Question

If \(\bar{x}_1, \bar{x}_2, \bar{x}_3, \dots, \bar{x}_n\) are the means of n groups with \(n_1, n_2, n_3, \dots, n_n\) numbers of observations respectively. Then the mean \(\bar{x}\) of all the groups taken together is given by :

• A. \(\sum_{i=1}^{n} n_i \bar{x}_i\)
• B. \(\frac{\sum_{i=1}^{n} n_i \bar{x}_i}{n^2}\)
• C. \(\frac{\sum_{i=1}^{n} n_i \bar{x}_i}{\sum_{i=1}^{n} n_i}\)
• D. \(\frac{\sum_{i=1}^{n} n_i \bar{x}_i}{2n}\)

Hint:

The combined mean (or overall mean) is like a weighted average. 1. For each group, find the total sum of its observations: Sum = (Number of observations in group) \(\times\) (Mean of group) = \(n_i \bar{x}_i\). 2. Add up these sums from all groups to get the grand total sum: \(\sum n_i \bar{x}_i\). 3. Add up the number of observations from all groups to get the grand total number of observations: \(\sum n_i\). 4. Combined Mean = (Grand Total Sum) / (Grand Total Number of Observations) = \(\frac{\sum n_i \bar{x}_i}{\sum n_i}\).

Answer 1

The Correct Option is C

Concept: This question asks for the formula for the combined mean (or weighted average) of several groups when the mean and the number of observations for each group are known. Step 1: Understanding the mean of a single group For any group \(i\), its mean \(\bar{x}_i\) is defined as the sum of observations in that group divided by the number of observations in that group (\(n_i\)). Let \(S_i\) be the sum of observations in group \(i\). Then, \(\bar{x}_i = \frac{S_i}{n_i}\). This implies that the sum of observations in group \(i\) is \(S_i = n_i \bar{x}_i\). Step 2: Calculating the combined mean for all groups The combined mean (\(\bar{x}\)) of all groups taken together is found by dividing the total sum of all observations across all groups by the total number of observations across all groups. Total sum of all observations = Sum of observations in group 1 + Sum of observations in group 2 + ... + Sum of observations in group n Total sum = \(S_1 + S_2 + \dots + S_n\) Using \(S_i = n_i \bar{x}_i\), this becomes: Total sum = \(n_1 \bar{x}_1 + n_2 \bar{x}_2 + \dots + n_n \bar{x}_n = \sum_{i=1}^{n} n_i \bar{x}_i\). Total number of observations = Number of observations in group 1 + Number of observations in group 2 + ... + Number of observations in group n Total number of observations = \(n_1 + n_2 + \dots + n_n = \sum_{i=1}^{n} n_i\). Step 3: Formulate the combined mean The combined mean \(\bar{x}\) is therefore: \[ \bar{x} = \frac{\text{Total sum of all observations}}{\text{Total number of all observations}} \] \[ \bar{x} = \frac{\sum_{i=1}^{n} n_i \bar{x}_i}{\sum_{i=1}^{n} n_i} \] Step 4: Compare with the given options The derived formula for the combined mean is \(\frac{\sum_{i=1}^{n} n_i \bar{x}_i}{\sum_{i=1}^{n} n_i}\). This matches option (3).