Difference between revisions of "Interquartile range"

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{{ref improve|date=May 2012}}
 
{{ref improve|date=May 2012}}
In [[descriptive statistics]], the '''interquartile range''' ('''IQR'''), also called the '''midspread''' or '''middle fifty''', is a measure of [[statistical dispersion]], being equal to the difference between the upper and lower [[quartile]]s,<ref name=Upton/><ref name= ZK/> IQR = ''Q''<sub>3</sub>&nbsp;−&nbsp; ''Q''<sub>1</sub>
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[[Image:Boxplot vs PDF.svg|250px|thumb|[[Boxplot]] (with an interquartile range) and a [[probability density function]] (pdf) of a Normal {{maths|N(0,σ<sup>2</sup>)}} Population]]
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 +
In [[descriptive statistics]], the '''interquartile range''' ('''IQR'''), also called the '''midspread''' or '''middle fifty''', is a measure of [[statistical dispersion]], being equal to the difference between the upper and lower [[quartile]]s,<ref name=Upton/><ref name= ZK/> IQR = ''Q''<sub>3</sub>&nbsp;−&nbsp; ''Q''<sub>1</sub>. In other words, the IQR is the 1st quartile subtracted from the 3rd quartile; these quartiles can be clearly seen on a [[box plot]] on the data. It is a [[trimmed estimator]], defined as the 25% trimmed [[mid-range]], and is the most significant basic [[robust measures of scale|robust measure of scale]].
  
 
==Use==
 
==Use==
Unlike  (total) [[range (statistics)|range]], the interquartile range is a [[robust statistic]], having a [[breakdown point]] of 25%, and is thus often preferred to the total range.
+
Unlike  (total) [[range (statistics)|range]], the interquartile range has a [[breakdown point]] of 25%, and is thus often preferred to the total range.
  
The IQR is used to build [[box plot]]s, simple graphical representations of a [[probability distribution]].
+
The IQR is used to build box plots, simple graphical representations of a [[probability distribution]].
  
 
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).
 
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 [[median]] is the corresponding measure of [[central tendency]].
 +
 +
Identification of [[outlier]]s (see [[Interquartile_range#Interquartile_range_and_outliers|below]]).
  
 
==Examples==
 
==Examples==
 
[[Image:Boxplot vs PDF.svg|250px|thumb|[[Boxplot]] (with an interquartile range) and a [[probability density function]] (pdf) of a Normal {{maths|N(0,σ<sup>2</sup>)}} Population]]
 
 
 
===Data set in a table===
 
===Data set in a table===
:{| class="wikitable"
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:{| class="wikitable" style="text-align:center;"
 
|-
 
|-
! i
+
! width="40px"| i
! x[i]
+
! width="40px" |x[i]
 
! Quartile
 
! Quartile
 
|-
 
|-
| 1|| 102
+
| 1|| 102 ||
 
|-
 
|-
| 2|| 104
+
| 2|| 104 ||
 
|-
 
|-
 
| 3|| 105|| Q<sub>1</sub>
 
| 3|| 105|| Q<sub>1</sub>
 
|-
 
|-
| 4|| 107
+
| 4|| 107 ||
 
|-
 
|-
| 5|| 108
+
| 5|| 108||
 
|-
 
|-
| 6|| 109|| Q<sub>2</sub> (median)
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| 6|| 109|| Q<sub>2</sub><br /> (median)
 
|-
 
|-
| 7|| 110
+
| 7|| 110 ||
 
|-
 
|-
| 8|| 112
+
| 8|| 112 ||
 
|-
 
|-
 
| 9|| 115|| Q<sub>3</sub>
 
| 9|| 115|| Q<sub>3</sub>
 
|-
 
|-
| 10|| 116
+
| 10|| 116 ||
 +
|-
 +
| 11|| 118 ||
 
|-
 
|-
| 11|| 118
 
 
|}
 
|}
  
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| [[Normal distribution|Normal]]
 
| [[Normal distribution|Normal]]
 
| μ
 
| μ
| 2 Φ<sup>&minus;1</sup>(0.75) ≈ 1.349<math>\sigma\,</math>
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| 2 Φ<sup>&minus;1</sup>(0.75)σ ≈ 1.349σ ≈ (27/20)σ
 +
 
 
|-
 
|-
 
| [[Laplace distribution|Laplace]]
 
| [[Laplace distribution|Laplace]]
 
| μ
 
| μ
| 2''b''&nbsp;ln(2)
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| 2''b''&nbsp;ln(2) ≈ 1.386''b''
 
|-
 
|-
 
| [[Cauchy distribution|Cauchy]]
 
| [[Cauchy distribution|Cauchy]]
 
| μ
 
| μ
|<math>2\gamma\,</math>
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|
 
|}
 
|}
  
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:<math>Q_3 = (\sigma \, z_3) + X</math>
 
:<math>Q_3 = (\sigma \, z_3) + X</math>
  
If the actual values of the first or third quartiles differ substantially from the calculated values, ''P'' is not normally distributed.
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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.
 +
 
 +
==Interquartile range and outliers==
 +
 
 +
[[File:Box-Plot mit Interquartilsabstand.png|thumb|Figure 3. Box-and-whisker plot with four close and one far away extreme values, defined as outliers above Q3 + 1.5(IQR) and Q3 + 3(IQR), respectively.]]
 +
The interquartile range is often used to find [[outliers]] in data. Outliers are 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 drawn as bar of the ''whiskers'', and the outliers as individual points.
  
 
==See also==
 
==See also==
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* [[Interdecile range]]
 
* [[Interdecile range]]
 
* [[Robust measures of scale]]
 
* [[Robust measures of scale]]
* [[Outlier]]
 
  
 
==References==
 
==References==
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[[Category:Scale statistics]]
 
[[Category:Scale statistics]]
 
[[Category:Wikipedia articles with ASCII art]]
 
[[Category:Wikipedia articles with ASCII art]]
 
[[ca:Rang interquartílic]]
 
[[es:Rango intercuartílico]]
 
[[eu:Kuartil arteko ibiltarte]]
 
[[fa:دامنه بین چارکی]]
 
[[fr:Écart interquartile]]
 
[[it:Scarto interquartile]]
 
[[he:טווח בין רבעוני]]
 
[[nl:Interkwartielafstand]]
 
[[pl:Rozstęp ćwiartkowy]]
 
[[simple:Interquartile range]]
 
[[sl:Kvartilni razmik]]
 
[[tr:Çeyrekler açıklığı]]
 
[[vi:Độ trải giữa]]
 
[[zh:四分差]]
 

Latest revision as of 20:06, 5 January 2015

Template:Ref improve

Boxplot (with an interquartile range) and a probability density function (pdf) of a Normal Template:Maths Population

In descriptive statistics, the interquartile range (IQR), also called the midspread or middle fifty, is a measure of statistical dispersion, being equal to the difference between the upper and lower quartiles,[1][2] IQR = Q3 −  Q1. In other words, the IQR is the 1st quartile subtracted from the 3rd quartile; these quartiles can be clearly seen on a box plot on the data. It is a trimmed estimator, defined as the 25% trimmed mid-range, and is the most significant basic robust measure of scale.

Use

Unlike (total) range, the interquartile range has a breakdown point of 25%, and is thus often preferred to the total range.

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

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.

Identification of outliers (see below).

Examples

Data set in a table

i x[i] Quartile
1 102
2 104
3 105 Q1
4 107
5 108
6 109 Q2
(median)
7 110
8 112
9 115 Q3
10 116
11 118

For the data in this table the interquartile range is IQR = 115 − 105 = 10.

Data set in a plain-text box plot

                    
                             +-----+-+     
   o           *     |-------|     | |---|
                             +-----+-+    
                    
 +---+---+---+---+---+---+---+---+---+---+---+---+   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

Interquartile range of distributions

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:

where CDF−1 is the quantile function.

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

Distribution Median IQR
Normal μ 2 Φ−1(0.75)σ ≈ 1.349σ ≈ (27/20)σ
Laplace μ 2b ln(2) ≈ 1.386b
Cauchy μ

Interquartile range test for normality of distribution

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 = X and standard deviation = σ for P, if P is normally distributed, the first quartile

and the third quartile

If the actual values of the first or third quartiles differ substantiallyTemplate:Clarify from the calculated values, P is not normally distributed.

Interquartile range and outliers

Figure 3. Box-and-whisker plot with four close and one far away extreme values, defined as outliers above Q3 + 1.5(IQR) and Q3 + 3(IQR), respectively.

The interquartile range is often used to find outliers in data. Outliers are 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 drawn as bar of the whiskers, and the outliers as individual points.

See also

References

  1. {{#invoke:citation/CS1|citation |CitationClass=book }}
  2. Zwillinger, D., Kokoska, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, CRC Press. ISBN 1-58488-059-7 page 18.

Template:Statistics