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 = Q_{3} − Q_{1}. 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 midrange, and is the most significant basic robust measure of scale.
Contents

Use 1

Examples 2

Data set in a table 2.1

Data set in a plaintext box plot 2.2

Interquartile range of distributions 3

Interquartile range test for normality of distribution 3.1

Interquartile range and outliers 4

See also 5

References 6
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

Q_{1}

4

107


5

108


6

109

Q_{2}
(median)

7

110


8

112


9

115

Q_{3}

10

116


11

118


For the data in this table the interquartile range is IQR = 115 − 105 = 10.
Data set in a plaintext box plot
+++
o *   
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+++++++++++++ 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 Q_{1} = 7

median (second quartile) Q_{2} = 8.5

upper (third) quartile Q_{3} = 9

interquartile range, IQR = Q_{3} − Q_{1} = 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_{1}, is a number such that integral of the PDF from ∞ to Q_{1} equals 0.25, while the upper quartile, Q_{3}, is such a number that the integral from ∞ to Q_{3} equals 0.75; in terms of the CDF, the quartiles can be defined as follows:

Q_1 = \text{CDF}^{1}(0.25) ,

Q_3 = \text{CDF}^{1}(0.75) ,
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

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. If P is normally distributed, then the standard score of the first quartile, z_{1}, is 0.67, and the standard score of the third quartile, z_{3}, is +0.67. Given mean = X and standard deviation = σ for P, if P is normally distributed, the first quartile

Q_1 = (\sigma \, z_1) + X
and the third quartile

Q_3 = (\sigma \, z_3) + X
If the actual values of the first or third quartiles differ substantially from the calculated values, P is not normally distributed.
Interquartile range and outliers
Figure 3. Boxandwhisker 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

^ Upton, Graham; Cook, Ian (1996). Understanding Statistics. Oxford University Press. p. 55.

^ Zwillinger, D., Kokoska, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, CRC Press. ISBN 1584880597 page 18.
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