Problem 13 - A Complete Resource Book in Mathematics for JEE Main

Question

The S.D. of a variate \(x\) is \(\sigma\). The S.D. of the variate \(\frac{a x+b}{c}\) where \(a, b, c\) are constants, is (A) \(\left(\frac{a}{c}\right) \sigma\) (B) \(\left|\frac{a}{c}\right| \sigma_{\mid}\) (C) \(\left(\frac{a^{2}}{c^{2}}\right) \sigma\) (D) None of these

Step-by-Step Solution

Answer

(B) \(\left|\frac{a}{c}\right| \sigma\)

Step 1: Understand Standard Deviation Transformation

The standard deviation (S.D.) of a variate \(x\) is given by \(\sigma\). When a linear transformation is applied to a variate in the form \(\frac{ax+b}{c}\), the S.D. of the transformed variate is affected by the scaling factor \(\frac{a}{c}\). The addition of a constant does not affect the S.D., only scaling does.

Step 2: Calculate the Scaling Factor

In the expression \(\frac{ax+b}{c}\), the scaling factor is \(\frac{a}{c}\). Therefore, the new S.D. after transformation will be the product of the original S.D. \(\sigma\), and the absolute value of the scaling factor, \(\left| \frac{a}{c} \right| \). Mathematically, it becomes \(\left| \frac{a}{c} \right| \sigma \).

Step 3: Verify with Given Options

Review the provided options: (A) \(\left(\frac{a}{c}\right) \sigma\) (B) \(\left|\frac{a}{c}\right| \sigma\) (C) \(\left(\frac{a^{2}}{c^{2}}\right) \sigma\) (D) None of theseFrom Step 2, we found that the correct transformation of S.D. is \(\left| \frac{a}{c} \right| \sigma\). Therefore, option (B) is correct.

Key Concepts

Linear Transformation

When analyzing data, a linear transformation is a common method applied to variates to change their scale or position. This transformation typically takes the form \( \frac{ax + b}{c} \), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the original variate. The goal of a linear transformation is to adjust the data for better analysis while maintaining its inherent properties.
Linear transformations allow you to either stretch or compress the data depending on the values of the constants. Particularly important is the fact that while changes in \(b\) will shift the data up or down, they do not affect the spread or variability of the data. The real impact on spread and variability comes from the scaling factors \(a\) and \(c\).
This leads us to our next topic, which is the scaling factor, a crucial element in standard deviation transformations when a linear transformation is applied.

Scaling Factor

The scaling factor in a linear transformation is a critical element that affects the spread of the data. In the expression \( \frac{ax+b}{c} \), the scaling factor is \( \frac{a}{c} \). It determines how much the data is stretched or compressed when a linear transformation is applied.
Here's why the scaling factor matters:

• Effect on Standard Deviation: The standard deviation (S.D.), which measures the spread of data, is directly multiplied by the absolute value of the scaling factor \( \left| \frac{a}{c} \right| \). Thus, if the scaling factor is greater than 1, the S.D. increases, and if it is less than 1, the S.D. decreases.
• Interpretation: The absolute value ensures that we are focusing on the magnitude, as the standard deviation is inherently non-negative. This reflects the actual change in variability or spread, independent of direction.

By understanding the scaling factor, we can accurately predict how transformations affect the variability of data, which is crucial in many statistical analyses.

Absolute Value in Transformations

In transformations, particularly in dealing with standard deviation, the absolute value is a crucial concept. When calculating the transformed standard deviation using a scaling factor, the expression becomes \( \left| \frac{a}{c} \right| \sigma \). Here, taking the absolute value ensures that the transformed standard deviation remains a non-negative number, as standard deviation inherently measures variation as a distance.

• Purpose: The absolute value removes any negative sign, focusing purely on the magnitude of the scaling factor. This is essential because the "size" of scaling directly impacts spread, regardless of direction.
• Consistency: By using the absolute value, analysts ensure consistency in interpretation across various transformations. It guarantees that standard deviations are always understood as positive values, reflecting true data variability.

In conclusion, understanding the role of absolute values in transformations is crucial to maintaining the integrity of statistical measures such as standard deviation. This understanding helps ensure that data analysis remains accurate and meaningful in various contexts.