Problem 18 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The variance of \(\alpha, \beta\) and \(\gamma\) is 9 , then variance of \(5 \alpha, 5 \beta\) and \(5 \gamma\) is (A) 45 (B) \(9 / 5\) (C) \(5 / 9\) (D) 225

Step-by-Step Solution

Answer

The variance of \(5\alpha, 5\beta, 5\gamma\) is 225 (Option D).

Step 1: Recall the Variance Formula

The variance of a set of values is a measure of how much the values differ from their mean. It is given by the formula: \( \text{Variance} = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2 \), where \( x_i \) are the values and \( \bar{x} \) is the mean of the values. For three numbers \( \alpha, \beta, \gamma \), this simplifies to the given variance as 9.

Step 2: Understand the Effect of Scaling on Variance

If each element of a data set is multiplied by a constant \( c \), then the variance is multiplied by \( c^2 \). For example, if \( \text{Variance}(\alpha, \beta, \gamma) = 9 \), then \( \text{Variance}(5\alpha, 5\beta, 5\gamma) = 5^2 \times 9 \).

Step 3: Calculate the New Variance

Since scaling affects the variance, calculate the variance of \( 5\alpha, 5\beta, 5\gamma \) using the relationship: \( \text{Variance}(5\alpha, 5\beta, 5\gamma) = 25 \times 9 = 225 \).

Step 4: Complete the Exercise

Therefore, the variance of \( 5\alpha, 5\beta, 5\gamma \) is given by 225, which corresponds to option (D).

Key Concepts

Understanding the Mean

The mean is a fundamental concept in statistics that helps to find the average value of a set of numbers. It is calculated by summing all the numbers in a dataset and then dividing by the number of values in that dataset.

• For example, if you have three numbers, say \( \alpha, \beta, \gamma \), the mean \( \bar{x} \) is given by \( \bar{x} = \frac{\alpha + \beta + \gamma}{3} \).

The mean provides a central value around which the numbers are distributed. It is a useful measure of central tendency, helping us understand how typical a value is within a dataset.
The variance, as mentioned in the problem, measures how far each number in the set is from this mean. It is important to note that variance is a measure of spread or dispersion. Thus, while the mean gives you a central point, the variance tells you about the data's spread around that central point.

Exploring Scaling Effects on Variance

Scaling refers to the effect of multiplying all values in a dataset by a constant factor. When this happens, the variance of the dataset changes significantly.
Here's why: variance is not just about the individual data points themselves, but about their distances from the mean.

• If you multiply each value by a constant \( c \), the variance of the set is multiplied by \( c^2 \).
• This is because each distance from the mean also gets multiplied by \( c \), and variance is a squared measure.

In the example from the exercise, the variance of \( \alpha, \beta, \gamma \) was 9. By scaling each of these by 5 (i.e., \( 5\alpha, 5\beta, 5\gamma \)), the variance becomes \( 5^2 \times 9 \). This results in a new variance of 225.
Understanding scaling effects is crucial when analyzing transformed data sets and helps in predicting changes in variability and spread.

Constant Multiplication and Its Implications

In mathematics and statistics, constant multiplication is an operation where every element in a set is multiplied by the same number. This action is called a 'constant' because the multiplier is a fixed number, identical for all elements in the dataset.
When analyzing data, this operation impacts statistical measures like variance and mean in different ways.

• The mean of scaled data \((c\alpha, c\beta, c\gamma)\) becomes \(c\) times the original mean \(\bar{x}\).
• However, and importantly for variance, the multiplication by a constant changes the variance by the square of that constant.

This characteristic explains why, in our exercise, the variance was multiplied by \( 5^2 \), resulting in 225. It's a key concept in transforming data sets and understanding how linear transformations, such as constant multiplication, affect statistical properties.
Awareness of this effect makes it easier to work with scaled data and make informed interpretations of the results.