Multiple choice

If $\bar{x_1} , \bar{x_2} , \bar{x_3} , ... , \bar{x_n}$ are the means of n groups with $n_1, n_2, ... , n_n$ number of observations respectively, then the mean x of all the groups taken together is given by :

  1. $\sum_{i=1}^{n}n_i\bar{x_i}$
  2. $\frac{\sum_{i=1}^{n}n_i\bar{x_i}}{n^2}$
  3. $\frac{\sum_{i=1}^{n}n_i\bar{x_i}}{\sum_{i=1}^{n}n_{i}}$
  4. $\frac{\sum_{i=1}^{n}n_i\bar{x_i}}{2n}$
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C Correct answer
Explanation

The combined mean is the sum of all observations divided by the total number of observations. The sum of observations in group i is n_i * mean_i. Total sum = sum(n_i * mean_i). Total count = sum(n_i).

AI explanation

To find the combined mean of multiple groups, you divide the total sum of all observations by the total number of observations across all groups. Since the sum of observations in a group equals its size times its mean, the total sum is the sum of n_i times x_i from i equals 1 to n. The total number of observations is the sum of n_i from i equals 1 to n, making the combined mean the sum of n_i times x_i divided by the sum of n_i.

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