Animated Solution for Mathematics - Statistics: If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?

Understanding Variance

Welcome, fellow traveler on the JEE journey. Today, we aren't just solving a statistics problem; we are learning to dance with data.

When we look at a set of numbers like $\{2,3,a,11\}$, we aren't just looking at digits. We are looking at a 'spread'—a measure of how much these numbers deviate from their center. This is the heart of standard deviation, denoted by $\sigma$.

The Variance Shortcut

We are given $\sigma =3.5$. Now, I want you to pause. Working with square roots is rarely fun.

In the world of competitive exams, we prefer to work with the square of the standard deviation, which we call Variance (${\sigma }^2$). By squaring $3.5$, we get $12.25$.

This small shift in perspective—moving from the root to the square—is the first step in mastering statistical problems. It clears the path for cleaner algebra.

We will use the computational formula for variance:

This formula is your best friend. It separates the 'sum of squares' from the 'square of the mean,' allowing us to handle the unknown variable $a$ with precision.

The Heavy Lifting

Let's break this down into two manageable tasks. First, the arithmetic mean, $\bar{x}$. The mean is simply the sum of all observations divided by the count, $n=4$.

Next, we need the sum of the squares of our observations, $\sum x_i^2$. This is where students often make a mistake by forgetting to square the unknown $a$. Let's be methodical:

See how clean that is? We have our mean and our sum of squares. Now, we are ready to assemble the puzzle.

The Algebraic Dance

Now, we substitute our findings into the variance formula:

This looks intimidating, but don't panic. We have a denominator of $4$ in the first term and a denominator of $16$ (since $4^2=16$) in the second.

To clear these fractions, we multiply the entire equation by $16$. This is a classic JEE strategy—clear the denominators early to avoid messy fraction arithmetic.

Now, let's expand carefully. Remember the identity $(x+y{)}^2=x^2+2xy+y^2$.

Here is the moment of truth. Distribute that negative sign across the entire bracket. If you miss this, the whole equation collapses.

Now, we group our terms. We have $4a^2-a^2$, which gives us $3a^2$. We have the linear term $-32a$.

For the constants, $536-256=280$.

Finally, subtract $196$ from both sides to set the equation to zero:

And there it is. The elegance of the result matches the effort you put in. We have arrived at the quadratic equation that defines the relationship for $a$.

You didn't just solve a problem; you navigated a logical structure. Keep this methodical approach in your toolkit, and no statistical problem will ever intimidate you again.