Problem 57 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

In a set of 2 n observations, half of them are equal to 'a' and the remaining half are equal to '-a'. If the standard deviation of all the observations is 2 ; then the value of \(|a|\) is : [Online April 25, 2013] (a) 2 (b) \(\sqrt{2}\) (c) 4 (d) \(2 \sqrt{2}\)

Step-by-Step Solution

Answer

|a| = 2

Step 1: Understand the Structure of the Data

We are given a set of \(2n\) observations. Half of these observations are equal to 'a', and the other half are equal to '-a'. This implies we have \(n\) observations of 'a' and \(n\) observations of '-a'.

Step 2: Calculate the Mean of the Data

The mean (average) of all observations can be calculated using the formula: \[ \mu = \frac{n \cdot a + n \cdot (-a)}{2n} = \frac{na - na}{2n} = \frac{0}{2n} = 0. \] Hence, the mean of the observations is 0.

Step 3: Use the Standard Deviation Formula

The standard deviation formula is given by: \[ \sigma = \sqrt{\frac{1}{2n} \sum_{i=1}^{2n} (x_i - \mu)^2}. \] Since the mean \(\mu\) is 0, the formula simplifies to: \[ \sigma = \sqrt{\frac{1}{2n} (n(a-0)^2 + n(-a-0)^2)} = \sqrt{\frac{1}{2n} (na^2 + na^2)} = \sqrt{\frac{2na^2}{2n}} = \sqrt{a^2}. \] Since we know the standard deviation \(\sigma\) is given as 2, we set \(\sqrt{a^2} = 2\).

Step 4: Solve for |a|

From the equation \(\sqrt{a^2} = 2\), we know that \(|a| = 2\). Because the square root function produces only non-negative results, we identify \(|a|\) with positive values only.

Key Concepts

Mean

The mean is a fundamental concept in statistics that represents the average of a set of numbers. It provides a single value that summarizes the entire dataset. To calculate the mean, you add up all the numbers in the dataset and then divide by the number of observations. In our example, the set consists of two equal groups of numbers: positive 'a' and negative '-a'. This structured balance means the values effectively cancel each other out when added. Hence, the mean of this specific set, calculated as \( \mu = \frac{n \cdot a + n \cdot (-a)}{2n} = 0 \), is zero. This reflects the perfect symmetry in the dataset where each positive value is countered by a corresponding negative value.

Set of Observations

A set of observations is simply a collection of data points that are subject to analysis. It is crucial to understand the composition of this set to conduct proper statistical calculations. For our exercise, the set of observations consists of \( 2n \) data points. These are evenly divided into two groups, where half of the observations are the value 'a' and the other half are '-a'. This creates a symmetric dataset where each value 'a' is mirrored by a corresponding '-a'.

• This symmetry ensures that certain calculations, such as the mean, result in simple outcomes (like zero in this case).
• Analyzing such a set highlights the importance of understanding the dataset's structure before diving into further calculations.

Recognizing patterns in distribution and symmetry can simplify complex statistical problems.

Mathematical Calculations

Mathematical calculations are the backbone of statistical analysis, allowing us to derive meaningful insights from raw data. In our problem, the essential calculations include finding the mean and using the standard deviation formula:

• The mean was derived simply due to the symmetrical nature of the dataset, which led to an average of zero.
• The standard deviation is calculated using the formula \( \sigma = \sqrt{\frac{1}{2n} \sum_{i=1}^{2n} (x_i - \mu)^2} \approx \sqrt{a^2} \).

Given the problem states that the standard deviation equals 2, this directly allows us to solve \( \sqrt{a^2} = 2 \), leading to \(|a| = 2\). Understanding these calculations is crucial for interpreting the spread and variability of data within your set of observations. This process of calculation sheds light on not only how dispersed the data points are, but also emphasizes the importance of the assumptions made (like data symmetry) in simplifying the computations.