Problem 35 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If both the mean and the standard deviation of 50 observations \(x_{1}, x_{2}, \ldots ., x_{50}\) are equal to 16, then the mean of \(\left(x_{1}-4\right)^{2},\left(x_{2}-4\right)^{2}, \ldots,\left(x_{50}-4\right)^{2}\) is: \(\quad\) [April 10, \(\mathbf{2 0 1 9}\) (II)] (a) 400 (b) 380 (c) 525 (d) 480

Step-by-Step Solution

Answer

The mean of the transformed observations is 400, matching option (a).

Step 1: Calculate the New Observations

The given observations are \(x_1, x_2, \ldots, x_{50}\). We need to find the mean of the transformed observations \((x_1-4)^2, (x_2-4)^2, \ldots, (x_{50}-4)^2\). First, note that each new observation is derived from the original observation by shifting it by -4 and squaring the result.

Step 2: Relate Original Mean and Standard Deviation

We're given that the mean \(\mu\) and the standard deviation \(\sigma\) of the original set are both 16. Recall that standard deviation is the square root of variance. Therefore, the variance \(\sigma^2\) is also equal to 256, since \(16^2 = 256\).

Step 3: Calculate Mean of New Observations

Considering the shift, the mean of the new observations can be calculated using the formula for expectation: \[E((X-4)^2) = Var(X) + (E(X) - 4)^2\]Substitute \(Var(X) = 256\) and \(E(X) = 16\):\[= 256 + (16-4)^2 = 256 + 144 = 400\]

Step 4: Choose the Correct Answer

The calculated mean of the transformed observations is 400. This matches option (a).

Key Concepts

Variance

Variance is a measure of how much the values in a data set differ from the mean, or average, of the data set. It is calculated by taking the average of the squared differences between each data point and the mean.

For example, if you have a set of observations, the variance tells you how spread out the values are. It provides a sense of the variability within the set. In mathematical terms, if the mean of these values is denoted by \( \mu \) and each data point by \( x_i \), the variance \( \sigma^2 \) is calculated as:
\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 \]
Where \( n \) is the number of observations.

• High variance indicates that the data points are very spread out from the mean and from each other.
• Low variance suggests that the data points tend to be very close to the mean.

The variance is crucial for understanding the spread of a data set, and it is directly related to the standard deviation, which is the square root of the variance.

Transformed Observations

Transformed observations refer to data that have been altered using a mathematical function or operation. This transformation is often done to simplify the analysis or to meet the assumptions of specific statistical tests.

In the exercise provided, each original observation \( x_i \) has been transformed to \((x_i - 4)^2\). This is a two-step transformation:

• First, subtract 4 from each observation (shifting each value).
• Then square the result (to emphasize differences from 4).

This particular transformation affects the mean and variance of the data set. The transformation shifts and scales the data, which can alter its statistical properties. Understanding how transformations impact the data helps in forming correct conclusions from the analysis.

Transformations like these are common in statistics to stabilize variance, normalize data, or simply for mean deviation analysis.

Mathematical Expectation

Mathematical expectation, also known as the expected value, represents the average outcome of a random variable based on its probability distribution. It is a fundamental concept in probability and statistics, giving us the long-run average value of repetitions of the experiment it represents.

When computing the mean of a transformed observation, mathematical expectation helps us simplify the process. It uses properties of expected values to determine the mean after a transformation.
For any random variable \( X \), the expectation \( E(X) \) is given by:
\[ E(X) = \sum x_i P(x_i) \] Where \( x_i \) represents possible values of \( X \) and \( P(x_i) \) is the probability of \( X = x_i \).

• In the given problem, the expectation formula is leveraged to compute the mean of transformed observations \((X-4)^2\).
• By understanding mathematical expectation, we find that \( E((X-4)^2) \) incorporates both variance and the square of the shift \((16 - 4)^2\).

By utilizing mathematical expectation, you streamline calculations and gain insights into how data behaves under certain transformations.