Find the mean of the following:
`10(1)/(4), 9, 4 (3)/(4), 8, 2 (2)/(3), 12 "and" 2(1)/(3)`

Answer

To find the mean of the given observations, we will follow these steps:

### Step 1: Convert mixed numbers to improper fractions
The given observations are:
- \(10 \frac{1}{4}\)
- \(9\)
- \(4 \frac{3}{4}\)
- \(8\)
- \(2 \frac{2}{3}\)
- \(12\)
- \(2 \frac{1}{3}\)

Let's convert the mixed numbers to improper fractions:
1. \(10 \frac{1}{4} = \frac{10 \times 4 + 1}{4} = \frac{40 + 1}{4} = \frac{41}{4}\)
2. \(4 \frac{3}{4} = \frac{4 \times 4 + 3}{4} = \frac{16 + 3}{4} = \frac{19}{4}\)
3. \(2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}\)
4. \(2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}\)

So, the observations in improper fractions are:
- \(\frac{41}{4}\)
- \(9\) (which can be written as \(\frac{9}{1}\))
- \(\frac{19}{4}\)
- \(8\) (which can be written as \(\frac{8}{1}\))
- \(\frac{8}{3}\)
- \(12\) (which can be written as \(\frac{12}{1}\))
- \(\frac{7}{3}\)

### Step 2: Find a common denominator
The denominators we have are \(4\), \(1\), \(3\). The least common multiple (LCM) of these denominators is \(12\).

### Step 3: Convert all fractions to have a common denominator
Now, we will convert each observation to have a denominator of \(12\):
1. \(\frac{41}{4} = \frac{41 \times 3}{4 \times 3} = \frac{123}{12}\)
2. \(\frac{9}{1} = \frac{9 \times 12}{1 \times 12} = \frac{108}{12}\)
3. \(\frac{19}{4} = \frac{19 \times 3}{4 \times 3} = \frac{57}{12}\)
4. \(\frac{8}{1} = \frac{8 \times 12}{1 \times 12} = \frac{96}{12}\)
5. \(\frac{8}{3} = \frac{8 \times 4}{3 \times 4} = \frac{32}{12}\)
6. \(\frac{12}{1} = \frac{12 \times 12}{1 \times 12} = \frac{144}{12}\)
7. \(\frac{7}{3} = \frac{7 \times 4}{3 \times 4} = \frac{28}{12}\)

### Step 4: Sum all the observations
Now we will sum all the fractions:
\[ \frac{123}{12} + \frac{108}{12} + \frac{57}{12} + \frac{96}{12} + \frac{32}{12} + \frac{144}{12} + \frac{28}{12} \]
Combine the numerators:
\[ 123 + 108 + 57 + 96 + 32 + 144 + 28 = 588 \]
So, the total sum is:
\[ \frac{588}{12} \]

### Step 5: Calculate the mean
Now, we find the mean by dividing the total sum by the number of observations:
\[ \text{Mean} = \frac{\text{Total Sum}}{\text{Number of Observations}} = \frac{588/12}{7} = \frac{588}{12 \times 7} = \frac{588}{84} = 7 \]

### Final Answer
The mean of the observations is \(7\).