Find the mean of the following numbers.
` " 16 14 12 8 23 26 " `
` "18 20 23 9 11 22 " `
` " 27 18 and 20 " `
State the value of the mean when each of these numbers is :
(i) increased by 4.
(ii) decreased by 6.

Answer

To find the mean of the given numbers and determine the new means after adjustments, we can follow these steps:

### Step 1: List the numbers
The numbers provided are:
- 16, 14, 12, 8, 23, 26
- 18, 20, 23, 9, 11, 22
- 27, 18, 20

### Step 2: Calculate the total sum of the numbers
We will add all the numbers together:
\[ 16 + 14 + 12 + 8 + 23 + 26 + 18 + 20 + 23 + 9 + 11 + 22 + 27 + 18 + 20 \]

Calculating this step-by-step:
- First group: \(16 + 14 + 12 + 8 + 23 + 26 = 99\)
- Second group: \(18 + 20 + 23 + 9 + 11 + 22 = 113\)
- Third group: \(27 + 18 + 20 = 65\)

Now, summing these results:
\[ 99 + 113 + 65 = 277 \]

### Step 3: Count the total number of observations
We count how many numbers there are:
- There are 15 numbers in total.

### Step 4: Calculate the mean
The mean (average) is calculated by dividing the total sum by the number of observations:
\[ \text{Mean} = \frac{\text{Total Sum}}{\text{Number of Observations}} = \frac{277}{15} \]
Calculating this gives:
\[ \text{Mean} = 18.47 \text{ (approximately)} \]

### Step 5: Calculate the new mean when each number is increased by 4
According to the properties of mean:
\[ \text{New Mean} = \text{Old Mean} + 4 \]
Thus:
\[ \text{New Mean} = 18.47 + 4 = 22.47 \]

### Step 6: Calculate the new mean when each number is decreased by 6
Similarly:
\[ \text{New Mean} = \text{Old Mean} - 6 \]
Thus:
\[ \text{New Mean} = 18.47 - 6 = 12.47 \]

### Final Answers:
- The mean of the numbers is approximately **18.47**.
- When each number is increased by 4, the new mean is approximately **22.47**.
- When each number is decreased by 6, the new mean is approximately **12.47**.