State the expression for variance of two
combined series.
Answers
Answer:
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}
Answer:
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