Let `alpha, beta in R`. If the mean and the variable of 6 observation, `-3, 4, 7, -6, alpha, beta` be 2 and 23 respectively then mean deviation about the mean of the 6 observation is

Options

A. `(11)/3`
B. `(16)/3`
C. `(13)/3`
D. `(14)/3`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the Mean
Given the observations: \(-3, 4, 7, -6, \alpha, \beta\), we know that the mean is given by:

\[ \text{Mean} = \frac{-3 + 4 + 7 - 6 + \alpha + \beta}{6} = 2 \]

Calculating the sum of the known values:

\[ -3 + 4 + 7 - 6 = 2 \]

Thus, we can rewrite the equation:

\[ \frac{2 + \alpha + \beta}{6} = 2 \]

Multiplying both sides by 6 gives:

\[ 2 + \alpha + \beta = 12 \]

So, we have:

\[ \alpha + \beta = 10 \quad \text{(Equation 1)} \]

### Step 2: Calculate the Variance
The variance is given as 23. The formula for variance is:

\[ \text{Variance} = \frac{\sum (x_i^2)}{n} - \left(\text{Mean}\right)^2 \]

Calculating \(\sum (x_i^2)\):

\[ (-3)^2 + 4^2 + 7^2 + (-6)^2 + \alpha^2 + \beta^2 = 9 + 16 + 49 + 36 + \alpha^2 + \beta^2 = 110 + \alpha^2 + \beta^2 \]

Substituting into the variance formula:

\[ 23 = \frac{110 + \alpha^2 + \beta^2}{6} - 2^2 \]

Calculating \(2^2\):

\[ 23 = \frac{110 + \alpha^2 + \beta^2}{6} - 4 \]

Adding 4 to both sides:

\[ 27 = \frac{110 + \alpha^2 + \beta^2}{6} \]

Multiplying both sides by 6:

\[ 162 = 110 + \alpha^2 + \beta^2 \]

Thus, we have:

\[ \alpha^2 + \beta^2 = 52 \quad \text{(Equation 2)} \]

### Step 3: Solve for \(\alpha\) and \(\beta\)
We have two equations now:

1. \(\alpha + \beta = 10\)
2. \(\alpha^2 + \beta^2 = 52\)

Using the identity:

\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]

Substituting Equation 1 into this identity:

\[ 52 = 10^2 - 2\alpha\beta \]

Calculating \(10^2\):

\[ 52 = 100 - 2\alpha\beta \]

Rearranging gives:

\[ 2\alpha\beta = 100 - 52 = 48 \]

Thus:

\[ \alpha\beta = 24 \quad \text{(Equation 3)} \]

### Step 4: Find \(\alpha\) and \(\beta\)
Now we can solve for \(\alpha\) and \(\beta\) using the equations:

1. \(\alpha + \beta = 10\)
2. \(\alpha\beta = 24\)

Let \(\alpha\) and \(\beta\) be the roots of the quadratic equation:

\[ x^2 - (10)x + 24 = 0 \]

Calculating the discriminant:

\[ D = 10^2 - 4 \cdot 1 \cdot 24 = 100 - 96 = 4 \]

The roots are:

\[ x = \frac{10 \pm \sqrt{4}}{2} = \frac{10 \pm 2}{2} \]

Thus, we have:

\[ x = 6 \quad \text{or} \quad x = 4 \]

So, \(\alpha = 6, \beta = 4\) or \(\alpha = 4, \beta = 6\).

### Step 5: Calculate the Mean Deviation
The mean deviation about the mean (which is 2) is given by:

\[ \text{Mean Deviation} = \frac{\sum |x_i - \text{Mean}|}{n} \]

Calculating each term:

1. \(|-3 - 2| = 5\)
2. \(|4 - 2| = 2\)
3. \(|7 - 2| = 5\)
4. \(|-6 - 2| = 8\)
5. \(|6 - 2| = 4\)
6. \(|4 - 2| = 2\)

Summing these:

\[ 5 + 2 + 5 + 8 + 4 + 2 = 26 \]

Now, dividing by the number of observations (6):

\[ \text{Mean Deviation} = \frac{26}{6} = \frac{13}{3} \]

### Final Answer
The mean deviation about the mean of the 6 observations is:

\[ \frac{13}{3} \]