Problem 53 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(x_{1}, x_{2} \ldots x_{n}\) be \(n\) observations, and let \(\bar{x}\) be their arithmetic mean and \(\sigma^{2}\) be their variance. Statement-1: Variance of \(2 x_{1}, 2 x_{2} \ldots 2 x_{n}\) is \(4 \sigma^{2}\). Statement-2: Arithmetic mean of \(2 x_{1}, 2 x_{2} \ldots 2 x_{n}\) is 4 \(\bar{x}\) (A) Statement- 1 is false, statement- 2 is true (B) Statement-1 is true, statement- 2 is true; statement 2 is a correct explanation for statement 1 (C) Statement-1 is true, statement- 2 is true; statement 2 is not a correct explanation for statement 1 (D) Statement- 1 is true, statement- 2 is false

Step-by-Step Solution

Answer

(D) Statement-1 is true, statement-2 is false.

Step 1: Understanding Variance

The variance \( \sigma^2 \) is given by \( \sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n} \). When each observation is multiplied by 2, i.e., considering the data set \(2x_1, 2x_2, \ldots, 2x_n\), each deviation \((x_i - \bar{x})\) is also multiplied by 2. The new deviations are therefore \(2(x_i - \bar{x})\), so their squared values become \(4(x_i - \bar{x})^2\). Hence, the variance of \(2x_1, 2x_2, \ldots, 2x_n\) is \(4\sigma^2\). Statement-1 is true.

Step 2: Understanding Arithmetic Mean

The arithmetic mean \( \bar{x} \) is given by \( \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \). When all observations are multiplied by 2, i.e., from the data set \(2x_1, 2x_2, \ldots, 2x_n\), arithmetic mean also gets multiplied by 2. Therefore, the new mean is \(2\bar{x}\), not \(4\bar{x}\) as Statement-2 suggests. Thus, Statement-2 is false.

Step 3: Conclusion

Based on the analysis above, Statement-1 is true as the variance of the multiplied set is \(4\sigma^2\). Statement-2 is false since the arithmetic mean of the multiplied set should be \(2\bar{x}\), not \(4\bar{x}\). Therefore, the correct choice is (D): Statement-1 is true, statement-2 is false.

Key Concepts

Arithmetic Mean

The arithmetic mean, often referred to as the "average," is a fundamental concept in statistics. It is calculated by summing up all the observations and then dividing the total by the number of observations. Mathematically, it is expressed as follows:
\[\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\]
This formula signifies that each observation equally contributes to the final average. It's a simple yet powerful way to summarize a large set of data with a single number. For instance, if you have observations like 3, 7, and 10, your arithmetic mean will be:
\[\bar{x} = \frac{3 + 7 + 10}{3} = 6.67\]
However, when you multiply each observation by a constant factor, like 2, the arithmetic mean also gets multiplied by that same factor. This is because:

• For a new dataset of 2 times each original observation, the new mean becomes \(2\bar{x}\).
• This means multiplying the entire set by 2 scales the mean by the same factor (not by the square, as might be mistakenly assumed in statement analysis).

Mathematical Statements

Mathematical statements provide significant insights into how changes in data affect statistical measures like mean and variance. When manipulating such statements, understanding the underlying mathematical operations is crucial. They achieve various purposes, such as:

• Conveying relationships between variables and constants.
• Providing instructions for data manipulation.
• Describing the expected outcomes of mathematical operations.

In the context of the exercise, statements regarding mean and variance demonstrate this. Statement-1 accurately describes how variance behaves when each data element is multiplied by 2, as variance depends on the square of deviations. Meanwhile, Statement-2 incorrectly suggests multiplying by 2 affects the arithmetic mean proportionally by 4, which is inaccurate. Clear comprehension of mathematical statements helps prevent such misconceptions.

Variance Calculation

Variance is a critical statistical measure that shows how much data points in a set differ from the mean. It helps us understand the spread or dispersion of a dataset. The formula for variance is:
\[\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}\]
This equation calculates the average of the squared differences between each data point and the mean. Squaring the differences ensures that variance accounts for all deviations, whether they are above or below the mean.
When each observation in the dataset is multiplied by a constant (like 2 in our exercise), the variance changes.

• Each deviation becomes 2 times larger, impacting the squared differences by a factor of 4: \((2(x_i - \bar{x}))^2 = 4(x_i - \bar{x})^2\).
• This results in the variance of the transformed dataset being four times the original variance: \(4\sigma^2\).

Understanding how various transformations affect variance is vital for correctly interpreting statistical results and ensuring accurate data analysis.