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Data Modelling & AIData Structure & Algorithm

Find combined mean and variance of two series

Shaida Kate Naidoo
By Shaida Kate Naidoo
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Given two different series arr1[n] and arr2[m] of size n and m. The task is to find the mean and variance of combined series.
Examples : 
 

Input : arr1[] = {3, 5, 1, 7, 8, 5}
        arr2[] = {5, 9, 7, 1, 5, 4, 7, 3}
Output : Mean1: 4.83333   mean2: 5.125
         StandardDeviation1: 5.47222   
         StandardDeviation2: 5.60938
         Combined Mean: 5
         d1_square: 0.0277777  
         d2_square: 0.015625
         Combined Variance: 5.57143

Input : arr1[] = {23, 45, 34, 78, 12, 76, 34}
        arr2[] = {65, 67, 34, 23, 45}
Output : Mean1: 43.1429   mean2: 46.8
         StandardDeviation1: 548.694   
         StandardDeviation2: 294.56
         Combined Mean: 44.6667
         d1_square: 2.32199  
         d2_square: 4.55112
         Combined Variance: 446.056

 

Approach: 
The variance \alpha ^{2} of the combined series is given by 
\alpha ^{2} = \frac{1}{n_{1} + n_{2}}\left [ n_{1} \left ( \alpha _{1}^{2} + d_{1}^{2} \right ) + n_{2} \left ( \alpha _{2}^{2} + d_{2}^{2} \right ) \right ]
Where d_{1} = \bar{x_{1}} - \bar{x} , d_{2} = \bar{x_{2}} - \bar{x}
and \bar{x} = \frac{n_{1} \bar{x_{1}} + n_{2} \bar{x_{2}}}{n_{1} + n_{2}} , is the mean of combined series. 
\sigma _{1}^{2} = \frac{1}{n_{1}} \sum_{i = 1}^{n_{1}}\left ( x_{1i} - \bar{x_{1}} \right )^{2}
\sigma _{2}^{2} = \frac{1}{n_{2}} \sum_{j = 1}^{n_{2}}\left ( x_{2j} - \bar{x_{2}} \right )^{2}
\bar{x_{1}} , \bar{x_{2}} are the means and \sigma _{1} , \sigma _{2} are the standard deviations of two series. 
\bar{x_{1}} = \frac{1}{n_{1}} \sum_{i = 1}^{n_{1}}x_{1i}
\bar{x_{2}} = \frac{1}{n_{2}} \sum_{j = 1}^{n_{2}}x_{2j}
Below is the implementation of above formula: 
 

C++




// C++ program to find combined mean
// and variance of two series.
#include <bits/stdc++.h>
using namespace std;
 
// Function to find mean of series.
float mean(int arr[], int n)
{
    int sum = 0;
    for (int i = 0; i < n; i++)
        sum = sum + arr[i];
     
    float mean = (float)sum / n;
    return mean;
}
 
// Function to find the standard
// deviation of series.
float sd(int arr[], int n)
{
    float sum = 0;
    for (int i = 0; i < n; i++)
        sum = sum + (arr[i] - mean(arr, n)) *
              (arr[i] - mean(arr, n));
     
    float sdd = sum / n;
    return sdd;
}
 
// Function to find combined variance
// of two different series.
float combinedVariance(int arr1[], int arr2[],
                                int n, int m)
{
    // mean1 and mean2 are the mean
    // of two arrays.
    float mean1 = mean(arr1, n);
    float mean2 = mean(arr2, m);
     
    cout << "Mean1: " << mean1
         << " mean2: " << mean2 << endl;
 
    // sd1 and sd2 are the standard
    // deviation of two array.
    float sd1 = sd(arr1, n);
    float sd2 = sd(arr2, m);
     
    cout << "StandardDeviation1: " << sd1
         << " StandardDeviation2: " << sd2
         << endl;
 
    // combinedMean is variable to store
    // the combined mean of both array.
    float combinedMean = (float)(n * mean1 +
                         m * mean2) / (n + m);
     
    cout << "Combined Mean: " << combinedMean
         << endl;
 
    // d1_square and d2_square are
    // the combined mean deviation.
    float d1_square = (mean1 - combinedMean) *
                      (mean1 - combinedMean);
    float d2_square = (mean2 - combinedMean) *
                      (mean2 - combinedMean);
     
    cout << "d1 square: " << d1_square
         << " d2_square: " << d2_square
         << endl;
 
    // combinedVar is variable to store
    // combined variance of both array.
    float combinedVar = (n * (sd1 + d1_square) + m *
                       (sd2 + d2_square)) / (n + m);
    return combinedVar;
}
 
// Driver function.
int main()
{
    int arr1[] = { 23, 45, 34, 78, 12, 76, 34 };
    int arr2[] = { 65, 67, 34, 23, 45 };
    int n = sizeof(arr1) / sizeof(arr1[0]);
    int m = sizeof(arr2) / sizeof(arr2[0]);
     
    // Function call to combined mean.
    cout << "Combined Variance: "
         << combinedVariance(arr1, arr2, n, m);
    return 0;
}
 

















Java


















 






 




 



























// Java program to find combined mean


// and variance of two series.


 


import java.io.*;


 


class GFG {


     


    // Function to find mean of series.


    static float mean(int arr[], int n)


    {


        int sum = 0;


         


        for (int i = 0; i < n; i++)


            sum = sum + arr[i];


         


        float mean = (float)sum / n;


         


        return mean;


    }


     


    // Function to find the standard 


    // deviation of series.


    static float sd(int arr[], int n)


    {


        float sum = 0;


        for (int i = 0; i < n; i++)


            sum = sum + (arr[i] - mean(arr, n)) *


                            (arr[i] - mean(arr, n));


         


        float sdd = sum / n;


         


        return sdd;


    }


     


    // Function to find combined variance


    // of two different series.


    static float combinedVariance(int arr1[],


                        int arr2[], int n, int m)


    {


        // mean1 and mean2 are the mean 


        // of two arrays.


        float mean1 = mean(arr1, n);


        float mean2 = mean(arr2, m);


         


        System.out.print("Mean1: " + mean1 + " ") ;


        System.out.println("Mean2: " + mean2) ;


     


        // sd1 and sd2 are the standard 


        // deviation of two array.


        float sd1 = sd(arr1, n);


        float sd2 = sd(arr2, m);


         


        System.out.print("StandardDeviation1: "


                                   + sd1 + " ") ;


                                    


        System.out.println("StandardDeviation2: "


                                    + sd2 + " ");


     


     


        // combinedMean is variable to store


        // the combined mean of both array.


        float combinedMean = (float)(n * mean1 +


                                m * mean2) / (n + m);


         


        System.out.println( "Combined Mean: "


                            + combinedMean + " "); 


             


     


        // d1_square and d2_square are 


        // the combined mean deviation.


        float d1_square = (mean1 - combinedMean) *


                            (mean1 - combinedMean);


         


        float d2_square = (mean2 - combinedMean) * 


                            (mean2 - combinedMean);


         


        System.out.print("d1 square: " + d1_square + " " ); 


        System.out.println("d2_square: " + d2_square); 


     


        // combinedVar is variable to store 


        // combined variance of both array.


        float combinedVar = (n * (sd1 + d1_square)


                    + m * (sd2 + d2_square)) / (n + m);


         


        return combinedVar;


    }


     


       // Driver function.


        public static void main (String[] args) {


         


        int arr1[] = { 23, 45, 34, 78, 12, 76, 34 };


        int arr2[] = { 65, 67, 34, 23, 45 };


        int n = arr1.length;


        int m = arr2.length;


         


        // Function call to combined mean.


            System.out.println("Combined Variance: "


                + combinedVariance(arr1, arr2, n, m));


        }


}


 


 


// This code is contributed by vt_m.








































Python3


















 






 




 



























# Python3 program to find 


# combined mean and variance


# of two series.


 


# Function to find


# mean of series.


def mean(arr, n):


    sum = 0;


    for i in range(n):


        sum = sum + arr[i];


    mean = sum / n;


    return mean;


 


# Function to find the 


# standard deviation 


# of series.


def sd(arr, n):


    sum = 0;


    for i in range(n):


        sum = sum + ((arr[i] - mean(arr, n)) *


                     (arr[i] - mean(arr, n)));


    sdd = sum / n;


    return sdd;


 


# Function to find combined 


# variance of two different


# series.


def combinedVariance(arr1, arr2, n, m):


     


    # mean1 and mean2 are 


    # the mean of two arrays.


    mean1 = mean(arr1, n);


    mean2 = mean(arr2, m);


     


    print("Mean1: ", round(mean1, 2), 


          " mean2: ", round(mean2, 2));


 


    # sd1 and sd2 are the standard


    # deviation of two array.


    sd1 = sd(arr1, n);


    sd2 = sd(arr2, m);


     


    print("StandardDeviation1: ", round(sd1, 2), 


          " StandardDeviation2: ", round(sd2, 2));


     


    # combinedMean is variable 


    # to store the combined 


    # mean of both array.


    combinedMean = (n * mean1 +


                    m * mean2) / (n + m);


    print("Combined Mean: ", 


           round(combinedMean, 2));


     


    # d1_square and d2_square are


    # the combined mean deviation.


    d1_square = ((mean1 - combinedMean) *


                 (mean1 - combinedMean));


    d2_square = ((mean2 - combinedMean) *


                 (mean2 - combinedMean));


    print("d1 square: ", round(d1_square, 2),


          " d2_square: ", round(d2_square, 2));


     


    # combinedVar is variable to 


    # store combined variance of 


    # both array.


    combinedVar = (n * (sd1 + d1_square) +


                   m * (sd2 + d2_square)) / (n + m);


    print("Combined Variance: ", 


         round(combinedVar, 2));


 


# Driver Code


arr1 = [ 23, 45, 34, 78, 12, 76, 34 ];


arr2 = [ 65, 67, 34, 23, 45 ];


n = len(arr1);


m = len(arr2);


     


# Function call to combined mean.


combinedVariance(arr1, arr2, n ,m);


 


# This code is contributed by mits








































C#


















 






 




 



























// C# program to find combined mean


// and variance of two series.


using System;


   


class GFG {


       


    // Function to find mean of series.


    static float mean(int []arr, int n)


    {


        int sum = 0;


           


        for (int i = 0; i < n; i++)


            sum = sum + arr[i];


           


        float mean = (float)sum / n;


           


        return mean;


    }


       


    // Function to find the standard 


    // deviation of series.


    static float sd(int []arr, int n)


    {


        float sum = 0;


        for (int i = 0; i < n; i++)


            sum = sum + (arr[i] - mean(arr, n)) *


                        (arr[i] - mean(arr, n));


           


        float sdd = sum / n;


        return sdd;


    }


       


    // Function to find combined variance


    // of two different series.


    static float combinedVariance(int []arr1,


                                     int []arr2, 


                                     int n, int m)


    {


           


        // mean1 and mean2 are the 


        // mean of two arrays.


        float mean1 = mean(arr1, n);


        float mean2 = mean(arr2, m);


           


        Console.Write("Mean1: " + mean1 + " ") ;


        Console.WriteLine("Mean2: " + mean2) ;


       


        // sd1 and sd2 are the standard 


        // deviation of two array.


        float sd1 = sd(arr1, n);


        float sd2 = sd(arr2, m);


           


        Console.Write("StandardDeviation1: "


                                + sd1 + " ") ;


                                       


        Console.WriteLine("StandardDeviation2: "


                                + sd2 + " ");


       


       


        // combinedMean is variable to store


        // the combined mean of both array.


        float combinedMean = (float)(n * mean1 +


                              m * mean2) / (n + m);


           


        Console.WriteLine("Combined Mean: "


                           + combinedMean + " "); 


               


       


        // d1_square and d2_square are 


        // the combined mean deviation.


        float d1_square = (mean1 - combinedMean) *


                             (mean1 - combinedMean);


           


        float d2_square = (mean2 - combinedMean) * 


                          (mean2 - combinedMean);


           


        Console.Write("d1 square: " + 


                       d1_square + " " ); 


        Console.WriteLine("d2_square: " + 


                           d2_square); 


       


        // combinedVar is variable to store 


        // combined variance of both array.


        float combinedVar = (n * (sd1 + d1_square)    +


                             m * (sd2 + d2_square)) /


                            (n + m);


           


        return combinedVar;


    }


       


// Driver code


public static void Main ()


{


        int []arr1 = {23, 45, 34, 78, 12, 76, 34};


        int []arr2 = {65, 67, 34, 23, 45};


        int n = arr1.Length;


        int m = arr2.Length;


           


        // Function call to combined mean.


        Console.WriteLine("Combined Variance: " + 


                           combinedVariance(arr1, arr2,


                           n, m));


        }


}


   


// This code is contributed by vt_m.








































PHP


















 






 




 



























<?php


// PHP program to find combined mean


// and variance of two series.


   


// Function to find mean of series.


function mean($arr, $n)


{


    $sum = 0;


    for ($i = 0; $i < $n; $i++)


        $sum = $sum + $arr[$i];


       


    $mean = (float)($sum / $n);


    return $mean;


}


   


// Function to find the standard 


// deviation of series.


function sd($arr, $n)


{


    $sum = 0;


    for ($i = 0; $i < $n; $i++)


        $sum = $sum + ($arr[$i] - mean($arr, $n)) *


                      ($arr[$i] - mean($arr, $n));


       


    $sdd = $sum / $n;


    return $sdd;


}


   


// Function to find combined variance


// of two different series.


function combinedVariance($arr1, $arr2, 


                                 $n, $m)


{


    // mean1 and mean2 are the mean 


    // of two arrays.


    $mean1 = mean($arr1, $n);


    $mean2 = mean($arr2, $m);


       


    echo("Mean1: " . round($mean1, 2) . " " . 


         " mean2: " . round($mean2, 2));


   


    // sd1 and sd2 are the standard 


    // deviation of two array.


    $sd1 = sd($arr1, $n);


    $sd2 = sd($arr2, $m);


       


    echo("\nStandardDeviation1: " . round($sd1, 2) . " "


        . " StandardDeviation2: " . round($sd2, 2));


   


    // combinedMean is variable to store


    // the combined mean of both array.


    $combinedMean = (float)($n * $mean1 + $m *


                            $mean2) / ($n + $m);


       


    echo("\nCombined Mean: " . 


          round($combinedMean, 2));


   


    // d1_square and d2_square are 


    // the combined mean deviation.


    $d1_square = ($mean1 - $combinedMean) *


                 ($mean1 - $combinedMean);


    $d2_square = ($mean2 - $combinedMean) * 


                 ($mean2 - $combinedMean);


       


    echo("\nd1 square: " . round($d1_square, 2) . " "


        . " d2_square: " . round($d2_square, 2));


   


    // combinedVar is variable to store 


    // combined variance of both array.


    $combinedVar = ($n * ($sd1 + $d1_square) + 


                    $m * ($sd2 + $d2_square)) / 


                                     ($n + $m);


    return $combinedVar;


}


   


// Driver Code


$arr1 = array( 23, 45, 34, 


               78, 12, 76, 34 );


$arr2 = array( 65, 67, 34, 23, 45 );


$n = sizeof($arr1);


$m = sizeof($arr2);


       


// Function call to combined mean.


echo("\nCombined Variance: "


        . round(combinedVariance($arr1, $arr2, 


                                 $n, $m), 2));


   


// This code is contributed by Ajit.


?>








































Javascript


















 






 




 



























<script>


 


// JavaScript program to find combined mean


// and variance of two series.


 


       


    // Function to find mean of series.


    function mean(arr,  n)


    {


        var sum = 0;


           


        for (var i = 0; i < n; i++)


            sum = sum + arr[i];


           


         var mean = sum / n;


           


        return mean;


    }


       


    // Function to find the standard 


    // deviation of series.


     


    function sd(arr, n)


    {


        var sum = 0;


        for (var i = 0; i < n; i++)


            sum = sum + (arr[i] - mean(arr, n)) *


            (arr[i] - mean(arr, n));


           


        var sdd = sum / n;


        return sdd;


    }


       


    // Function to find combined variance


    // of two different series.


    function combinedVariance(arr1,arr2,n,m)


    {


           


        // mean1 and mean2 are the 


        // mean of two arrays.


         


        var mean1 = mean(arr1, n);


        var mean2 = mean(arr2, m);


           


        document.write("Mean1: " + mean1.toFixed(4) + " ") ;


        document.write("Mean2: " + mean2 + "<br>") ;


       


        // sd1 and sd2 are the standard 


        // deviation of two array.


        var sd1 = sd(arr1, n);


        var sd2 = sd(arr2, m);


           


        document.write("StandardDeviation1: " + 


        sd1.toFixed(3) + " ") ;


                                       


        document.write("StandardDeviation2: " + 


        sd2 + " " + "<br>");


       


       


        // combinedMean is variable to store


        // the combined mean of both array.


         


        var combinedMean = (n * mean1 + m * mean2) / (n + m);


           


        document.write("Combined Mean: " + 


        combinedMean.toFixed(4) + " " + "<br>"); 


               


       


        // d1_square and d2_square are 


        // the combined mean deviation.


        var d1_square = (mean1 - combinedMean) * 


        (mean1 - combinedMean);


           


        var d2_square = (mean2 - combinedMean) *  


        (mean2 - combinedMean);


           


        document.write("d1 square: " +  


        d1_square.toFixed(3) + " " ); 


         


        document.write("d2_square: " +  


        d2_square.toFixed(4) + "<br>"); 


       


        // combinedVar is variable to store 


        // combined variance of both array.


         


        var combinedVar = (n * (sd1 + d1_square)  +  


        m * (sd2 + d2_square)) / (n + m);


           


        return combinedVar;


    }


       


// Driver code


 


        var arr1 = [23, 45, 34, 78, 12, 76, 34 ]


        var arr2 = [65, 67, 34, 23, 45 ]


        var n = arr1.length;


        var m = arr2.length;


           


           


        // Function call to combined mean.


   document.write("Combined Variance: " +  


   combinedVariance(arr1, arr2,n, m).toFixed(3));


    


</script>


 


 


  










































Output:  






Mean1: 43.1429 mean2: 46.8
StandardDeviation1: 548.694 StandardDeviation2: 294.56
Combined Mean: 44.6667
d1 square: 2.322 d2_square: 4.5511
Combined Variance: 446.056


 








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