Let X1, X2, ..., X18 be eighteen observation such that Σ(Xi - α) = 36 and Σ(Xi - β)² = 90, where α a

Q: Wait, why can't I assume $\alpha$ is the mean? The problem doesn't say that!

Excellent catch! You are absolutely right. $\alpha$ is just a parameter. If you assumed it was the mean, you might get lucky, but you would be building your house on sand. We treat $\alpha$ as a variable and let the algebra reveal its relationship to the mean. Never assume properties that aren't given; let the equations speak for themselves.

Q: Why does the $18$ cancel out so cleanly in the variance formula?

That is the beauty of JEE problems! They are designed with internal symmetry. When you divide the sum of squares by $n=18$, the coefficients align perfectly. If you do not see the cancellation, re-check your expansion of $(X_i - \beta)^2$. It is a test of your algebraic discipline, not just your statistical knowledge.

Q: Why do we reject the root where $\alpha - \beta = 0$?

The problem explicitly states that $\alpha$ and $\beta$ are *distinct* real numbers. If $\alpha - \beta = 0$, then $\alpha = \beta$, which violates the given condition. Always check your constraints! In JEE, the difference between a correct answer and a trap is often found in the fine print.

Question

Let $X_1, X_2, \dots, X_{18}$ be eighteen observation such that $\sum_{i=1}^{18} (X_i - \alpha) = 36$ and $\sum_{i=1}^{18} (X_i - \beta)^2 = 90$, where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observations is 1, then the value of $|\alpha - \beta|$ is

Sigma Percentile · Accepted Answer

Welcome, JEE warriors! Today, we are going to dismantle a problem that looks like a statistical nightmare but is actually a beautifully choreographed dance of algebra.

When you first look at this problem, you see eighteen observations, two unknown parameters $\alpha$ and $\beta$, and a standard deviation of one. It is easy to feel overwhelmed. But remember, in the world of JEE Advanced, complexity is often just a mask for elegance.

Let us peel back that mask together.

### Analyzing the Hidden Mean

We are given the condition:
$$\sum_{i=1}^{18} (X_i - \alpha) = 36$$

Your instinct might be to panic because you do not know the values of $X_i$. But you do not need them! We are dealing with the properties of summation.

We can distribute the summation operator:
$$\sum X_i - \sum \alpha = 36$$

Since $\alpha$ is a constant, adding it eighteen times is simply $18\alpha$. Thus, we have $\sum X_i - 18\alpha = 36$, which gives us the sum of all observations:
$$\sum X_i = 18(\alpha + 2)$$

This is our first anchor. We have successfully translated a cryptic statement into a concrete algebraic expression.

### The Variance Trap

Next, we tackle the second condition:
$$\sum_{i=1}^{18} (X_i - \beta)^2 = 90$$

This is where many students stumble. Do not try to guess the values of $X_i$. Instead, expand the perfect square:
$$(X_i - \beta)^2 = X_i^2 - 2\beta X_i + \beta^2$$

Now, apply the summation to each term:
$$\sum X_i^2 - 2\beta \sum X_i + 18\beta^2 = 90$$

Notice how the pieces are fitting together? We already know $\sum X_i = 18(\alpha + 2)$. Let us substitute that in:
$$\sum X_i^2 - 2\beta(18(\alpha + 2)) + 18\beta^2 = 90$$

Rearranging this, we find an expression for the sum of squares:
$$\sum X_i^2 = 90 + 36\beta(\alpha + 2) - 18\beta^2$$

### The Algebraic Symphony

Now, we invoke the definition of variance. We know $\sigma = 1$, so $\sigma^2 = 1$. The formula for variance is:
$$\sigma^2 = \frac{\sum X_i^2}{n} - (\bar{X})^2$$

Here, $n=18$, and the mean $\bar{X}$ is:
$$\bar{X} = \frac{\sum X_i}{18} = \frac{18(\alpha + 2)}{18} = \alpha + 2$$

Substituting everything into the variance formula:
$$1 = \frac{90 + 36\beta(\alpha + 2) - 18\beta^2}{18} - (\alpha + 2)^2$$

Let us simplify this. Dividing the numerator by $18$, we get:
$$1 = 5 + 2\beta(\alpha + 2) - \beta^2 - (\alpha + 2)^2$$

Expanding the terms:
$$1 = 5 + 2\alpha\beta + 4\beta - \beta^2 - (\alpha^2 + 4\alpha + 4)$$

Simplifying further:
$$1 = 5 + 2\alpha\beta + 4\beta - \beta^2 - \alpha^2 - 4\alpha - 4$$

The $5$ and $-4$ combine to $1$, which cancels the $1$ on the left side. We are left with:
$$0 = -\alpha^2 - \beta^2 + 2\alpha\beta - 4\alpha + 4\beta$$

Multiply by $-1$ to make it look cleaner:
$$\alpha^2 + \beta^2 - 2\alpha\beta + 4\alpha - 4\beta = 0$$

### The Final Reveal

Look closely at that expression. $\alpha^2 + \beta^2 - 2\alpha\beta$ is exactly $(\alpha - \beta)^2$. And $4\alpha - 4\beta$ is $4(\alpha - \beta)$.

So, our equation is:
$$(\alpha - \beta)^2 + 4(\alpha - \beta) = 0$$

Factoring this, we get:
$$(\alpha - \beta)(\alpha - \beta + 4) = 0$$

Since the problem states $\alpha$ and $\beta$ are distinct, $\alpha - \beta 
eq 0$. Therefore, $\alpha - \beta + 4 = 0$, which means $\alpha - \beta = -4$.

The absolute value $|\alpha - \beta|$ is $|-4| = 4$.

**The final answer is 4.**

Enter Numerical Value:

Visualized Solution

Given Data

Expanding the First Summation

Isolating $\sum X_{i}$

Expanding the Second Summation

Distributing the Summation

Substituting $\sum X_{i}$

Variance Formula

Substituting into Variance

Simplifying the Equation

Grouping Terms

Factoring the Expression

Final Conclusion

The Sigma Insight: Variance and Standard Deviation

Analyzing the Hidden Mean

The Variance Trap

The Algebraic Symphony

The Final Reveal

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