A data consists of n observations: x₁, x₂, ....., xₙ. If Σ(xᵢ+1)² = 9n and Σ(xᵢ-1)² = 5n, then the s

Q: Wait, why do we reject the negative root here?

Standard deviation $\sigma$ is defined as the square root of the variance $\sigma^2$. Since variance is a measure of spread (the average of squared deviations), it must be non-negative. Therefore, $\sigma$ is the positive square root of the variance. We are looking for the magnitude of dispersion, which is always positive.

Q: Why can't we just find the individual values of $x_i$?

That is a great question! We have $n$ unknown variables ($x_1, x_2, \dots, x_n$) but only two equations. It is impossible to solve for each individual $x_i$. However, in statistics, we don't need the individual values; we only need the aggregate sums $\sum x_i^2$ and $\sum x_i$ to calculate the variance. The problem is designed to give us exactly what we need.

Q: Why is the sum of $1$ equal to $n$?

Remember that $\sum_{i=1}^n 1$ means adding the number $1$ to itself $n$ times. So, $1 + 1 + \dots + 1$ ($n$ times) is simply $n \times 1 = n$. This is a common point of confusion, but it is a fundamental property of summations.

Question

A data consists of $n$ observations: $x_1, x_2, ....., x_n$. If $\sum_{i=1}^n (x_i+1)^2 = 9n$ and $\sum_{i=1}^n (x_i-1)^2 = 5n$, then the standard deviation of this data is :

Options:

(A) 5

(B) $\sqrt{5}$

(C) $\sqrt{7}$

(D) 2

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer! Today, we are going to peel back the layers of a seemingly simple statistics problem. We are given two equations involving a dataset $x_1, x_2, \dots, x_n$:

$$\sum_{i=1}^n (x_i+1)^2 = 9n$$

$$\sum_{i=1}^n (x_i-1)^2 = 5n$$

Our mission is to find the standard deviation. Think of this as a detective story where we must reconstruct the 'mean of squares' and the 'square of the mean' to determine the variance.

### Phase 1: The Algebraic Expansion

Let's start by expanding our first clue using the identity $(a+b)^2 = a^2 + 2ab + b^2$:

$$\sum_{i=1}^n (x_i^2 + 2x_i + 1) = 9n$$

Distributing the summation operator across each term, we recall that $\sum_{i=1}^n 1 = n$. This simplifies the expression to:

$$\sum x_i^2 + 2\sum x_i + n = 9n$$

Subtracting $n$ from both sides yields our first elegant equation:

$$\sum x_i^2 + 2\sum x_i = 8n \quad 	ext{--- (Equation 1)}$$

Now, we repeat the process for the second clue, $\sum_{i=1}^n (x_i-1)^2 = 5n$. Expanding this, we get:

$$\sum x_i^2 - 2\sum x_i + n = 5n$$

Simplifying this leads to our second equation:

$$\sum x_i^2 - 2\sum x_i = 4n \quad 	ext{--- (Equation 2)}$$

### Phase 2: The Power of Elimination

We now have a system of two equations:

$$\sum x_i^2 + 2\sum x_i = 8n$$

$$\sum x_i^2 - 2\sum x_i = 4n$$

If we add these two equations, the $2\sum x_i$ and $-2\sum x_i$ terms cancel out, leaving us with:

$$2\sum x_i^2 = 12n \implies \frac{\sum x_i^2}{n} = 6$$

Next, subtracting Equation 2 from Equation 1 causes the $\sum x_i^2$ terms to vanish:

$$4\sum x_i = 4n \implies \bar{x} = \frac{\sum x_i}{n} = 1$$

### Phase 3: The Final Bridge

We now have all the pieces of the puzzle. The variance $\sigma^2$ is defined as the mean of squares minus the square of the mean:

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

Substituting our calculated values into the formula:

$$\sigma^2 = 6 - (1)^2 = 6 - 1 = 5$$

Finally, the standard deviation $\sigma$ is the square root of the variance:

$$\sigma = \sqrt{5}$$

You have successfully navigated the data, eliminated the noise, and arrived at the truth. The final answer is **$\sqrt{5}$**.

Animated Solution for Mathematics - Statistics: A data consists of $n$ observations: $x_{1}, x_{2}, ....., x_{n}$ . If $\sum_{i = 1}^{n} (x_{i} + 1)^{2} = 9 n$ and $\sum_{i = 1}^{n} (x_{i} - 1)^{2} = 5 n$ , then the standard deviation of this data is :

Select Answer:

Visualized Solution

Given Data & Objective

Expanding the First Equation

Simplifying Equation 1

Expanding the Second Equation

Simplifying Equation 2

Adding Equations (Eq. 1 + Eq. 2)

Calculating Mean of Squares

Subtracting Equations (Eq. 1 - Eq. 2)

Calculating the Mean $(\overset{x}{ˉ})$

The Variance Formula

Calculating the Variance

Finding the Standard Deviation

The Sigma Insight: Variance and Standard Deviation

Analyzing the Setup

Phase 1

Phase 2

Phase 3

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