To solve the problem step by step, we will follow the given information about the observations, mean, and variance.
### Step 1: Understand the given data
We have:
- Mean of 5 observations = 5
- Variance of the observations = 12.4
- Three observations are 1, 2, and 6.
### Step 2: Find the sum of all observations
The mean of the observations is given by the formula:
\[
\text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}
\]
Let the two unknown observations be \(x\) and \(y\). Thus, we can write:
\[
\frac{1 + 2 + 6 + x + y}{5} = 5
\]
Calculating the sum of the known observations:
\[
1 + 2 + 6 = 9
\]
Substituting this into the equation:
\[
\frac{9 + x + y}{5} = 5
\]
Multiplying both sides by 5:
\[
9 + x + y = 25
\]
Thus, we have:
\[
x + y = 16 \quad \text{(Equation 1)}
\]
### Step 3: Use the variance to find another equation
The variance is given by:
\[
\text{Variance} = \frac{\sum (x_i - \bar{x})^2}{N}
\]
Where \(N\) is the number of observations, and \(\bar{x}\) is the mean. We know:
\[
\text{Variance} = 12.4
\]
Substituting the values:
\[
\frac{(1 - 5)^2 + (2 - 5)^2 + (6 - 5)^2 + (x - 5)^2 + (y - 5)^2}{5} = 12.4
\]
Calculating the squared differences for the known observations:
\[
(1 - 5)^2 = 16, \quad (2 - 5)^2 = 9, \quad (6 - 5)^2 = 1
\]
So, we have:
\[
\frac{16 + 9 + 1 + (x - 5)^2 + (y - 5)^2}{5} = 12.4
\]
Calculating the sum of the known squared differences:
\[
16 + 9 + 1 = 26
\]
Substituting this into the variance equation:
\[
\frac{26 + (x - 5)^2 + (y - 5)^2}{5} = 12.4
\]
Multiplying both sides by 5:
\[
26 + (x - 5)^2 + (y - 5)^2 = 62
\]
Thus:
\[
(x - 5)^2 + (y - 5)^2 = 36 \quad \text{(Equation 2)}
\]
### Step 4: Substitute \(y\) in terms of \(x\)
From Equation 1, we have \(y = 16 - x\). Substituting this into Equation 2:
\[
(x - 5)^2 + ((16 - x) - 5)^2 = 36
\]
This simplifies to:
\[
(x - 5)^2 + (11 - x)^2 = 36
\]
Expanding both squares:
\[
(x^2 - 10x + 25) + (121 - 22x + x^2) = 36
\]
Combining like terms:
\[
2x^2 - 32x + 146 = 36
\]
Subtracting 36 from both sides:
\[
2x^2 - 32x + 110 = 0
\]
Dividing the entire equation by 2:
\[
x^2 - 16x + 55 = 0
\]
### Step 5: Solve the quadratic equation
Using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 1\), \(b = -16\), and \(c = 55\):
\[
x = \frac{16 \pm \sqrt{(-16)^2 - 4 \cdot 1 \cdot 55}}{2 \cdot 1}
\]
Calculating the discriminant:
\[
x = \frac{16 \pm \sqrt{256 - 220}}{2} = \frac{16 \pm \sqrt{36}}{2} = \frac{16 \pm 6}{2}
\]
Thus:
\[
x = \frac{22}{2} = 11 \quad \text{or} \quad x = \frac{10}{2} = 5
\]
If \(x = 11\), then \(y = 5\). If \(x = 5\), then \(y = 11\).
### Step 6: Calculate the mean deviation
Now we have the observations: 1, 2, 6, 5, and 11. The mean is still 5.
The formula for mean deviation is:
\[
\text{Mean Deviation} = \frac{\sum |x_i - \bar{x}|}{N}
\]
Calculating the absolute deviations:
\[
|1 - 5| = 4, \quad |2 - 5| = 3, \quad |6 - 5| = 1, \quad |5 - 5| = 0, \quad |11 - 5| = 6
\]
Summing these absolute deviations:
\[
4 + 3 + 1 + 0 + 6 = 14
\]
Now, divide by the number of observations (5):
\[
\text{Mean Deviation} = \frac{14}{5} = 2.8
\]
### Final Answer
The mean deviation from the mean of the data is **2.8**.
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