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) is a measure of statistical dispersion, which is the spread of the data. The IQR may also be called the midspread, middle 50%, or H‑spread. It is defined as the difference between the 75th and 25th percentiles of the data. To calculate the IQR, the data set is divided into quartiles, or four rank-ordered even parts via linear interpolation. These quartiles are denoted by Q₁ (also called the lower quartile), Q₂ (the median), and Q₃ (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.

The IQR is an example of a trimmed estimator, defined as the 25% trimmed range, which enhances the accuracy of dataset statistics by dropping lower contribution, outlying points. It is also used as a robust measure of scale It can be clearly visualized by the box on a Box plot.

Use

The primary use of the IQR is to represent the difference between the upper and lower quartiles of a data set. This can be used as an indicator for variability of the dataset.

It is also used to build box plots, which are a graphical representation of probability distribution. In the box plot, the IQR is the height of the box itself, and the whiskers have a length of 1.5*IQR. Any data point located outside of the whiskers is referred to as an outlier (see below).

IQR is often used as a preferred measurement or variability to total range or median absolute deviation because it has a lower breakdown point: 25% compared to MAD's 50%.

The IQR has been practically used in a number of recent studies. Some of these uses include:

• Sampling for Design Space Exploration
• Predicting Stock Returns
• Image Denoising

Algorithm

Discrete Variables

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 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.

The second quartile Q₂ is the same as the ordinary median.

Continuous Variables

The interquartile range of a continuous distribution can be calculated by integrating the probability density function over specific intervals. 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: where CDF⁻¹ is the quantile function.

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. The median, minimum, maximum, and the first and third quartile constitute the Five-number summary.

Distributions

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

If both the median and mean of a distribution fall inside the interquartile range, the distribution is considered to be reasonably symmetrical.

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. A fence is used to identify and categorize types of outliers from the data, or on a box plot. There are four relevant fences:

• Lower Inner Fence: Q₁ - 1.5 * IQR
• Upper Inner Fence: Q₃ + 1.5 * IQR
• Lower Outer Fence: Q₁ - 3 * IQR
• Upper Outer Fence: Q₃ + 3 * IQR

Any data points that fall between the inner and outer fences are called mild outliers. Points that fall beyond the outer fences are called extreme outliers.

See also

• Interdecile range
• Midhinge
• Probable error
• Robust measures of scale

External links

• Media related to Interquartile range at Wikimedia Commons