Interquartile Range (IQR)

The quartiles of a ranked set of data values are three points which divide the data into exactly four equal parts, each part comprising of quarter data.

Q1 is defined as the middle number between the smallest number and the median of the data set.
Q2 is the median of the data.
Q3 is the middle value between the median and the highest value of the data set.

The interquartile range IQR tells us the range 
where the bulk of the values lie. The interquartile 
range is calculated by subtracting the first quartile
from the third quartile. 
IQR = Q3 - Q1

Uses
1. Unlike range, IQR tells where the majority of data lies and is thus preferred over range.
2. IQR can be used to identify outliers in a data set.
3. Gives the central tendency of the data.

Examples:

Input : 1, 19, 7, 6, 5, 9, 12, 27, 18, 2, 15
Output : 13
The data set after being sorted is 
1, 2, 5, 6, 7, 9, 12, 15, 18, 19, 27
As mentioned above Q2 is the median of the data. 
Hence Q2 = 9
Q1 is the median of lower half, taking Q2 as pivot.
So Q1 = 5
Q3 is the median of upper half talking Q2 as pivot. 
So Q3 = 18
Therefore IQR for given data=Q3-Q1=18-5=13 

Input : 1, 3, 4, 5, 5, 6, 7, 11
Output : 3

C++

// CPP program to find IQR of a data set 
#include <bits/stdc++.h> 
using namespace std; 
  
// Function to give index of the median 
int median(int* a, int l, int r) 
{ 
    int n = r - l + 1; 
    n = (n + 1) / 2 - 1; 
    return n + l; 
} 
  
// Function to calculate IQR 
int IQR(int* a, int n) 
{ 
    sort(a, a + n); 
  
    // Index of median of entire data 
    int mid_index = median(a, 0, n); 
  
    // Median of first half 
    int Q1 = a[median(a, 0, mid_index)]; 
  
    // Median of second half 
    int Q3 = a[median(a, mid_index + 1, n)]; 
  
    // IQR calculation 
    return (Q3 - Q1); 
} 
  
// Driver Function 
int main() 
{ 
    int a[] = { 1, 19, 7, 6, 5, 9, 12, 27, 18, 2, 15 }; 
    int n = sizeof(a)/sizeof(a[0]); 
    cout << IQR(a, n); 
    return 0; 
} 

Java

// Java program to find  
// IQR of a data set 
import java.io.*; 
import java .util.*; 
  
class GFG 
{ 
      
// Function to give  
// index of the median 
static int median(int a[],  
                  int l, int r) 
{ 
    int n = r - l + 1; 
    n = (n + 1) / 2 - 1; 
    return n + l; 
} 
  
// Function to  
// calculate IQR 
static int IQR(int [] a, int n) 
{ 
    Arrays.sort(a); 
  
    // Index of median  
    // of entire data 
    int mid_index = median(a, 0, n); 
  
    // Median of first half 
    int Q1 = a[median(a, 0,  
                      mid_index)]; 
  
    // Median of second half 
    int Q3 = a[median(a,  
               mid_index + 1, n)]; 
  
    // IQR calculation 
    return (Q3 - Q1); 
} 
  
// Driver Code 
public static void main (String[] args)  
{ 
    int []a = {1, 19, 7, 6, 5, 9,  
               12, 27, 18, 2, 15}; 
    int n = a.length; 
    System.out.println(IQR(a, n)); 
} 
} 
  
// This code is contributed  
// by anuj_67. 

Python3

# Python3 program to find IQR of  
# a data set 
  
# Function to give index of the median 
def median(a, l, r): 
    n = r - l + 1
    n = (n + 1) // 2 - 1
    return n + l 
  
# Function to calculate IQR 
def IQR(a, n): 
  
    a.sort() 
  
    # Index of median of entire data 
    mid_index = median(a, 0, n) 
  
    # Median of first half 
    Q1 = a[median(a, 0, mid_index)] 
  
    # Median of second half 
    Q3 = a[median(a, mid_index + 1, n)] 
  
    # IQR calculation 
    return (Q3 - Q1) 
  
# Driver Function 
if __name__=='__main__': 
    a = [1, 19, 7, 6, 5, 9, 12, 27, 18, 2, 15] 
    n = len(a) 
    print(IQR(a, n)) 
  
# This code is contributed by 
# Sanjit_Prasad 

C#

// C# program to find  
// IQR of a data set 
using System; 
  
class GFG 
{ 
      
// Function to give  
// index of the median 
static int median(int []a,  
                int l, int r) 
{ 
    int n = r - l + 1; 
    n = (n + 1) / 2 - 1; 
    return n + l; 
} 
  
// Function to  
// calculate IQR 
static int IQR(int [] a, int n) 
{ 
    Array.Sort(a); 
  
    // Index of median  
    // of entire data 
    int mid_index = median(a, 0, n); 
  
    // Median of first half 
    int Q1 = a[median(a, 0,  
                    mid_index)]; 
  
    // Median of second half 
    int Q3 = a[median(a,  
            mid_index + 1, n)]; 
  
    // IQR calculation 
    return (Q3 - Q1); 
} 
  
// Driver Code 
public static void Main ()  
{ 
    int []a = {1, 19, 7, 6, 5, 9,  
            12, 27, 18, 2, 15}; 
    int n = a.Length; 
    Console.WriteLine(IQR(a, n)); 
} 
} 
  
// This code is contributed  
// by anuj_67. 

PHP

<?php 
// PHP program to find IQR of a data set 
  
// Function to give index of the median 
function median($a, $l, $r) 
{ 
    $n = $r - $l + 1; 
    $n = (int)(($n + 1) / 2) - 1; 
    return $n + $l; 
} 
  
// Function to calculate IQR 
function IQR($a, $n) 
{ 
    sort($a); 
  
    // Index of median of entire data 
    $mid_index = median($a, 0, $n); 
  
    // Median of first half 
    $Q1 = $a[median($a, 0, $mid_index)]; 
  
    // Median of second half 
    $Q3 = $a[median($a, $mid_index + 1, $n)]; 
  
    // IQR calculation 
    return ($Q3 - $Q1); 
} 
  
// Driver Function 
$a = array( 1, 19, 7, 6, 5, 9, 
            12, 27, 18, 2, 15 ); 
$n = count($a); 
echo IQR($a, $n); 
  
// This code is contributed by mits 
?> 

Output:

Reference
https://en.wikipedia.org/wiki/Interquartile_range

This article is contributed by Vineet Joshi. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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