Datasets with the Same Mean

Let's consider two datasets, Dataset A and Dataset B. We are told that they have the same mean. Let's denote the number of observations in Dataset A as \(n_A\) and in Dataset B as \(n_B\). Let the mean of Dataset A be \(\mu_A\) and the mean of Dataset B be \(\mu_B\).

The question states that \(\mu_A = \mu_B\). Let's call this common mean \(\mu\). So, \(\mu_A = \mu_B = \mu\).

The mean of a dataset is calculated as the sum of all observations divided by the number of observations.

For Dataset A, the sum of observations is \(\text{Sum}_A = n_A \cdot \mu_A\). Since \(\mu_A = \mu\), we have \(\text{Sum}_A = n_A \cdot \mu\).

For Dataset B, the sum of observations is \(\text{Sum}_B = n_B \cdot \mu_B\). Since \(\mu_B = \mu\), we have \(\text{Sum}_B = n_B \cdot \mu\).

Analyzing the Options

Let's examine each option based on this information.

• Option 1: Sum of the observations in A = Sum of the observations in B.

  From our formulas, \(\text{Sum}_A = n_A \cdot \mu\) and \(\text{Sum}_B = n_B \cdot \mu\). For \(\text{Sum}_A\) to equal \(\text{Sum}_B\), we would need \(n_A \cdot \mu = n_B \cdot \mu\). If \(\mu \neq 0\), this would imply \(n_A = n_B\). However, the problem does not state that the number of observations in the two datasets must be equal (\(n_A = n_B\)). For example, Dataset A could be {10} (\(n_A=1\), \(\mu_A=10\)) and Dataset B could be {10, 10} (\(n_B=2\), \(\mu_B=10\)). Here, the means are equal (\(\mu=10\)), but \(\text{Sum}_A = 10\) and \(\text{Sum}_B = 20\). The sums are not equal. Therefore, this statement is not necessarily true.

• Option 2: Mean of the squares of the observations in A = Mean of the squares of the observations in B.

  Let the observations in Dataset A be \(a_1, a_2, \dots, a_{n_A}\) and in Dataset B be \(b_1, b_2, \dots, b_{n_B}\). The mean of the squares of observations in A is \(\frac{\sum_{i=1}^{n_A} a_i^2}{n_A}\) and in B is \(\frac{\sum_{j=1}^{n_B} b_j^2}{n_B}\). Having the same mean (\(\frac{\sum a_i}{n_A} = \frac{\sum b_j}{n_B}\)) does not provide enough information to conclude anything about the mean of the squares. For example, Dataset A could be {5, 15} (\(n_A=2\), \(\mu_A=10\)) and Dataset B could be {10} (\(n_B=1\), \(\mu_B=10\)). The means are equal. Mean of squares for A is \(\frac{5^2 + 15^2}{2} = \frac{25 + 225}{2} = \frac{250}{2} = 125\). Mean of squares for B is \(\frac{10^2}{1} = 100\). The means of squares are not equal. Therefore, this statement is not necessarily true.

• Option 3: If the two datasets are combined, then the mean of the combined dataset = mean of A + mean of B.

  When the two datasets are combined, the total number of observations is \(n_{combined} = n_A + n_B\), and the total sum of observations is \(\text{Sum}_{combined} = \text{Sum}_A + \text{Sum}_B\).

  The mean of the combined dataset is \(\mu_{combined} = \frac{\text{Sum}_{combined}}{n_{combined}} = \frac{\text{Sum}_A + \text{Sum}_B}{n_A + n_B}\).

  We know \(\text{Sum}_A = n_A \cdot \mu\) and \(\text{Sum}_B = n_B \cdot \mu\).

  So, \(\mu_{combined} = \frac{n_A \cdot \mu + n_B \cdot \mu}{n_A + n_B} = \frac{\mu (n_A + n_B)}{n_A + n_B}\).

  Assuming \(n_A + n_B \neq 0\) (which is true since we have datasets), we can cancel \(n_A + n_B\):

  \(\mu_{combined} = \mu\).

  The option states that \(\mu_{combined} = \mu_A + \mu_B\). Since \(\mu_A = \mu_B = \mu\), this would mean \(\mu_{combined} = \mu + \mu = 2\mu\). We found that \(\mu_{combined} = \mu\). This statement would only be true if \(\mu = 2\mu\), which implies \(\mu = 0\). The mean doesn't have to be 0. For example, if \(\mu=10\), the combined mean is 10, not \(10+10=20\). Therefore, this statement is false.

• Option 4: If the two datasets are combined, then the mean of the combined dataset = mean of A.

  From our calculation for Option 3, we found that the mean of the combined dataset is \(\mu_{combined} = \mu\). We also know that the mean of Dataset A is \(\mu_A = \mu\). Therefore, \(\mu_{combined} = \mu_A\). This statement is always true when the two datasets have the same mean.

Conclusion

When two datasets have the same mean, the mean of the combined dataset will be equal to that common mean, which is also equal to the mean of A (and the mean of B).