Find the mean of 8, 12, 16, 22, 10 and 4. Find the resulting mean, if each of the observations, given above, be divided by 2.

Answer

To find the mean of the given numbers and the resulting mean after dividing each observation by 2, we can follow these steps:

### Step-by-Step Solution:

**Step 1: List the Observations**
The given observations are:
\[ 8, 12, 16, 22, 10, 4 \]

**Step 2: Calculate the Sum of the Observations**
Add all the observations together:
\[ 8 + 12 + 16 + 22 + 10 + 4 \]

Calculating the sum:
\[ 8 + 12 = 20 \]
\[ 20 + 16 = 36 \]
\[ 36 + 22 = 58 \]
\[ 58 + 10 = 68 \]
\[ 68 + 4 = 72 \]
So, the total sum is:
\[ \text{Sum} = 72 \]

**Step 3: Count the Number of Observations**
The number of observations (n) is:
\[ n = 6 \]

**Step 4: Calculate the Mean**
The mean is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{n} \]
Substituting the values:
\[ \text{Mean} = \frac{72}{6} = 12 \]

**Step 5: Divide Each Observation by 2**
Now, we divide each observation by 2:
\[ \frac{8}{2} = 4, \quad \frac{12}{2} = 6, \quad \frac{16}{2} = 8, \quad \frac{22}{2} = 11, \quad \frac{10}{2} = 5, \quad \frac{4}{2} = 2 \]
So the new observations are:
\[ 4, 6, 8, 11, 5, 2 \]

**Step 6: Calculate the Sum of the New Observations**
Add the new observations together:
\[ 4 + 6 + 8 + 11 + 5 + 2 \]

Calculating the sum:
\[ 4 + 6 = 10 \]
\[ 10 + 8 = 18 \]
\[ 18 + 11 = 29 \]
\[ 29 + 5 = 34 \]
\[ 34 + 2 = 36 \]
So, the total sum of the new observations is:
\[ \text{Sum} = 36 \]

**Step 7: Count the Number of New Observations**
The number of new observations (n) remains:
\[ n = 6 \]

**Step 8: Calculate the New Mean**
The new mean is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of New Observations}}{n} \]
Substituting the values:
\[ \text{Mean} = \frac{36}{6} = 6 \]

### Final Answer:
The mean of the original observations is **12**, and the resulting mean after dividing each observation by 2 is **6**.