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{{ref improve|date=May 2012}}
[[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]]


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==
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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 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]].
 
Filtering of [[outlier]]s (see [[Interquartile_range#Interquartile_range_and_outliers|below]]).
 
==Examples==
===Data set in a table===
:{| class="wikitable" style="text-align:center;"
|-
! width="40px"| i
! width="40px" |x[i]
! Quartile
|-
| 1|| 102 ||
|-
| 2|| 104 ||
|-
| 3|| 105|| Q<sub>1</sub>
|-
| 4|| 107 ||
|-
| 5|| 108||
|-
| 6|| 109|| Q<sub>2</sub><br /> (median)
|-
| 7|| 110 ||
|-
| 8|| 112 ||
|-
| 9|| 115|| Q<sub>3</sub>
|-
| 10|| 116 ||
|-
| 11|| 118 ||
|-
|}
 
For the data in this table the interquartile range is IQR&nbsp;=&nbsp;115&nbsp;&minus;&nbsp;105 = 10.
 
===Data set in a plain-text box plot===
<pre style="font-family:monospace">
                   
                            +-----+-+   
  o          *    |-------|    | |---|
                            +-----+-+   
                   
+---+---+---+---+---+---+---+---+---+---+---+---+  number line
0  1  2  3  4  5  6  7  8  9  10  11  12 
</pre>
 
For the data set in this [[box plot]]:
* lower (first) quartile ''Q''<sub>1</sub> = 7
* median (second quartile) ''Q''<sub>2</sub> = 8.5
* upper (third) quartile  ''Q''<sub>3</sub> = 9
* interquartile range, IQR = ''Q''<sub>3</sub>&nbsp;−&nbsp; ''Q''<sub>1</sub> = 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, ''Q''<sub>1</sub>, is a number such that integral of the PDF from -∞ to ''Q''<sub>1</sub> equals 0.25, while the upper quartile, ''Q''<sub>3</sub>, is such a number that the integral from -∞ to ''Q''<sub>3</sub> equals 0.75; in terms of the CDF, the quartiles can be defined as follows:
 
:<math>Q_1 = \text{CDF}^{-1}(0.25) ,</math>
 
:<math>Q_3 = \text{CDF}^{-1}(0.75) ,</math>
 
where CDF<sup>−1</sup> is the [[quantile function]].
 
The interquartile range and median of some common distributions are shown below
 
{| class="wikitable"
|-
! Distribution
! Median
! IQR
|-
| [[Normal distribution|Normal]]
| μ
| 2 Φ<sup>&minus;1</sup>(0.75)σ ≈ 1.349σ
|-
| [[Laplace distribution|Laplace]]
| μ
| 2''b''&nbsp;ln(2) ≈ 1.386''b''
|-
| [[Cauchy distribution|Cauchy]]
| μ
|2γ
|}
 
===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 distribution|Gaussian]]. If ''P'' is normally distributed, then the [[standard score]] of the first quartile, ''z''<sub>1</sub>, is -0.67, and the standard score of the third quartile, ''z''<sub>3</sub>, is +0.67. Given ''mean''&nbsp;=&nbsp;''X'' and ''standard&nbsp;deviation''&nbsp;=&nbsp;σ for ''P'', if ''P'' is normally distributed, the first quartile
 
:<math>Q_1 = (\sigma \, z_1) + X</math>
 
and the third quartile
 
:<math>Q_3 = (\sigma \, z_3) + X</math>
 
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==
* [[Midhinge]]
* [[Interdecile range]]
* [[Robust measures of scale]]
 
==References==
{{reflist|refs=
 
<ref name=Upton>{{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=http://books.google.com/books?id=vXzWG09_SzAC&pg=PA55&dq=interquartile+range#v=onepage&q=interquartile%20range&f=false}}</ref>
 
<ref name= ZK>Zwillinger, D., Kokoska, S. (2000) ''CRC Standard Probability and Statistics Tables and Formulae'', CRC Press. ISBN 1-58488-059-7 page 18.</ref>
 
}}
 
{{Statistics|descriptive}}
 
{{DEFAULTSORT:Interquartile Range}}
[[Category:Scale statistics]]
[[Category:Wikipedia articles with ASCII art]]

Latest revision as of 16:21, 31 December 2014


Advertising and marketing is costly. This is not information but what exactly is usually astonishing tome is how the business owner overlooks himself as a method of promotion and advertising. I'm-not referring to walking around with a sandwich-board or wearing tee-shirts emblazoned with the business logo. And I'm-not discussing turning up at every conceivable festival show with a wallet high in business cards. I am speaking about thinking of the dog owner as a product of the company and selling the product like every other in the company point. I'd like to explain.

Banners on the billboard are very valuable. Of course, you have to determine whether this technique of Ad Agency fits inside your budget, since it is somewhat pricey. Furthermore, the use of billboard advertising additionally is determined by the scale of the function. You needn't set up an advertisement for a garage sale over a billboard. Billboard advertising is another one of the things that vouches for the event as being a great, genuine one, if it's a sizable-scale event you're having, such as a food festival, a sports event, or even a music festival.



But, I'm-not just meaning one guide, one movie, and on occasion even one website. You may need to totally dominate the area online search engine results. To do that, you need to get your true information (articles, videos, etc) on several sites, all providing the capacity to get listed highly in the Searchengines (Google is actually a search engine). Buying "Featured Listings" in online sites is one such waste of profit.

"The offer that is up, I do believe, is performed with a boat load of taste," said Wamp, a representative from Chattanooga. "It makes assertions that may be painful, but it's all accurate. I would not state it is anything besides a normal tv professional. It has a fellow talking about one of many other applicants, but I believe it is done with a huge quantity of class.

The Republican primary for governor is Aug. 5, and the present piece at hand is an ad byWampthat could be the most strong slap by one campaign at an opponent inthe raceso far, and early voting started on Friday. And it appears the fact is in the eye of the beholder.

"Weare pleased we've had more contributors contribute more income than any campaign beforehand in Tennessee during this period, and we believe that's a testament to the wide, serious support Bill has over the state, and that's seldom buying the election," Ingram said.

List characteristics. McDonald's Big Mac had "two all-beef patties, special sauce, lettuce, cheese, pickles, onions on a sesame-seed bun?" The same process has been used by nyquil.

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