Median

Median is the middle value of a set of data. To determine the median value in a sequence of numbers, the numbers must first be arranged in ascending order.

  • If there is an odd amount of numbers, the median value is the number that is in the middle, with the same amount of numbers below and above.
  • If there is an even amount of numbers in the list, the median is the average of the two middle values.

Fact about Median :

  1. Median is joined by the mean and the mode to create a grouping called measures of central tendency.
  2. Median is an important measure (compared to mean) for distorted data, because median is not so easily distorted. For example, median of {1, 2, 2, 5, 100) is 2 and mean is 22.
  3. If user add a constant to every value, the mean and median increase by the same constant.
  4. If user multiply every value by a constant, the mean and the median will also be multiplied by that constant.

Formula of Median of ungrouped data :

Formula of Median of grouped data :

How to find a median of an unsorted array ?

Naive solution:
Given n size unsorted array, find its median.



Median of a sorted array of size n is defined as below :

It is middle element when n is odd and average of middle two elements when n is even. Since the array is not sorted here, we sort the array first, then apply above formula.

Examples:

Input  : {1, 3, 4, 2, 6, 5, 8, 7}
Output : Median = 4.5

Since number of elements are even, median
is average of 4th and 5th largest elements.
which means (4 + 5)/2 = 4.5

Input  : {4, 4, 4, 4, 4}
Output : Median = 4         

Below is the code implementation:

C++

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// CPP program to find median
#include <bits/stdc++.h>
using namespace std;
  
// Function for calculating median
double findMedian(int a[], int n)
{
    // First we sort the array
    sort(a, a + n);
  
    // check for even case
    if (n % 2 != 0)
        return (double)a[n / 2];
  
    return (double)(a[(n - 1) / 2] + a[n / 2]) / 2.0;
}
  
// Driver program
int main()
{
    int a[] = { 1, 3, 4, 2, 7, 5, 8, 6 };
    int n = sizeof(a) / sizeof(a[0]);
    cout << "Median = " << findMedian(a, n) << endl;
    return 0;
}

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Java

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// Java program to find median
import java.util.*;
  
class GFG {
  
    // Function for calculating median
    public static double findMedian(int a[], int n)
    {
        // First we sort the array
        Arrays.sort(a);
  
        // check for even case
        if (n % 2 != 0)
            return (double)a[n / 2];
  
        return (double)(a[(n - 1) / 2] + a[n / 2]) / 2.0;
    }
  
    // Driver program
    public static void main(String args[])
    {
        int a[] = { 1, 3, 4, 2, 7, 5, 8, 6 };
        int n = a.length;
        System.out.println("Median = " + findMedian(a, n));
    }
}

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Python3

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# Python3 program to find median
  
# Function for calculating median
def findMedian(a, n):
  
    # First we sort the array
    sorted(a)
  
    # check for even case
    if n % 2 != 0:
        return float(a[n // 2])
      
    return float((a[int((n-1)/2)] +
                  a[int(n / 2)])/2.0)
  
# Driver program
a = [ 1, 3, 4, 2, 7, 5, 8, 6 ]
n = len(a)
print("Median =", findMedian(a, n))

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C#

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// C# program to find median
using System;
  
class GFG {
  
    // Function for
    // calculating median
    public static double findMedian(int[] a, int n)
    {
        // First we sort
        // the array
        Array.Sort(a);
  
        // check for
        // even case
        if (n % 2 != 0)
            return (double)a[n / 2];
  
        return (double)(a[(n - 1) / 2] + a[n / 2]) / 2.0;
    }
  
    // Driver Code
    public static void Main()
    {
        int[] a = { 1, 3, 4, 2,
                    7, 5, 8, 6 };
        int n = a.Length;
        Console.Write("Median = " + findMedian(a, n) + "\n");
    }
}

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PHP

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<?php 
// PHP program to find median
  
// Function for 
// calculating median
function findMedian(&$a, $n)
{
    // First we sort the array
    sort($a);
  
    // check for even case
    if ($n % 2 != 0)
    return (double)$a[$n / 2];
      
    return (double)($a[($n - 1) / 2] +
                    $a[$n / 2]) / 2.0;
}
  
// Driver Code
$a = array(1, 3, 4, 2, 
           7, 5, 8, 6);
$n = sizeof($a);
echo "Median = "
      findMedian($a, $n); 
?>

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Output:

Median = 4.5

Time Complexity to find median = O(n Log n) as we need to sort the array first. Note that we can find median in O(n) time using methods discussed here and here.

Basic Program related to Median:

More problems related to Median:

Recent Articles on Median!

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