Problem 54 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\overline{\mathrm{x}}, \mathrm{M}\) and \(\sigma^{2}\) be respectively the mean, mode and variance of \(n\) observations \(x_{1}, x_{2}, \ldots, x_{n}\) and \(d_{i}=-x_{i}-a\), \(\mathrm{i}=1,2, \ldots, \mathrm{n}\), where \(\mathrm{a}\) is any number. Statement I: Variance of \(\mathrm{d}_{1}, \mathrm{~d}_{2}, \ldots \mathrm{d}_{\mathrm{n}}\) is \(\sigma^{2}\). Statement II: Mean and mode of \(\mathrm{d}_{1}, \mathrm{~d}_{2}, \ldots . \mathrm{d}_{\mathrm{n}}\) are \(-\overline{\mathrm{x}}-\mathrm{a}\) and \(-\mathrm{M}-\mathrm{a}\), respectively. [Online April 19, 2014] (a) Statement I and Statement II are both false (b) Statement I and Statement II are both true (c) Statement Iis true and Statement II is false (d) Statement I is false and Statement II is true

Step-by-Step Solution

Answer

Both Statement I and Statement II are true.

Step 1: Calculate the Mean of the Transformed Data

The mean of the original observations is \( \overline{x} \). The transformed observations are given by \( d_i = -x_i - a \). The mean of the transformed observations is \( \overline{d} = -\overline{x} - a \). This matches the statement given in Statement II about the mean.

Step 2: Determine the Mode of the Transformed Data

The mode of the original observations is \( M \). For the transformed data, the mode becomes \( -M - a \) since the transformation applied is \( -x_i - a \). Hence, the mode of the transformed data is \( -M - a \), matching Statement II.

Step 3: Evaluate the Variance of the Transformed Data

The variance \( \sigma^2 \) of the original observations is unaffected by adding or subtracting a constant. The transformation \( d_i = -x_i - a \) involves adding \(-a\), which does not change the variance. Therefore, the variance of the transformed data \( d_1, d_2, \ldots, d_n \) remains \( \sigma^2 \). This confirms Statement I.

Key Concepts

Mean

In statistics, the mean is a measure of central tendency that outlines the average value of a dataset. To find the mean, you sum up all the observations and divide by the number of observations. It’s a straightforward and widely-used statistical metric that gives you a sense of the overall level of the data. For example, if you have observations \( x_1, x_2, \ldots, x_n \), the mean \( \overline{x} \) is calculated as:\[ \overline{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} \]

• The mean is useful for understanding the general magnitude of numbers in a dataset.
• It can be skewed by extremely high or low values, known as outliers.
• When transforming data by adding or subtracting a constant, the mean changes by that constant.

In the case of the transformed data equations \( d_i = -x_i - a \), the mean of the transformed observations is adjusted accordingly, resulting in \( -\overline{x} - a \). This showcases how arithmetic operations affect the mean.

Mode

The mode is another important measure of central tendency, specified as the value that appears most frequently in a dataset. Unlike the mean, the mode is not calculated through arithmetic. It simply reflects either the most common observation or the most repeated item in the data. This can provide insights into the most typical values occurring in your dataset.

• The mode can be particularly useful in categorical data or with numbers that repeat frequently.
• Datasets can be unimodal (one mode), bimodal (two modes) or multimodal (many modes).
• It remains unchanged when the entire dataset is shifted by adding or subtracting a constant.

When transforming the dataset with the equation \( d_i = -x_i - a \), the mode becomes \( -M - a \). This transformation does not affect the core concept of the mode, as it is simply shifted by the same constant applied to the dataset.

Variance

Variance is a statistical measure that captures the spread or dispersion of a dataset. It answers the question: how much do the numbers in the dataset differ from the mean? The formula for variance of a dataset \( x_1, x_2, \ldots, x_n \) is:\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \overline{x})^2 \]Variance gives us crucial insights into how far each observation is from the mean. More variance means more spread in the dataset.

• Variance is fundamental in understanding data variability.
• It is used to calculate standard deviation, another measure of dispersion.
• Adding or subtracting a constant does not affect variance.

In the transformation \( d_i = -x_i - a \), variance remains unchanged, staying as \( \sigma^2 \). This is because variance is only affected by differences in values, not by the addition of constants. Hence, despite transformations, the variance remains consistent, confirming the statement in the problem exercise.