Let the mean of 6 observations 1, 2, 4, 5, x and y be 5 and their variance be 10. Then their mean de

Q: Wait, why do we use the absolute value in mean deviation? Can't we just sum the differences?

That is a brilliant question! If you simply sum the differences $(x_i - \bar{x})$, the positive and negative deviations will cancel each other out, always resulting in zero. The absolute value ensures we are measuring the 'distance' of each point from the mean, regardless of whether it is above or below it. It captures the true magnitude of the spread.

Q: Is there a faster way to find the sum of squares without manually squaring each number?

While you can use the shortcut $\sum x_i^2 = n(\sigma^2 + \bar{x}^2)$, it is essentially the same as the formula we used. For a small set of numbers like this, manual calculation is actually safer and less prone to memory errors. Always prioritize clarity over speed in the exam hall!

Q: How do I know if I should reject one of the roots for x and y?

In this specific problem, the system is symmetric. Whether $x=8$ and $y=10$ or $x=10$ and $y=8$, the set of observations remains the same: $1, 2, 4, 5, 8, 10$. Since the mean deviation depends on the set of values, the order of $x$ and $y$ does not change the final result. You don't need to reject anything!

Question

Let the mean of 6 observations 1, 2, 4, 5, $x$ and $y$ be 5 and their variance be 10. Then their mean deviation about the mean is equal to

Options:

(A) $\frac{7}{3}$

(B) 3

(C) $\frac{8}{3}$

(D) $\frac{10}{3}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are presented with a set of six observations: $1, 2, 4, 5, x,$ and $y$. We are given that the mean is $5$ and the variance is $10$. Our objective is to determine the mean deviation about the mean.

### Unmasking the Variables

We start with the fundamental concept of the mean, defined as the balance point of the data. The formula is:

$$\bar{x} = \frac{\sum x_i}{n}$$

Given $n=6$ and $\bar{x}=5$, the sum of all observations must be $30$. Adding our known values, $1 + 2 + 4 + 5 = 12$, we establish the following relationship:

$$12 + x + y = 30 \Rightarrow x + y = 18$$

Next, we utilize the variance formula to measure the spread of our data:

$$\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$$

Substituting our known values into the equation:

$$10 = \frac{1^2 + 2^2 + 4^2 + 5^2 + x^2 + y^2}{6} - 5^2$$

Since $5^2 = 25$, adding $25$ to $10$ gives $35$. Multiplying by $6$ yields $210 = 46 + x^2 + y^2$, which simplifies to:

$$x^2 + y^2 = 164$$

### The Algebraic Bridge

We now possess a system of two equations: $x + y = 18$ and $x^2 + y^2 = 164$. We use the algebraic identity $(x+y)^2 = x^2 + y^2 + 2xy$ to find the product of the variables:

$$18^2 = 164 + 2xy \Rightarrow 324 = 164 + 2xy$$

Solving for $xy$, we find $2xy = 160$, which implies $xy = 80$. We require two numbers that sum to $18$ and multiply to $80$.

By inspection, the factors of $80$ that satisfy this condition are $8$ and $10$. Thus, our missing observations are $8$ and $10$.

### Final Calculation

With our complete data set identified as $\{1, 2, 4, 5, 8, 10\}$, we calculate the mean deviation about the mean. This is defined as the average of the absolute distances of each point from the mean of $5$.

The absolute deviations are:

$$|1-5|=4, \quad |2-5|=3, \quad |4-5|=1, \quad |5-5|=0, \quad |8-5|=3, \quad |10-5|=5$$

Summing these distances, we get $4 + 3 + 1 + 0 + 3 + 5 = 16$. Dividing by the total number of observations, $n=6$, we arrive at the final result:

$$M.D. = \frac{16}{6} = \frac{8}{3}$$

The mean deviation about the mean is **$\frac{8}{3}$**.

Animated Solution for Mathematics - Statistics: Let the mean of 6 observations 1, 2, 4, 5, $x$ and $y$ be 5 and their variance be 10. Then their mean deviation about the mean is equal to

Select Answer:

Visualized Solution

Analyze the Given Data

Apply the Mean Formula

Simplify the Mean Equation

Introduce the Variance Formula

Substitute into Variance Formula

Simplify the Variance Equation

Find the Sum of Squares

Solve for x and y

Identify the Unknowns

Define Mean Deviation

Calculate Absolute Deviations

Final Computation

The Sigma Insight: Measures of Dispersion

Analyzing the Setup

Unmasking the Variables

The Algebraic Bridge

Final Calculation

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