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Interquartile range

Boxplot (with an interquartile range) and a probability density function (pdf) of a Normal N(0,σ2) Population

In descriptive statistics, the interquartile range (IQR), also called the midspread, middle 50%, or H‑spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles,[1][2] IQR = Q3 −  Q1. 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. It is a trimmed estimator, defined as the 25% trimmed range, and is a commonly used robust measure of scale.

The IQR is a measure of variability, based on dividing a data set into quartiles. Quartiles divide a rank-ordered data set into four equal parts. The values that separate parts are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

Use

Unlike total range, the interquartile range has a breakdown point of 25%,[3] and is thus often preferred to the total range.

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

The IQR is used in businesses as a marker for their income rates.

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.

The IQR can be used to identify outliers (see below).

The quartile deviation or semi-interquartile range is defined as half the IQR.[4][5]

Algorithm

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[6] calculated as follows.

Given an even 2n or odd 2n+1 number of values

first quartile Q1 = median of the n smallest values
third quartile Q3 = median of the n largest values[6]

The second quartile Q2 is the same as the ordinary median.[6]

Examples

Data set in a table

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 upper half, from row 1 to 6)
2 7
3 31
4 31
5 47
6 75
7 87
8 115
Q3=119
(median of lower 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.

Data set in a plain-text box plot


+−−−−−+−+
* |−−−−−−−−−−−|     | |−−−−−−−−−−−|
+−−−−−+−+

+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+   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 Q1 = 7
• median (second quartile) Q2 = 8.5
• upper (third) quartile Q3 = 9
• interquartile range, IQR = Q3 - Q1 = 2
• lower 1.5*IQR whisker = Q1 - 1.5 * IQR = 7 - 3 = 4. (If there is no data point at 4, then the lowest point greater than 4.)
• upper 1.5*IQR whisker = Q3 + 1.5 * IQR = 9 + 3 = 12. (If there is no data point at 12, then the highest point less than 12.)

This means the 1.5*IQR whiskers can be uneven in lengths.

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, Q1, is a number such that integral of the PDF from -∞ to Q1 equals 0.25, while the upper quartile, Q3, is such a number that the integral from -∞ to Q3 equals 0.75; in terms of the CDF, the quartiles can be defined as follows:

${\displaystyle Q_{1}={\text{CDF}}^{-1}(0.25),}$
${\displaystyle Q_{3}={\text{CDF}}^{-1}(0.75),}$

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

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, z1, is −0.67, and the standard score of the third quartile, z3, is +0.67. Given mean = X and standard deviation = σ for P, if P is normally distributed, the first quartile

${\displaystyle Q_{1}=(\sigma \,z_{1})+X}$

and the third quartile

${\displaystyle Q_{3}=(\sigma \,z_{3})+X}$

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 as Q-Q plot would be indicated here.

Outliers

Box-and-whisker plot with four mild outliers and one extreme outlier. In this chart, outliers are defined as mild above Q3 + 1.5 IQR and extreme above Q3 + 3 IQR.

The interquartile range is often used to find outliers 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 by whiskers of the box (frequently with an additional bar at the end of the whisker) and any outliers as individual points.