Interquartile range#Quartile deviation

Unlike total range, the interquartile range has a breakdown point of 25%{{cite news |title=Explicit Scale Estimators with High Breakdown Point |first1=Peter J. |last1=Rousseeuw |first2=Christophe |last2=Croux |work=L1-Statistical Analysis and Related Methods |editor=Y. Dodge |location=Amsterdam |publisher=North-Holland |year=1992 |pages=77–92 |url=https://feb.kuleuven.be/public/u0017833/PDF-FILES/l11992.pdf}} and is thus often preferred to the total range.

The IQR is used to build box plots, simple graphical representations of a probability distribution.

The IQR is used in businesses as a marker for their income rates.

For a symmetric distribution (where the median equals the midhinge, the average of the first and third quartiles), half the IQR equals the median absolute deviation (MAD).

The median is the corresponding measure of central tendency.

The IQR can be used to identify outliers (see below). The IQR also may indicate the skewness of the dataset.

{{Anchor|Quartile deviation}} The quartile deviation or semi-interquartile range is defined as half the IQR.{{cite book |first=G. Udny |last=Yule |title=An Introduction to the Theory of Statistics |url=https://archive.org/details/in.ernet.dli.2015.223539 |publisher=Charles Griffin and Company |date=1911 |pages=[https://archive.org/details/in.ernet.dli.2015.223539/page/n170 147]–148}}

The IQR of a set of values is calculated as the difference between the upper and lower quartiles, Q₃ and Q₁. Each quartile is a median{{Cite book|title=Beta [beta] mathematics handbook : concepts, theorems, methods, algorithms, formulas, graphs, tables|last=Bertil.|first=Westergren|date=1988|publisher=Studentlitteratur|isbn=9144250517|oclc=18454776|page=348}} calculated as follows.

Given an even 2n or odd 2n+1 number of values

:first quartile Q₁ = median of the n smallest values

:third quartile Q₃ = median of the n largest values

The second quartile Q₂ is the same as the ordinary median.

The following table has 13 rows, and follows the rules for the odd number of entries.

For the data in this table the interquartile range is IQR = Q₃ − Q₁ = 119 - 31 = 88.

+−−−−−+−+
* |−−−−−−−−−−−|     | |−−−−−−−−−−−|
+−−−−−+−+
+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+   Number line
0   1   2   3   4   5   6   7   8   9   10  11  12

For the data set in this box plot:

• Lower (first) quartile Q₁ = 7
• Median (second quartile) Q₂ = 8.5
• Upper (third) quartile Q₃ = 9
• Interquartile range, IQR = Q₃ - Q₁ = 2
• Lower 1.5*IQR whisker = Q₁ - 1.5 * IQR = 7 - 3 = 4. (If there is no data point at 4, then the lowest point greater than 4.)
• Upper 1.5*IQR whisker = Q₃ + 1.5 * IQR = 9 + 3 = 12. (If there is no data point at 12, then the highest point less than 12.)
• Pattern of latter two bullet points: If there are no data points at the true quartiles, use data points slightly "inland" (closer to the median) from the actual quartiles.

This means the 1.5*IQR whiskers can be uneven in lengths. The median, minimum, maximum, and the first and third quartile constitute the Five-number summary.Dekking, Kraaikamp, Lopuhaä & Meester, pp. 235–237

The interquartile range of a continuous distribution can be calculated by integrating the probability density function (which yields the cumulative distribution function—any other means of calculating the CDF will also work). The lower quartile, Q₁, is a number such that integral of the PDF from -∞ to Q₁ equals 0.25, while the upper quartile, Q₃, is such a number that the integral from -∞ to Q₃ equals 0.75; in terms of the CDF, the quartiles can be defined as follows:

:$Q\_1\; =\; \backslash text\{CDF\}^\{-1\}(0.25)\; ,$

:$Q\_3\; =\; \backslash text\{CDF\}^\{-1\}(0.75)\; ,$

where CDF⁻¹ is the quantile function.

The interquartile range and median of some common distributions are shown below

The IQR, mean, and standard deviation of a population P can be used in a simple test of whether or not P is normally distributed, or Gaussian. If P is normally distributed, then the standard score of the first quartile, z₁, is −0.67, and the standard score of the third quartile, z₃, is +0.67. Given mean = $\backslash bar\{P\}$ and standard deviation = σ for P, if P is normally distributed, the first quartile

:$Q\_1\; =\; (\backslash sigma\; \backslash ,\; z\_1)\; +\; \backslash bar\{P\}$

and the third quartile

:$Q\_3\; =\; (\backslash sigma\; \backslash ,\; z\_3)\; +\; \backslash bar\{P\}$

If the actual values of the first or third quartiles differ substantially{{Clarify|date=December 2012}} from the calculated values, P is not normally distributed. However, a normal distribution can be trivially perturbed to maintain its Q1 and Q2 std. scores at 0.67 and −0.67 and not be normally distributed (so the above test would produce a false positive). A better test of normality, such as Q–Q plot would be indicated here.

File:Box-Plot mit Interquartilsabstand.png with four mild outliers and one extreme outlier. In this chart, outliers are defined as mild above Q3 + 1.5 IQR and extreme above Q3 + 3 IQR.]]

The interquartile range is often used to find outliers in data. Outliers here are defined as observations that fall below Q1 − 1.5 IQR or above Q3 + 1.5 IQR. In a boxplot, the highest and lowest occurring value within this limit are indicated by whiskers of the box (frequently with an additional bar at the end of the whisker) and any outliers as individual points.

• {{Annotated link|Interdecile range}}
• {{Annotated link|Midhinge}}
• {{Annotated link|Probable error}}
• {{Annotated link|Robust measures of scale}}

{{reflist|refs=

{{cite book |title=Understanding Statistics |first1=Graham|last1=Upton|first2=Ian|last2= Cook|year=1996 |publisher=Oxford University Press |isbn=0-19-914391-9 |page=55 |url=https://books.google.com/books?id=vXzWG09_SzAC&q=interquartile+range&pg=PA55}}

Zwillinger, D., Kokoska, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, CRC Press. {{ISBN|1-58488-059-7}} page 18.

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{{Statistics|descriptive}}

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Category:Scale statistics

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