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

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, option (C).

Step 1: Understand the Problem

We have two sets of observations, each of length \( n \). The standard deviation of the first set is 4, and the standard deviation of the second set is 3. We need to find the standard deviation of the differences of corresponding observations from these two sets.

Step 2: Use the Standard Deviation Formula for Differences

To calculate the standard deviation of the differences \( x_i - y_i \), we can use the formula for the standard deviation of the sum or difference of two independent random variables: \( \sigma_{(x-y)} = \sqrt{ \sigma_x^2 + \sigma_y^2} \). Here, \( \sigma_x = 4 \) and \( \sigma_y = 3 \).

Step 3: Calculate the Combined Variance

First, calculate the variances: \( \sigma_x^2 = 4^2 = 16 \) and \( \sigma_y^2 = 3^2 = 9 \). The variance of the differences \( x_i - y_i \) is the sum of these two variances: \( 16 + 9 = 25 \).

Step 4: Determine the Standard Deviation of Differences

Now, take the square root of the combined variance to find the standard deviation: \( \sigma_{(x-y)} = \sqrt{25} = 5 \).

Step 5: Choose the Correct Option

Compare the calculated standard deviation with the options given: 1, \( \frac{2}{\sqrt{3}} \), 5, and 'Data insufficient'. Since we found the standard deviation is 5, the correct option is (C) 5.

Key Concepts

Variance

Variance is an essential concept in statistics. It measures how spread out a set of data points are around their mean.
It is the average of the squared differences from the mean. In mathematical terms, for a data set \( x_1, x_2, \ldots, x_n \), the variance \( s^2 \) is calculated as follows:\[ s^2 = \frac{1}{n}\sum\limits_{i=1}^n (x_i - \bar{x})^2 \]where \( \bar{x} \) is the mean of the data set.

• The larger the variance, the more spread out the data points are.
• A smaller variance indicates that the data points are closer to the mean.

When dealing with standard deviation, it's important to recognize that it's the square root of the variance,
meaning that the standard deviation provides a direct insight into the spread of data in its original units.Understanding variance and how it relates to standard deviation helps us grasp the concept of the variability of our data.

Independent Random Variables

In probability and statistics, random variables are often classified as dependent or independent.
Two random variables are considered independent if the outcome of one does not influence the outcome of the other.

• If you have two independent variables, their joint probability is the product of their individual probabilities.
• This independence is crucial in calculations that involve combining different data sets.

When working with independent random variables, one important application is the calculation of the variance and standard deviation of their sums or differences.
For independent random variables \( X \) and \( Y \), the variance of their sum or difference is determined by the formula:\[ \text{var}(X \pm Y) = \text{var}(X) + \text{var}(Y) \]This means the variances simply add up,
which significantly simplifies the analysis of such data.

Difference of Observations

The concept of the difference of observations is critical when comparing two sets of data.
It involves subtracting corresponding elements in each set, resulting in a new set of differences.

• For example, if you have data sets \( X = [x_1, x_2, \ldots] \) and \( Y = [y_1, y_2, \ldots] \), their differences are \( X - Y = [x_1 - y_1, x_2 - y_2, \ldots] \).
• These differences help us understand variability between paired data points.

In terms of standard deviation, the variance of these differences assumes the data sets are composed of independent random variables.
Calculating the variance of differences uses the formula \( \text{var}(X - Y) = \text{var}(X) + \text{var}(Y) \),
which helps determine how spread out the differences are. The square root of this variance gives the standard deviation of the differences.
This provides insight into the consistency or variability between the two sets of observations.