Problem 6 - A Complete Resource Book in Mathematics for JEE Main

Question

If Q.D. is 16, the most likely value of S.D. will be (A) 24 (B) 42 (C) 10 (D) None of these

Step-by-Step Solution

Answer

The most likely value of S.D. is 24.

Step 1: Understanding Q.D. and S.D.

Quartile Deviation (Q.D.) is half the distance between the first quartile \(Q_1\) and the third quartile \(Q_3\). The Standard Deviation (S.D.) is a measure of the amount of variation or dispersion of a set of values. These are two different descriptive statistics used in data analysis.

Step 2: Relating Q.D. to S.D.

For a normal distribution, there exists an empirical relationship between Q.D. and S.D. approximately given by the formula: \( S.D. = 1.35 \times Q.D. \). This relationship helps us estimate the standard deviation when we know the quartile deviation.

Step 3: Substituting Q.D. into the Formula

Given that \( Q.D. = 16 \), substitute this value into the empirical relationship: \[ S.D. = 1.35 \times 16 \]

Step 4: Calculating S.D.

Perform the multiplication to find the standard deviation: \[ S.D. = 1.35 \times 16 = 21.6 \]

Step 5: Comparing with Options

Compare the calculated S.D. value of 21.6 with the provided options: (A) 24, (B) 42, (C) 10, (D) None of these. Since 21.6 is closest to option (A) 24, we consider this as the most likely value of S.D.

Key Concepts

Standard Deviation

Standard Deviation (SD) is a statistical measurement that tells us how much values in a data set deviate from the average (mean). It is a way to understand the spread of the data. When the standard deviation is low, it indicates that the values in a data set are closely clustered around the mean. Conversely, a high standard deviation indicates that the values are spread out over a wide range.

In practical terms, SD quantifies the uncertainty or risk associated with a particular dataset. It is widely used in finance, science, and many other fields where decision-making depends on data analysis.

• The formula to calculate standard deviation for a dataset: \( SD = \sqrt{\frac{\sum{(x_i - \overline{x})^2}}{n}} \), where \( x_i \) are data points, \( \overline{x} \) is the mean, and \( n \) is the number of data points.
• In the given problem, we use an alternative empirical approach which relates standard deviation to quartile deviation.

Empirical Relationship

The empirical relationship is a proven formula or rule-of-thumb that helps predict the behavior of one variable based on another, specifically when direct calculation is complex. In statistics, empirical relationships allow us to use simple formulas to estimate values of complex calculations.

In the context of Quartile Deviation (Q.D.) and Standard Deviation (S.D.), there is an empirical relationship that is particularly useful for normal distributions.

• This relationship is given by the formula: \( S.D. = 1.35 \times Q.D. \).
• It provides a quick estimate of the standard deviation when we know the quartile deviation, which can be easier to calculate in some situations.

This relationship highlights the link between central tendency measures and variability, thus saving time and effort when dealing with a normal distribution.

Normal Distribution

Normal distribution, also known as the Gaussian distribution, is a essential concept in statistics. It describes how the values of a variable are distributed. It is characterized by a bell-shaped curve, where most of the data points cluster around the mean, and the probabilities for values spread symmetrically around the mean.

Features of a normal distribution include:

• Symmetry: The shape of the distribution is symmetric around the mean.
• Bell-shaped Curve: Most occurrences take place around the mean, with fewer occurrences towards the edges.
• Empirical Rule: Approximately 68% of data falls within one standard deviation from the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations.

Understanding normal distribution helps statisticians and analysts make predictions about data, check for normality, and apply the appropriate statistical tests and empirical relationships like the one between Q.D. and S.D.