Problem 9 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the standard deviation of \(n\) observations \(x_{1}, x_{2}, \ldots\), \(x_{n}\) is 4 and another set of \(n\) observations \(y_{1}, y_{2}, \ldots, y_{n}\) is 3 . The standard deviation of \(n\) observations \(x_{1}-y_{1}\), \(x_{2}-y_{2}, \ldots, x_{n}-y_{n}\) is (A) 1 (B) \(\frac{2}{\sqrt{3}}\) (C) 5 (D) Data insufficient

Step-by-Step Solution

Answer

The standard deviation of the differences is 5.

Step 1: Understanding the Problem

We have two sets of observations, each of size \(n\). The standard deviation of the first set \(x_1, x_2, \ldots, x_n\) is 4, and the standard deviation of the second set \(y_1, y_2, \ldots, y_n\) is 3. We are required to find the standard deviation of the differences \(x_1-y_1, x_2-y_2, \ldots, x_n-y_n\).

Step 2: Using Variance Addition Principle

The variance of a set of data is the square of the standard deviation. Therefore, the variance of the \(x\)-set is \(4^2=16\) and the variance of the \(y\)-set is \(3^2=9\). The variance of the difference of normally distributed variables from two independent sets follows \(\text{Var}(x-y) = \text{Var}(x) + \text{Var}(y)\).

Step 3: Calculate the Combined Variance

Applying the formula, \(\text{Var}(x-y) = \text{Var}(x) + \text{Var}(y) = 16 + 9 = 25\). This is the variance of the differences of the observations.

Step 4: Find the Standard Deviation of the Differences

The standard deviation is the square root of the variance. Hence, the standard deviation of the differences \(x_1-y_1, x_2-y_2, \ldots, x_n-y_n\) is \(\sqrt{25} = 5\).

Step 5: Select the Correct Answer

The standard deviation of the differences is 5. Therefore, the correct option is (C) 5.

Key Concepts

Variance

Variance is a statistical measure that describes the spread or dispersion of a set of data points around their mean. It's essentially the average of the squared deviations from the mean, thus giving insight into how much the data varies.To compute variance, follow these steps:

• Find the mean (average) of the data set.
• Subtract the mean from each data point to find the deviation of each point from the average.
• Square each deviation to eliminate negative differences.
• Find the average of these squared deviations. This final value is the variance.

Mathematically, the variance of a data set \(x_1, x_2, \ldots, x_n\) is calculated as \( \sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2 \), where \( \mu \) is the mean of the data set.
This measure is crucial because it offers a quantifiable insight into the variability within a data set, forming the foundation for many statistical analyses.

Independent Sets

In statistics, independence between sets implies that the occurrence or variation of one set does not affect the other. This is a foundational assumption in many data analysis techniques.When two sets of data are independent, the variations or patterns in one do not influence or explain the variations in the other. This characteristic is often assumed in probability and statistics to simplify and enable cleaner analysis.For example, when sets \( X \) and \( Y \) are independent, their joint distribution can be expressed as the product of their individual distributions. This notion is crucial, especially when investigating combinations or differences between two sets, as in variance calculations.
Independence allows us to apply certain mathematical principles, such as the variance addition principle, without further adjustments.

Variance Addition Principle

The variance addition principle is a powerful concept used to determine the combined variability of two independent data sets.The principle states that if \( X \) and \( Y \) are two independent random variables, then the variance of their sum or difference is the sum of their variances. Mathematically, it's given by:\[\text{Var}(X \pm Y) = \text{Var}(X) + \text{Var}(Y)\]This becomes especially useful in situations involving the differences or sums of normally distributed variables, where you want to calculate the overall variability.In practical terms, if you have two independent sets of observations, you can calculate the variance of their difference without needing to account for any potential correlations or interactions between the sets. This simplifies complex data analyses involving multiple variables, facilitating clearer understanding of their combined effects.

Normally Distributed Variables

Normally distributed variables adhere to a probability distribution known for its symmetric bell-shaped curve, characterized by its mean and standard deviation. This distribution is ubiquitous in statistics due to the Central Limit Theorem, which states that the means of large samples of data taken from any distribution will approximate a normal distribution. Normally distributed variables showcase properties such as:

• The majority of data points cluster around the mean.
• Approximately 68% of data falls within one standard deviation of the mean.
• About 95% lies within two standard deviations, and nearly all within three.

Understanding that variables are normally distributed allows us to apply statistical concepts like the variance addition principle effectively. This is because the mathematical properties of normal distributions facilitate more straightforward computations and predictions under such schemes.
For instance, the standard deviation of differences between normally distributed independent variables can be confidently computed using these principles, offering a clear analytical advantage.