Let `barx` be the mean of `x_1,x_2,.....x_3` and `bary` be the mean of `y_1,y_2,....,y_n`. If `barz` is the mean of `x_1,x_2,.....,x_n,y_1,y_2,....y_n`, then `barz`=

Options

A. `(barx+bary)`
B. `1/2(barx+bary)`
C. `1/n(barx+bary)`
D. `1/(2n)(barx+bary)`

Answer

To find the mean \( \bar{z} \) of the combined observations \( x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n \), we can follow these steps:

### Step 1: Understand the means of the two sets
Let:
- \( \bar{x} \) be the mean of the observations \( x_1, x_2, \ldots, x_n \)
- \( \bar{y} \) be the mean of the observations \( y_1, y_2, \ldots, y_n \)

### Step 2: Express the means in terms of sums
The mean \( \bar{x} \) can be expressed as:
\[ \bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n} \]
Multiplying both sides by \( n \), we get:
\[ x_1 + x_2 + \ldots + x_n = n \bar{x} \]

Similarly, the mean \( \bar{y} \) can be expressed as:
\[ \bar{y} = \frac{y_1 + y_2 + \ldots + y_n}{n} \]
Multiplying both sides by \( n \), we get:
\[ y_1 + y_2 + \ldots + y_n = n \bar{y} \]

### Step 3: Combine the sums
Now, we need to find the total sum of all observations:
\[ \text{Total Sum} = (x_1 + x_2 + \ldots + x_n) + (y_1 + y_2 + \ldots + y_n) \]
Substituting the expressions from Step 2:
\[ \text{Total Sum} = n \bar{x} + n \bar{y} \]

### Step 4: Find the total number of observations
The total number of observations is:
\[ \text{Total Observations} = n + n = 2n \]

### Step 5: Calculate the mean \( \bar{z} \)
Now, we can find the mean \( \bar{z} \):
\[ \bar{z} = \frac{\text{Total Sum}}{\text{Total Observations}} = \frac{n \bar{x} + n \bar{y}}{2n} \]
This simplifies to:
\[ \bar{z} = \frac{\bar{x} + \bar{y}}{2} \]

### Final Result
Thus, the mean \( \bar{z} \) is given by:
\[ \bar{z} = \frac{\bar{x} + \bar{y}}{2} \]

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