If the mean of the data : 7, 8, 9, 7, 8, 7, λ, 8 is 8, then the variance of this data is :- (A) 2 (B

Q: Why can't we just use the standard deviation formula $\sqrt{\frac{\sum(x_i - \bar{x})^2}{n}}$?

You absolutely can! But in JEE Advanced, time is currency. The computational formula $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$ saves you from calculating the deviation $(x_i - \bar{x})$ for every single term, reducing the chance of a silly arithmetic error.

Q: What if $\lambda$ was a negative number? Would the variance change?

Great question! $\lambda$ is just a data point. If the math dictates it's negative, it's negative. Variance itself, however, is a measure of spread and will always be non-negative. Don't let the sign of the data point intimidate you; trust the algebra.

Q: Is there a shortcut to finding the sum of squares?

Not really, but you can group them! Notice you have three 7s, three 8s, one 9, and one 10. Instead of squaring individually, calculate $3(7^2) + 3(8^2) + 9^2 + 10^2$. It's faster and less prone to error.

Question

If the mean of the data : 7, 8, 9, 7, 8, 7, $\lambda$, 8 is 8, then the variance of this data is :-

Options:

(A) 2

(B) 7/8

(C) 9/8

(D) 1

Sigma Percentile · Accepted Answer

### The Mystery of the Missing Variable

Welcome, future engineers! Today, we are diving into a classic statistics problem. It might look like a simple list of numbers, but statistics is the language of uncertainty, and mastering it is crucial for your JEE journey.

We are given a dataset: $7, 8, 9, 7, 8, 7, \lambda, 8$. We know the mean is $8$, and our mission is to find the variance. But wait—there is a ghost in the machine! A variable $\lambda$ is hiding in our data. Before we can calculate the variance, we must unmask this variable.

### Phase 1: Balancing the Seesaw

Imagine the mean as a fulcrum on a seesaw. For the system to be in equilibrium, the sum of all values must be perfectly balanced.

The definition of the mean, $\bar{x}$, is the sum of all observations divided by the total count, $n$. Mathematically, we write this as:

$$
\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}
$$

Here, we have $8$ observations, and the mean is $8$. So, we set up our equation:

$$
\frac{7 + 8 + 9 + 7 + 8 + 7 + \lambda + 8}{8} = 8
$$

Adding the known constants, we get $54 + \lambda$ in the numerator. Cross-multiplying gives us $54 + \lambda = 64$.

Solving this, we find $\lambda = 10$. Just like that, the mystery is solved! Our dataset is complete: $7, 8, 9, 7, 8, 7, 10, 8$.

### Phase 2: The Power of the Computational Formula

Now, we need the variance. You might be tempted to subtract the mean from every single term, square the result, and then average them. While that works, it is a recipe for disaster in a high-pressure exam environment.

Instead, let us use the elegant computational formula:

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

This formula is a gift to students. It separates the sum of squares from the square of the mean, allowing us to compute the variance in one clean sweep. We already know $\bar{x} = 8$, so $(\bar{x})^2 = 64$.

### Phase 3: The Final Execution

Let us calculate $\sum x_i^2$. We square each term: $7^2=49$, $8^2=64$, $9^2=81$, $7^2=49$, $8^2=64$, $7^2=49$, $10^2=100$, and $8^2=64$.

Summing these up: $49 + 64 + 81 + 49 + 64 + 49 + 100 + 64 = 520$.

Now, we plug this into our variance formula:

$$
\sigma^2 = \frac{520}{8} - 64
$$

Performing the division, $520 \div 8 = 65$. Finally, we subtract: $65 - 64 = 1$.

The **variance is exactly 1**. It is beautiful, isn't it? A complex-looking problem reduced to a clean, single integer. Keep this methodical approach in your toolkit, and no statistics problem will ever stand in your way!

Animated Solution for Mathematics - Statistics: If the mean of the data : 7, 8, 9, 7, 8, 7, $λ$ , 8 is 8, then the variance of this data is :-

Select Answer:

Visualized Solution

Understanding the Data

The Mean Formula

Substituting the Values

Summing the Knowns

Solving for $λ$

The Variance Formula

Calculating Sum of Squares

Summing the Squares

Substituting into Variance

Final Calculation

The Sigma Insight: Variance and Standard Deviation

The Mystery of the Missing Variable

Phase 1

Phase 2

Phase 3

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Animated Solution for Mathematics - Statistics: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the mean of the data : 7, 8, 9, 7, 8, 7, λ, 8 is 8, then the variance of this data is :-