Interquartile range

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 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.

Examples

Data set in a table

The following table has 13 rows, and follows the rules for the odd number of entries.

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.)
• Pattern of latter two bullet points: If there are no data points at the true quartiles, use data points slightly "inland" (closer to the median) from the actual quartiles.

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 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

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 = ${\displaystyle \overline{P}}$ and standard deviation = σ for P, if P is normally distributed, the first quartile

${\displaystyle Q_1=(\sigma z_1)+\overline{P}}$

and the third quartile

${\displaystyle Q_3=(\sigma z_3)+\overline{P}}$

If the actual values of the first or third quartiles differ substantially 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

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.