Median

02 Jul, 2022
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 :  
Median is joined by the mean and the mode to create a grouping called measures of central tendency.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.If user add a constant to every value, the mean and median increase by the same constant.If user multiply every value by a constant, the mean and the median will also be multiplied by that constant.
Median is joined by the mean and the mode to create a grouping called measures of central tendency.
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.
If user add a constant to every value, the mean and median increase by the same constant.
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++
Java
Python3
C#
PHP
Javascript
// CPP program to find median#include <bits/stdc++.h>using namespace std; // Function for calculating mediandouble 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 programint 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;}
// Java program to find medianimport 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));    }}
# Python3 program to find median # Function for calculating mediandef 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 programa = [ 1, 3, 4, 2, 7, 5, 8, 6 ]n = len(a)print("Median =", findMedian(a, n))
// C# program to find medianusing 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");    }}
<?php// PHP program to find median // Function for// calculating medianfunction 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);?>
<script> // JavaScript program to find median // Function for calculating medianfunction findMedian(a, n){    // First we sort the array    a.sort();     // check for even case    if (n % 2 != 0)        return a[parseInt(n / 2)];     return (a[parseInt((n - 1) / 2)] + a[parseInt(n / 2)]) / 2.0;} // Driver program    let a = [ 1, 3, 4, 2, 7, 5, 8, 6 ];    let n = a.length;    document.write("Median = " + findMedian(a, n)); </script>
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.
Auxiliary Space: O(1)Basic Program related to Median:  
Maximize the median of an array
Minimum Increment / decrement to make array elements equal
Minimum sum of differences with an element in an array
Median of two sorted arrays of different sizes | Set 1 (Linear)
Median of two sorted arrays with different sizes in O(log(min(n, m)))
More problems related to Median:  
Median and Mode using Counting Sort
Minimum number of elements to add to make median equals x
Decode a median string to the original string
Median after K additional integers
Find median in row wise sorted matrix
Find median of BST in O(n) time and O(1) space
Median in a stream of integers (running integers)
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