Key Concepts and Formulas

This problem requires a solid understanding of three fundamental statistical measures: the mean, variance, and mean deviation about the mean. Let's recall their definitions and formulas:

1. Mean ($\bar{x}$): The arithmetic average of a set of numbers. It is calculated by summing all observations and dividing by the total number of observations. For a set of $n$ observations $x_1, x_2, \dots, x_n$: $\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$

2. Variance ($\sigma^2$): A measure of the spread or dispersion of a set of numbers from their mean. It quantifies how much the data points deviate from the average. The most common computational formula is: $\sigma^2 = \frac{\sum_{i=1}^n x_i^2}{n} - (\bar{x})^2$ This formula is generally preferred for calculations as it often simplifies the process compared to $\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n}$.

3. Mean Deviation about the Mean (M): This measures the average of the absolute differences between each observation and the mean. It provides an idea of the typical distance of data points from the mean, without the effects of squaring that variance introduces. $M = \frac{\sum_{i=1}^n |x_i - \bar{x}|}{n}$

Our primary goal is to first determine the unknown numbers $a$ and $b$ using the given mean and variance. Once we have all the numbers, we will calculate the mean deviation ($M$) and finally find the value of $25M$.

Step-by-Step Solution

We are given the numbers $a, b, 8, 5, 10$. The total number of observations is $n=5$.

Step 1: Utilize the Mean to find the sum of $a$ and $b$.

The mean ($\bar{x}$) of the numbers is given as 6. This is our first piece of information.

• Why this step? The mean formula directly relates the sum of all observations to the mean. Since $a$ and $b$ are unknown, using the mean will provide us with our first algebraic equation involving $a+b$. This is a crucial initial step in identifying the unknown values.

Applying the mean formula: $\bar{x} = \frac{a + b + 8 + 5 + 10}{5}$ Substitute the given mean $\bar{x}=6$: $6 = \frac{a + b + 23}{5}$ To isolate the sum $a+b$, multiply both sides by 5: $6 \times 5 = a + b + 23$ $30 = a + b + 23$ Now, subtract 23 from both sides to find $a+b$: $a + b = 30 - 23$ $a + b = 7 \quad \text{... (Equation 1)}$

Step 2: Utilize the Variance to find the sum of squares of $a$ and $b$.

The variance ($\sigma^2$) of the numbers is given as 6.8. We already know the mean $\bar{x}=6$.

• Why this step? We have one equation ($a+b=7$) but two unknowns ($a$ and $b$). To solve for $a$ and $b$ uniquely, we need a second independent equation. The variance formula involves the sum of squares of the observations ($\sum x_i^2$), which will yield an equation involving $a^2+b^2$. This second equation, combined with the first, will allow us to find the individual values of $a$ and $b$.

Applying the variance formula: $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$ Substitute the given variance $\sigma^2=6.8$ and mean $\bar{x}=6$, along with the squares of the known numbers: $6.8 = \frac{a^2 + b^2 + 8^2 + 5^2 + 10^2}{5} - (6)^2$ Calculate the squares of the known numbers: $8^2=64$, $5^2=25$, $10^2=100$. $6.8 = \frac{a^2 + b^2 + 64 + 25 + 100}{5} - 36$ Sum the squared known numbers: $64+25+100 = 189$. $6.8 = \frac{a^2 + b^2 + 189}{5} - 36$ Add 36 to both sides to move it away from the fraction: $6.8 + 36 = \frac{a^2 + b^2 + 189}{5}$ $42.8 = \frac{a^2 + b^2 + 189}{5}$ Multiply both sides by 5 to clear the denominator: $42.8 \times 5 = a^2 + b^2 + 189$ $214 = a^2 + b^2 + 189$ Subtract 189 from both sides to isolate $a^2+b^2$: $a^2 + b^2 = 214 - 189$ $a^2 + b^2 = 25 \quad \text{... (Equation 2)}$

Step 3: Solve for the unknown numbers $a$ and $b$.

We now have a system of two algebraic equations:

1. $a + b = 7$
2. $a^2 + b^2 = 25$

• Why this step? Before we can calculate the mean deviation, we need to know the exact values of all the numbers in our set. Solving this system of equations will reveal the specific values of $a$ and $b$.

We can use the algebraic identity $(a+b)^2 = a^2 + b^2 + 2ab$. Substitute the values from Equation 1 and Equation 2 into this identity: $(7)^2 = 25 + 2ab$ $49 = 25 + 2ab$ Subtract 25 from both sides to find $2ab$: $2ab = 49 - 25$ $2ab = 24$ Divide by 2 to find $ab$: $ab = 12$

Now we have $a+b=7$ and $ab=12$. We can find $a$ and $b$ by considering them as the roots of a quadratic equation. A quadratic equation whose roots are $x_1$ and $x_2$ is given by $x^2 - (x_1+x_2)x + x_1x_2 = 0$. So, for $a$ and $b$: $x^2 - (a+b)x + ab = 0$ Substitute the values: $x^2 - 7x + 12 = 0$ Factor the quadratic equation: $x^2 - 3x - 4x + 12 = 0$ $x(x-3) - 4(x-3) = 0$ $(x-3)(x-4) = 0$ This gives us two possible values for $x$: $x=3$ or $x=4$. Therefore, the values of $a$ and $b$ are 3 and 4 (the order does not matter for the set of numbers). The complete set of numbers is $3, 4, 8, 5, 10$.

Step 4: Calculate the Mean Deviation about the Mean (M).

Now that we know all the numbers ($3, 4, 8, 5, 10$) and the mean ($\bar{x}=6$), we can proceed to calculate the mean deviation about the mean.

• Why this step? This is a direct application of the mean deviation formula. We have all the necessary components (individual data points and the mean) to compute $M$.

First, we calculate the absolute deviation of each number from the mean $\bar{x}=6$:

• $|x_1 - \bar{x}| = |3 - 6| = |-3| = 3$
• $|x_2 - \bar{x}| = |4 - 6| = |-2| = 2$
• $|x_3 - \bar{x}| = |8 - 6| = |2| = 2$
• $|x_4 - \bar{x}| = |5 - 6| = |-1| = 1$
• $|x_5 - \bar{x}| = |10 - 6| = |4| = 4$

Next, sum these absolute deviations: $\sum |x_i - \bar{x}| = 3 + 2 + 2 + 1 + 4 = 12$

Finally, calculate the Mean Deviation (M) by dividing the sum of absolute deviations by the total number of observations $n=5$: $M = \frac{\sum |x_i - \bar{x}|}{n} = \frac{12}{5}$

Step 5: Calculate the final value of $25M$.

The problem asks for the value of $25M$.

• Why this step? This is the final calculation required to answer the specific question posed in the problem.

Substitute the calculated value of $M$: $25M = 25 \times \frac{12}{5}$ Simplify the expression: $25M = (5 \times 5) \times \frac{12}{5}$ $25M = 5 \times 12$ $25M = 60$

Common Mistakes & Tips

• Absolute Values in Mean Deviation: Always remember to take the absolute difference $|x_i - \bar{x}|$ for mean deviation. Forgetting the absolute value will lead to positive and negative deviations canceling out, often resulting in an incorrect mean deviation of zero.
• Variance Formula: Be precise with the variance formula. The term $(\bar{x})^2$ means the mean is calculated first and then squared, not the sum of squares of individual terms.
• Algebraic Accuracy: Be meticulous with algebraic manipulations, especially when solving simultaneous equations. A small calculation error in finding $a+b$ or $a^2+b^2$ will propagate and lead to an incorrect final answer.

Summary

This problem is an excellent exercise in applying fundamental statistical concepts. The solution hinges on a systematic approach: first, using the given mean and variance to determine the unknown data points $a$ and $b$ by solving a system of algebraic equations. Once all data points are known, the mean deviation about the mean is calculated by summing the absolute differences from the mean and dividing by the number of observations. Finally, the specific value requested, $25M$, is computed.

The final answer is $\boxed{\text{60}}$, which corresponds to option (A).