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Find combined mean and variance of two series in C++
Concept
With respect of two given two different series arr1[b] and arr2[a] of size b and a. Our task is to determine the mean and variance of combined series.
Input
Arr1[] = { 24, 46, 35, 79, 13, 77, 35 };
Arr2[] = { 66, 68, 35, 24, 46 };Output
Mean1: 44.1429 Mean2: 47.8 StandardDeviation1: 548.694 StandardDeviation2: 294.56 Combined Mean: 45.6667 d1 square: 2.322 d2_square: 4.5511 Combined Variance: 446.056
Method
Now suppose,
n1= No. of observations in 'region 1'
n2= No. of observations in 'region 1'
X1= mean of region 1.
X2=mean of region 2.
S1=standard deviation of region 1.
S2=standard deviation of region 2.
S12 = variance of region 1.
S22 = variance of region 2.
Let X=mean of total group
So d1=X – X1
and d2= X – X2
Calculate the mean of total group X as
(n1*X1+n2*X2)/(n1+n2)
Calculate the variance of total group as
n1*(S1 2+d1 2)+n2*(S2 2+d2 2)/(n1+n2)
Example
// C++ program to find combined mean
// and variance of two series.
#include <bits/stdc++.h>
using namespace std;
// Shows function to find mean of series.
float mean(int Arr[], int b){
int sum1 = 0;
for (int i = 0; i < b; i++)
sum1 = sum1 + Arr[i];
float mean = (float)sum1 / b;
return mean;
}
// Shows function to find the standard
// deviation of series.
float sd(int Arr[], int b){
float sum1 = 0;
for (int i = 0; i < b; i++)
sum1 = sum1 + (Arr[i] - mean(Arr, b)) *
(Arr[i] - mean(Arr, b));
float sdd = sum1 / b;
return sdd;
}
//Shows function to find combined variance
// of two different series.
float combinedVariance(int Arr1[], int Arr2[],
int b, int a){
// Here, mean1 and mean2 are the mean
// of two arrays.
float mean1 = mean(Arr1, b);
float mean2 = mean(Arr2, a);
cout << "Mean1: " << mean1
<< " mean2: " << mean2 << endl;
// Here, sd1 and sd2 are the standard
// deviation of two array.
float sd1 = sd(Arr1, b);
float sd2 = sd(Arr2, a);
cout << "StandardDeviation1: " << sd1
<< " StandardDeviation2: " << sd2
<< endl;
// Here, combinedMean is variable to store
// the combined mean of both array.
float combinedMean = (float)(b * mean1 +
a * mean2) / (b + a);
cout << "Combined Mean: " << combinedMean
<< endl;
// Here, 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;
// Here, combinedVar is variable to store
// combined variance of both array.
float combinedVar = (b * (sd1 + d1_square) + a *(sd2 + d2_square)) / (b + a);
cout << "Combined Variance: " << combinedVar;
}
// Driver function.
int main(){
int Arr1[] = { 24, 46, 35, 79, 13, 77, 35 };
int Arr2[] = { 66, 68, 35, 24, 46 };
int b = sizeof(Arr1) / sizeof(Arr1[0]);
int a = sizeof(Arr2) / sizeof(Arr2[0]);
// Shows function call to combined mean.
combinedVariance(Arr1, Arr2, b, a);
return 0;
}Output
Mean1: 44.1429 mean2: 47.8 StandardDeviation1: 548.694 StandardDeviation2: 294.56 Combined Mean: 45.6667 d1 square: 2.322 d2_square: 4.5511 Combined Variance: 446.056
Updated on: 24-Jul-2020
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