Why is variance completely unaffected by adding a constant shift $b$, but scaled by $a^2$ under $y = ax + b$?

Variance measures the average squared dispersion of data points from their mean, given by $\frac{1}{n}\sum (x_i - \bar{x})^2$. Adding $b$ shifts both individual points $x_i$ and the mean $\bar{x}$ by $b$, making the difference $(x_i + b) - (\bar{x} + b) = x_i - \bar{x}$ unchanged. However, multiplying by $a$ expands the difference to $a(x_i - \bar{x})$, which squares to $a^2(x_i - \bar{x})^2$, scaling variance by $a^2$.

Should I compute Mean Deviation about Mean or Mean Deviation about Median in JEE Main questions?

Unless the question explicitly specifies 'about the mean', always check if it asks for the minimum mean deviation. Mathematically, $\sum |x_i - a|$ reaches its absolute minimum value when $a$ equals the median. If a specific point is mentioned, strictly compute deviations about that requested point.

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Question #1JEE Main 2024

The mean and variance of $10$ observations are $8$ and $16$ , respectively. If each observation is multiplied by $3$ and then $4$ is added to each, what are the new mean and new variance?

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Step-by-Step Solution & Concept

Step 1

Identify the transformation equation $y_i = a x_i + b$ with scale parameter $a = 3$ , origin shift $b = 4$ , initial mean $\bar{x} = 8$ , and initial variance $\sigma_x^2 = 16$ .

Step 2

Calculate the new mean using linear transformation of central tendency: $\bar{y} = a \bar{x} + b = 3(8) + 4 = 24 + 4 = 28$ .

Step 3

Calculate the new variance using scale transformation: $\sigma_y^2 = a^2 \sigma_x^2 = 3^2 \times 16 = 9 \times 16 = 144$ . The constant addition $b = 4$ does not affect variance.

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Question #2JEE Main 2023

The mean and variance of $5$ observations are $4.4$ and $8.24$ , respectively. If three of the observations are $1$ , $2$ , and $6$ , find the remaining two observations.

Solution Locked

Step-by-Step Solution & Concept

Step 1

Let the two missing observations be $x$ and $y$ . Using the mean formula $\bar{x} = \frac{\sum x_i}{n}$ , we have $\frac{1 + 2 + 6 + x + y}{5} = 4.4 \implies 9 + x + y = 22 \implies x + y = 13$ .

Step 2

Apply the variance formula $\sigma^2 = \frac{1}{n}\sum x_i^2 - (\bar{x})^2$ . Substituting values: $8.24 = \frac{1^2 + 2^2 + 6^2 + x^2 + y^2}{5} - (4.4)^2 \implies 8.24 + 19.36 = \frac{41 + x^2 + y^2}{5} \implies 27.6 \times 5 = 41 + x^2 + y^2 \implies x^2 + y^2 = 97$ .

Step 3

Solve the algebraic system $x + y = 13$ and $x^2 + y^2 = 97$ . Using $(x+y)^2 = x^2 + y^2 + 2xy \implies 169 = 97 + 2xy \implies xy = 36$ . The integers with sum $13$ and product $36$ are $x = 4$ and $y = 9$ .

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Mastering Statistics: Formulas, Traps & Exam Strategy | ExamiFly

Chapter Overview

Core Subtopics & Syllabus Breakdown

Key Formulas & High-Yield Equations

Mean and Variance Formula

Linear Transformation of Data

Combined Variance of Two Groups

Variance of First n Natural Numbers

Mean Deviation Minimum Condition

Common Mistakes & Traps to Avoid

Past Year Question (PYQ) Trends

Solved High-Yield Sample Questions

Frequently Asked Questions

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