Indescriptive statistics, theinterquartile range (IQR) is a measure ofstatistical dispersion, which is the spread of the data.[1] The IQR may also be called themidspread,middle 50%,fourth spread, orH‑spread. It is defined as the difference between the 75th and 25thpercentiles of the data.[2][3][4] To calculate the IQR, the data set is divided intoquartiles, or four rank-ordered even parts via linear interpolation.[1] These quartiles are denoted byQ1 (also called the lower quartile),Q2 (themedian), andQ3 (also called the upper quartile). The lower quartile corresponds with the 25th percentile and the upper quartile corresponds with the 75th percentile, so IQR =Q3 − Q1[1].
The IQR is an example of atrimmed estimator, defined as the 25% trimmedrange, which enhances the accuracy of dataset statistics by dropping lower contribution, outlying points.[5] It is also used as arobust measure of scale[5] It can be clearly visualized by the box on abox plot.[1]
Unlike totalrange, the interquartile range has abreakdown point of 25%[6] and is thus often preferred to the total range.
The IQR is used to buildbox plots, simple graphical representations of aprobability distribution.
The IQR is used in businesses as a marker for theirincome rates.
For a symmetric distribution (where the median equals themidhinge, the average of the first and third quartiles), half the IQR equals themedian absolute deviation (MAD).
Themedian is the corresponding measure ofcentral tendency.
The IQR can be used to identifyoutliers (seebelow). The IQR also may indicate theskewness of the dataset.[1]
The quartile deviation or semi-interquartile range is defined as half the IQR.[7]
The IQR of a set of values is calculated as the difference between the upper and lower quartiles, Q3 and Q1. Each quartile is a median[8] calculated as follows.
Given an even2n or odd2n+1 number of values
Thesecond quartile Q2 is the same as the ordinary median.[8]
The following table has 13 rows, and follows the rules for the odd number of entries.
| i | x[i] | Median | Quartile |
|---|---|---|---|
| 1 | 7 | Q2=87 (median of whole table) | Q1=31 (median of lower half, from row 1 to 6) |
| 2 | 7 | ||
| 3 | 31 | ||
| 4 | 31 | ||
| 5 | 47 | ||
| 6 | 75 | ||
| 7 | 87 | ||
| 8 | 115 | Q3=119 (median of upper half, from row 8 to 13) | |
| 9 | 116 | ||
| 10 | 119 | ||
| 11 | 119 | ||
| 12 | 155 | ||
| 13 | 177 |
For the data in this table the interquartile range is IQR = Q3 − Q1 = 119 - 31 = 88.
+−−−−−+−+ * |−−−−−−−−−−−| | |−−−−−−−−−−−| +−−−−−+−+ +−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+ Number line 0 1 2 3 4 5 6 7 8 9 10 11 12
For the data set in thisbox plot:
This means the 1.5*IQR whiskers can be uneven in lengths. The median, minimum, maximum, and the first and third quartile constitute theFive-number summary.[9]
The interquartile range of a continuous distribution can be calculated by integrating theprobability density function (which yields thecumulative 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 -∞ toQ1 equals 0.25, while the upper quartile,Q3, is such a number that the integral from -∞ toQ3 equals 0.75; in terms of the CDF, the quartiles can be defined as follows:
where CDF−1 is thequantile 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 |
The IQR,mean, andstandard deviation of a populationP can be used in a simple test of whether or notP isnormally distributed, or Gaussian. IfP is normally distributed, then thestandard score of the first quartile,z1, is −0.67, and the standard score of the third quartile,z3, is +0.67. Givenmean = andstandard deviation = σ forP, ifP is normally distributed, the first quartile
and the third quartile
If the actual values of the first or third quartiles differ substantially[clarification needed] from the calculated values,P is not normally distributed. However, a normal distribution can be trivially perturbed to maintain its Q1 and Q2 std. scores at 0.67 and −0.67 and not be normally distributed (so the above test would produce a false positive). A better test of normality, such asQ–Q plot would be indicated here.
The interquartile range is often used to findoutliers in data. Outliers here are defined as 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 indicated bywhiskers of the box (frequently with an additional bar at the end of the whisker) and any outliers as individual points.
Media related toInterquartile range at Wikimedia Commons
