|Maximum value
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MATLAB:▼
In order to calculate quartiles in Matlab, the function ''quantile''(''A'',''p
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* [http://www.hackmath.net/en/calculator/quartile-q1-q2-q3-calculation Quartiles calculator] – simple quartiles calculator
* [http://www.vias.org/tmdatanaleng/cc_quartile.html Quartiles] – An example how to calculate it
* [https://quartilecalculator.net/ Quartiles Calculator] – online quartile and interquartile range calculator
[[Category:Summary statistics]]
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