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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.

S_{1}=standard deviation of region 1.

S_{2}=standard deviation of region 2.

S_{1}^{2} = variance of region 1.

S_{2}^{2} = variance of region 2.

Let X=mean of total group

So d_{1}=X – X1

and d_{2}= X – X2

Calculate the mean of total group X as

(n1*X1+n2*X2)/(n1+n2)

Calculate the variance of total group as

n1*(S_{1}
^{2}+d_{1}
^{2})+n2*(S_{2}
^{2}+d_{2}
^{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

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