用于计算前N个自然数的方差的程序

给定一个整数N ，任务是找到前N个自然数的方差。

Variance is used to determine how far the data is spread from their average value. It is generally represented by symbol σ² and the equation for finding the variance is generally given by the following equation: 
 

∑^N_{i = 1} (x_i– mean(x))² / N

where N is the total number of data

例子：

Input: 5
Output: 2
Explanation: 
Mean of the first 5 numbers = (1 + 2 + 3 + 4 + 5) / 5 = 3 
Therefore, Variance = ((1 – 3)² + (2 – 3)² + (3 – 3)²+(4 – 3)²+(5 – 3)²) / 5 = (4 + 1 + 0 + 1 + 4) / 5 = 10 / 5.

Input: 4
Output: 1.25
Explanation: 
Mean of first 4 numbers = (1 + 2 + 3 + 4) / 4 = 2.5 
Therefore, Variance = ((1 – 2.5)² + (2 – 2.5)² + (3 – 2.5)² + (4 – 2.5)²) / 4 = (2.25 + 0.25 + 0.25 + 2.25) / 4 = 5 / 4.

天真的方法：解决此问题的简单方法是，首先计算前N个自然数的均值，然后遍历范围[1，N]并计算方差。

时间复杂度： O(N)
辅助空间： O(1)

有效的解决方案：将上述溶液可以通过简化均值和方差的上述公式和使用所述第一N的自然数的总和的特性和所述第一n个自然数的平方和进行优化，如下所示。

。

因此，计算(N ² – 1)/ 12并将其打印为所需结果。

下面是上述方法的实现：

C++

// C++ Program to implement
// the above approach
 
#include 
using namespace std;
 
// Function to calculate Varaince
// of first N natural numbers
long double find_Variance(int n)
{
    long long int numerator = n * n - 1;
    long double ans = (numerator * 1.0) / 12;
    return ans;
}
 
// Driver Code
int main()
{
    int N = 5;
 
    cout << fixed << setprecision(6)
         << find_Variance(N);
}

Java

// Java program to implement
// the above approach
class GFG{
 
// Function to calculate Varaince
// of first N natural numbers
static double find_Variance(int n)
{
    long  numerator = n * n - 1;
    double ans = (numerator * 1.0) / 12;
    return ans;
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 5;
     
    System.out.println(find_Variance(N));
}
}
 
// This code is contributed by AnkThon

Python3

# Python3 program to implement
# the above approach
 
# Function to calculate Varaince
# of first N natural numbers
def find_Variance(n):
     
    numerator = n * n - 1
    ans = (numerator * 1.0) / 12
     
    return ans
 
# Driver Code
if __name__ == '__main__':
     
    N = 5
 
    a = find_Variance(N)
 
    print("{0:.6f}".format(a))
 
# This code is contributed by mohit kumar 29

C#

// C# program to implement
// the above approach
using System;
class GFG
{
 
    // Function to calculate Varaince
    // of first N natural numbers
    static double find_Variance(int n)
    {
        long  numerator = n * n - 1;
        double ans = (numerator * 1.0) / 12;
        return ans;
    }
     
    // Driver Code
    public static void Main(string[] args)
    {
        int N = 5;
        Console.WriteLine(find_Variance(N));
    }
}
 
// This code is contributed by AnkThon

Javascript

 
// Javascript program to implement
// the above approach
  
// Function to calculate Varaince
// of first N natural numbers
function find_Variance(n)
{
    var numerator = n * n - 1
    var ans = (numerator * 1.0) / 12
      
    return ans
}
   
// Driver Code
var N = 5;
 
document.write (find_Variance(N).toFixed(6));
 
// This code is contributed by bunnyram19
 

输出：

2.000000

时间复杂度： O(1)
辅助空间： O(1)