Problem 10 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(r\) be the range and \(S^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\) be the S.D. of a set of observations \(x_{1}, x_{2}, \ldots, x_{n}\), then (A) \(S \leq r \sqrt{\frac{n}{n-1}}\) (B) \(S=r \sqrt{\frac{n}{n-1}}\) (C) \(S \geq r \sqrt{\frac{n}{n-1}}\) (D) None of these

Step-by-Step Solution

Answer

Option (A) \( S \leq r \sqrt{\frac{n}{n-1}} \) is correct.

Step 1: Understand the Problem

We need to determine the correct relationship between the range \( r \) and the standard deviation \( S \) of a dataset. The options provided suggest comparing \( S \) with \( r \sqrt{\frac{n}{n-1}} \).

Step 2: Recall Definitions

The range \( r \) is defined as the difference between the maximum and minimum values in the dataset. The standard deviation \( S \) is given by \( S = \sqrt{S^2} \), where \( S^2 = \frac{1}{n-1} \sum_{i=1}^{n}(x_i - \bar{x})^2 \).

Step 3: Analyze Inequality

The inequality related to the standard deviation and range can be best compared using the empirical rule, which tells us properties about data spread. While the exact equality or inequality might not be derivable from first principles without specific data, we know that \( S \) should always be related to \( r \) but is typically less than or equal to some factor involving \( r \).

Step 4: Examine the Options

Given the statistical properties and normal behavior of data, the hints suggest an inequality relation rather than an exact equality. Therefore, option (A) \( S \leq r \sqrt{\frac{n}{n-1}} \) is likely correct because it proposes the standard deviation, which represents spread, being less or equal to a scaled version of the range.

Key Concepts

Range in Statistics

In the field of statistics, the range is a simple measure of variability or spread in a dataset. It is calculated by finding the difference between the highest and lowest values in a collection of numbers. For example, if we have the numbers 5, 10, and 20, the range will be 20 - 5 = 15.

The range helps us quickly understand the span of values in our dataset. However, it's important to note that while the range is easy to calculate, it only provides a limited view of variability.

• The range can be heavily influenced by outliers or extremes. A single very large or very small value can significantly skew the range.
• It does not give any information about how the values are distributed between the minimum and maximum values.

Despite its limitations, the range can serve as a useful starting point when comparing variability across different datasets.

Inequalities in Statistics

In statistics, inequalities often describe relationships between different measures of data spread, such as range and standard deviation. Inequality theorems provide insight into how these measures relate to each other without requiring exact computation.

One common inequality in statistics is between the standard deviation and range of a dataset. While specific data characteristics or distributions can determine exact values, we typically know that overall variability, shown by standard deviation, must adhere to mathematical relationships.

• A standard deviation indicates how much individual data points differ from the mean. It considers every data point, offering a comprehensive view of variability.
• Inequalities help predict reasonable bounds for the standard deviation based on other measures, like range. For instance, it's often expected that the standard deviation is less than or equal to a value involving the range, reflecting natural data properties.

By understanding these inequalities, we can make more informed guesses about data spreads and the relationships between different measures of variability.

Empirical Rule

The empirical rule is a straightforward statistical guideline that provides insight into the distribution of data, particularly in normal distributions. It states that in a normal distribution:

• Approximately 68% of the data points fall within one standard deviation of the mean.
• About 95% of the data will lie within two standard deviations.
• Nearly 99.7% of data points are found within three standard deviations.

These percentages allow us to predict the spread of data even without specific values, assuming the data distribution is normal.

This rule is incredibly helpful for understanding the probability of occurrence of certain data points and for identifying outliers. While it applies specifically to normal distributions, many real-world phenomena approximate these conditions, making the rule a versatile tool in statistics. By leveraging the empirical rule, you can gauge how well your data might be behaving normally and thus make predictions or judgments based on this assumption.