The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations ar

Q: Why can't I just solve for $x$ and $y$ using the quadratic formula?

You absolutely could! But in a high-stakes exam like JEE, time is your most precious resource. Solving for $x$ and $y$ individually involves finding roots of a quadratic, which is prone to calculation errors. Using the identity $(x-y)^2 = 2(x^2+y^2) - (x+y)^2$ is a 'pro-move' that bypasses the need to find the individual values, saving you time and reducing the chance of a slip-up.

Q: Why is the variance formula $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$ better than the definition $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$?

Great question! While both are mathematically equivalent, the first one is computationally superior. It allows you to separate the sum of squares from the mean, which is exactly what we did here. It turns a potentially messy subtraction problem into a clean algebraic manipulation.

Q: Does the order of $x$ and $y$ matter?

Not at all! The question asks for the absolute difference, $|x-y|$. Whether $x$ is larger than $y$ or vice versa, the distance between them on the number line remains the same. The symmetry of the equations $x+y$ and $x^2+y^2$ ensures that the result is invariant.

Question

The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is :

Options:

(A) $9$

(B) $3$

(C) $7$

(D) $5$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are a data scientist tasked with reconstructing a dataset. You have eight observations, but two of them have been erased by a smudge of ink. You know the mean is $\bar{x} = 10$ and the variance is $\sigma^2 = 13.5$.

The remaining six numbers are $5, 7, 10, 12, 14, 15$. Let the two missing numbers be $x$ and $y$. To solve for two variables, we need two independent equations derived from the pillars of statistics: the Mean and the Variance.

### Phase 1: The Mean as a Summation Constraint

The mean is the "center of gravity" of your data, defined as:

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

Given $\bar{x} = 10$ and $n = 8$, the total sum of all observations must be $8 	imes 10 = 80$. Summing the six known numbers ($5+7+10+12+14+15$) yields $63$.

Therefore, our first equation is:

$$x + y = 80 - 63 = 17$$

This serves as our first anchor point. We now know the sum of our missing pair is $17$.

### Phase 2: The Variance as a Power Constraint

The variance measures the spread of the data and is linked to the sum of the squares of the observations via the formula:

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

Rearranging this to solve for the sum of squares, we get:

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

Plugging in our known values:

$$\sum x_i^2 = 8(13.5 + 100) = 8(113.5) = 908$$

Next, we calculate the sum of squares for the six known numbers:

$$5^2 + 7^2 + 10^2 + 12^2 + 14^2 + 15^2 = 25 + 49 + 100 + 144 + 196 + 225 = 739$$

Subtracting this from the total, we find the sum of squares of our unknowns:

$$x^2 + y^2 = 908 - 739 = 169$$

### Phase 3: The Algebraic Bridge

We now have a system of two equations: $x + y = 17$ and $x^2 + y^2 = 169$. While substitution is possible, we can use algebraic symmetry to find the difference between the variables.

We utilize the identity:

$$(x - y)^2 = 2(x^2 + y^2) - (x + y)^2$$

Substituting our known values into this identity:

$$(x - y)^2 = 2(169) - (17)^2$$

$$(x - y)^2 = 338 - 289 = 49$$

Taking the square root, we find the absolute difference:

$$|x - y| = 7$$

The mystery is solved. By solving the system $x+y=17$ and $x-y=7$, we find the missing values are **$12$ and $5$**. This technique of using algebraic symmetry is a powerful tool to save time during examinations.

Animated Solution for Mathematics - Statistics: The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is :

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Visualized Solution

Let the unknowns be $x$ and $y$

The definition of Mean

Sum of all observations

Finding the first equation: $x + y = 17$

The definition of Variance

Sum of squares setup

Total sum of squares

Squares of the known values

Finding the second equation: $x^{2} + y^{2} = 169$

Using the algebraic identity

Calculating $(x - y)^{2}$

The absolute difference $∣ x - y ∣$

The Sigma Insight: Measures of Dispersion

Analyzing the Setup

Phase 1

Phase 2

Phase 3

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