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. 

1. Q1 is defined as the middle number between the smallest number and the median of the data set.
2. Q2 is the median of the data.
3. 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[mid_index + 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[mid_index + 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[mid_index + 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[mid_index + 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 = $a[$mid_index + 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
?>

Javascript

<script>

// javascript program to find
// IQR of a data set

// Function to give
// index of the median
function median(a, l , r)
{
    var n = r - l + 1;
    n = parseInt((n + 1) / 2) - 1;
    return parseInt(n + l);
}

// Function to
// calculate IQR
function IQR(a , n)
{
    a.sort((a,b)=>a-b);

    // Index of median
    // of entire data
    var mid_index = median(a, 0, n);

    // Median of first half
    var Q1 = a[median(a, 0,
                      mid_index)];

    // Median of second half
    var Q3 = a[mid_index + median(a,
               mid_index + 1, n)];

    // IQR calculation
    return (Q3 - Q1);
}

// Driver Code
var a = [1, 19, 7, 6, 5, 9,
         12, 27, 18, 2, 15];
var n = a.length;
document.write(IQR(a, n));


// This code contributed by Princi Singh
</script>

Output:

13

Time Complexity: O(1)

Auxiliary Space: O(1)
Reference
https://en.wikipedia.org/wiki/Interquartile_range
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