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Encyclopedia > Interquartile range

In descriptive statistics, the interquartile range (IQR), also called the midspread and middle fifty is the range between the third and first quartiles and is a measure of statistical dispersion. The interquartile range is a more stable statistic than the (total) range, and is often preferred to the latter statistic. Descriptive statistics are used to describe the basic features of the data in a study. ... In descriptive statistics, a quartile is any of the three values which divide the sorted data set into four equal parts, so that each part represents 1/4th of the sample or population. ... In descriptive statistics, statistical dispersion (also called statistical variability) is quantifiable variation of measurements of differing members of a population within the scale on which they are measured. ... In descriptive statistics, the range is the length of the smallest interval which contains all the data. ...


Since 25% of the data are less than or equal to the first quartile and 25% are greater than or equal to the third quartile, the IQR is expected to include about half of the data. The length of the IQR should be measured in the same units as the data.


One should note that, in ungrouped data(like in the example below), Q2 should be the median of the data. Following the Q2 (Q3 or Q4) the equation should be as such: **Q2x1.5** for Q3 and **Q3x0.5** for Q2.


Interquartile range is used to build Box plots, that can give a simple graphical representation of a probability distribution. Figure 1. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ...


Example

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
10 118
11 118

From this table, the length of the interquartile range is 115 - 105 = 10.


The median is the corresponding measure of central tendency. In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, central tendency is an average of a set of measurements, the word average being variously construed as mean, median, or other measure of location, depending on the context. ...


Interquartile range of distributions

The interquartile range of a continuous distribution can be calculated by integrating the Probability density function. The lower quartile, a, is the integral from minus infinity to a that equals 0.25, while the upper quartile, b, is the integral from b to infinity that equals 0.75. In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...


[insert equations here]


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

Distribution Median IQR
Normal μ -1(0.75)≈ 1.349
Laplace μ
Cauchy μ

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ...

See also

  • Boxplot

  Results from FactBites:
 
Stats: What is an interquartile range? (122 words)
The interquartile range (IQR) is the distance between the 75
The IQR is essentially the range of the middle 50% of the data.
Select halfway between the 11th and 12th observations (4.80 and 5.26) and halfway between the 3rd and 4th observations (3.49 and 3.68).
Establishing the arm's length price: gathering your own evidence (1360 words)
The median is the mid-point of the interquartile range.
The mid-point, or median of the interquartile range is 4.8%.
The mid-point or median of the interquartile range is 4.7%.
  More results at FactBites »

 
 

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