The mean square deviations of a set of...

MathematicsVariance and Standard DeviationJEE Advanced 1981Moderate

The mean square deviations of a set of observations $x_{1}, x_{2}, \dots, x_{n}$ about a point $c$ is defined to be $\frac{1}{n} \sum_{i = 1}^{n} (x_{i} - c)^{2}$ . The mean square deviations about $- 1$ and $+ 1$ of a set of observations are 7 and 3 respectively. Find the standard deviation of this set of observations.

Answer:

3

Visualized Solution (Hindi)

Understanding Mean Square Deviation

Mean Square Deviation (MSD) about a point $c$ is defined as:
MSD(c) = \frac{1}{n} \sum_{i=1}^n (x_i - c)^2
This function represents a parabola in terms of $c$ , with its minimum at the arithmetic mean $\overset{x}{ˉ}$ .

MSD about .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } c = - 1

Given MSD about $c = - 1$ is $7$ :
\frac{1}{n} \sum_{i=1}^n (x_i - (-1))^2 = 7
\frac{1}{n} \sum_{i=1}^n (x_i + 1)^2 = 7

Expanding the First Equation

Expand using $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ :
\frac{1}{n} \sum (x_i^2 + 2x_i + 1) = 7
\frac{1}{n} \sum x_i^2 + 2\left(\frac{\sum x_i}{n}\right) + \frac{n}{n} = 7
\frac{1}{n} \sum x_i^2 + 2\bar{x} + 1 = 7 \quad \dots (1)

MSD about .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } c = 1

Given MSD about $c = 1$ is $3$ :
\frac{1}{n} \sum_{i=1}^n (x_i - 1)^2 = 3

Expanding the Second Equation

Expand using $(a - b)^{2} = a^{2} - 2 ab + b^{2}$ :
\frac{1}{n} \sum (x_i^2 - 2x_i + 1) = 3
\frac{1}{n} \sum x_i^2 - 2\bar{x} + 1 = 3 \quad \dots (2)

Subtracting the Equations

Subtract equation $(2)$ from equation $(1)$ :
\left(\frac{1}{n} \sum x_i^2 + 2\bar{x} + 1\right) - \left(\frac{1}{n} \sum x_i^2 - 2\bar{x} + 1\right) = 7 - 3

Solving for the Mean .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \overset{x}{ˉ}

Simplify the subtraction:
4\bar{x} = 4
Divide by $4$ :
\bar{x} = 1

Adding the Equations

Add equation $(1)$ and equation $(2)$ :
\left(\frac{1}{n} \sum x_i^2 + 2\bar{x} + 1\right) + \left(\frac{1}{n} \sum x_i^2 - 2\bar{x} + 1\right) = 7 + 3

Solving for Mean of Squares

Simplify the addition:
2\left(\frac{1}{n} \sum x_i^2 + 1\right) = 10
\frac{1}{n} \sum x_i^2 + 1 = 5
\frac{1}{n} \sum x_i^2 = 4

The Variance Formula

The formula for Variance $(σ^{2})$ is:
\sigma^2 = \frac{1}{n} \sum x_i^2 - (\bar{x})^2

Calculating Variance

Substitute the values:
\sigma^2 = 4 - (1)^2
\sigma^2 = 4 - 1 = 3

Final Result: Standard Deviation

Standard Deviation $(σ)$ is the square root of Variance:
\sigma = \sqrt{\sigma^2}
\sigma = \sqrt{3}
Final Answer: The standard deviation is $3$ .

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Visualized Solution (Hindi)

Understanding Mean Square Deviation

MSD about .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } c=−1

Expanding the First Equation

MSD about .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } c=1

Expanding the Second Equation

Subtracting the Equations

Solving for the Mean .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } xˉ

Adding the Equations

Solving for Mean of Squares

The Variance Formula

Calculating Variance

Final Result: Standard Deviation

Conceptually Similar Problems

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a series of 2n observations, half of them equal a and remaining half equal -a. If the standard deviation of the observations is 2, then ∣a∣ equals.

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let x1​,x2​,…,xn​ be n observations such that ∑xi2​=400 and ∑xi​=80. Then the possible value of n among the following is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S be the standard deviation of n observations. Each of the n observations is multiplied by a constant c. Then the standard deviation of the resulting number is

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a series of 2n observations, half of them equal a and remaining half equal -a. If the standard deviation of the observations is 2, then ∣a∣ equals.

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If the mean deviation about the median of the numbers a, 2a,.......,50a is 50, then | a| equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let x1​,x2​,…,xn​ be n observations such that ∑xi2​=400 and ∑xi​=80. Then the possible value of n among the following is

MSD about $c = - 1$

MSD about $c = 1$

Solving for the Mean $\overset{x}{ˉ}$

Let $S$ be the standard deviation of $n$ observations. Each of the $n$ observations is multiplied by a constant $c$ . Then the standard deviation of the resulting number is

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Let $x_{1}, x_{2}, \dots, x_{n}$ be n observations such that $\sum x_{i}^{2} = 400$ and $\sum x_{i} = 80$ . Then the possible value of n among the following is