Interquartile range

inter-quartile rangebelowinterquartilequartile deviation
In descriptive statistics, the interquartile range (IQR), also called the midspread or middle 50%, or technically H-spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR = Q 3 − Q 1 .wikipedia
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Statistical dispersion

dispersionvariabilityspread
In descriptive statistics, the interquartile range (IQR), also called the midspread or middle 50%, or technically H-spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR = Q 3 − Q 1 . In other words, the IQR is the first quartile subtracted from the third quartile; these quartiles can be clearly seen on a box plot on the data.
Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range.

Box plot

boxplotbox and whisker plotadjusted boxplots
In descriptive statistics, the interquartile range (IQR), also called the midspread or middle 50%, or technically H-spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR = Q 3 − Q 1 . In other words, the IQR is the first quartile subtracted from the third quartile; these quartiles can be clearly seen on a box plot on the data. The IQR is used to build box plots, simple graphical representations of a probability distribution.
In addition to the points themselves, they allow one to visually estimate various L-estimators, notably the interquartile range, midhinge, range, mid-range, and trimean.

Robust measures of scale

Qn estimatorrobust estimator of dispersionrobust measure of scale
It is a trimmed estimator, defined as the 25% trimmed range, and is a commonly used robust measure of scale.
The most common such statistics are the interquartile range (IQR) and the median absolute deviation (MAD).

Trimmed estimator

trimmedtrimming
It is a trimmed estimator, defined as the 25% trimmed range, and is a commonly used robust measure of scale.
Interquartile range, the 25% trimmed range

Robust statistics

robustbreakdown pointrobustness
Unlike total range, the interquartile range has a breakdown point of 25%, and is thus often preferred to the total range.
The median absolute deviation and interquartile range are robust measures of statistical dispersion, while the standard deviation and range are not.

Quartile

quartileslower quartilelower and upper quartiles
In descriptive statistics, the interquartile range (IQR), also called the midspread or middle 50%, or technically H-spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR = Q 3 − Q 1 . In other words, the IQR is the first quartile subtracted from the third quartile; these quartiles can be clearly seen on a box plot on the data.
In the case of quartiles, the Interquartile Range (IQR) may be used to characterize the data when there may be extremities that skew the data; the interquartile range is a relatively robust statistic (also sometimes called "resistance") compared to the range and standard deviation.

Median

averagesample medianmedian-unbiased estimator
The median is the corresponding measure of central tendency.
When the median is used as a location parameter in descriptive statistics, there are several choices for a measure of variability: the range, the interquartile range, the mean absolute deviation, and the median absolute deviation.

Midhinge

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 midhinge is complemented by the H-spread, or interquartile range, which is the difference of the third and first quartiles and which is a measure of statistical dispersion, in sense that if one knows the midhinge and the interquartile range, one can find the first and third quartiles.

Central tendency

Localitycentral locationcentral point
The median is the corresponding measure of central tendency.
Interquartile mean: a truncated mean based on data within the interquartile range.

Range (statistics)

rangerangingsample range
It is a trimmed estimator, defined as the 25% trimmed range, and is a commonly used robust measure of scale. Unlike total range, the interquartile range has a breakdown point of 25%, and is thus often preferred to the total range.
Interquartile range

Outlier

outliersconservative estimateirregularities
The IQR can be used to identify outliers (see below).
Other methods flag observations based on measures such as the interquartile range.

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).
Interquartile range

Cauchy distribution

LorentzianCauchyLorentzian profile
\gamma is also equal to half the interquartile range and is sometimes called the probable error.

Interdecile range

Interdecile range
The interdecile range is a measure of statistical dispersion of the values in a set of data, similar to the range and the interquartile range, and can be computed from the (non-parametric) seven-number summary.

Descriptive statistics

descriptivedescriptive statisticstatistics
In descriptive statistics, the interquartile range (IQR), also called the midspread or middle 50%, or technically H-spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR = Q 3 − Q 1 . In other words, the IQR is the first quartile subtracted from the third quartile; these quartiles can be clearly seen on a box plot on the data.

Percentile

percentiles50th percentile85th percentile speed
In descriptive statistics, the interquartile range (IQR), also called the midspread or middle 50%, or technically H-spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR = Q 3 − Q 1 . In other words, the IQR is the first quartile subtracted from the third quartile; these quartiles can be clearly seen on a box plot on the data.

Probability distribution

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

Probability density function

probability densitydensity functiondensity
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).

Cumulative distribution function

distribution functionCDFcumulative probability distribution function
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).

Quantile function

quantileinverse distribution functionnormal quantile function
where CDF −1 is the quantile function.

Normal distribution

normally distributednormalGaussian
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.

Mean

mean valuepopulation meanaverage
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.

Standard deviation

standard deviationssample standard deviationsigma
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.

Standard score

normalizednormalisednormalized score
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.