# Interquartile range

1. Definition:
• The IQR represents the range for the middle 50% of your sample.
• It is the difference between the 75th percentile (Q3) and the 25th percentile (Q1) of the data.
• In other words, it includes the data points that lie between Q1 and Q3.
• Larger IQR values indicate that the central portion of the data spreads out further, while smaller values show that the middle values cluster more tightly.
2. Quartiles:
• Imagine dividing your data into quarters (quartiles).
• Statisticians label these quartiles as Q1, Q2 (the median), Q3, and Q4.
• Q1 covers the smallest quarter of values, and Q4 comprises the highest quarter.
• The IQR lies between Q1 and Q3.
• The IQR is preferred over the full data range because it is less affected by extreme values and outliers.
• It is robust and works well with skewed distributions.
• Unlike the mean and standard deviation, the IQR and median are not strongly influenced by outliers.
4. Calculation:
• The formula for finding the IQR is:
• IQR = Q3 - Q1
• Equivalently, it is the region between the 75th and 25th percentiles.

In summary, the IQR provides valuable insights into the variability of data within the central portion of a distribution. It helps us understand how tightly or widely the middle 50% of our data is spread out.

## Use

Unlike total range, the interquartile range has a breakdown point of 25%[1] and is thus often preferred to the total range.

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 IQR also may indicate the skewness of the dataset.[2]

The quartile deviation or semi-interquartile range is defined as half the IQR.[3]

## 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[4] 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[4]

The second quartile Q2 is the same as the ordinary median.[4]

## 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 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.

### 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.[5]

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

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

and the third quartile

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

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

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.