Problem 25 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

For the frequency distribution :where \(00\), the standard deviation cannot be: \(\quad\) [Sep. 03, 2020 (I)] (a) 4 (b) 1 (c) 6 (d) 2

Step-by-Step Solution

Answer

The standard deviation cannot be 6.

Step 1: Understand the properties of standard deviation

The standard deviation quantifies the amount of variation or dispersion in a set of values. For all values of a distribution to be greater than 0 and less than 10, the standard deviation provides a measure of how spread out these values are around the mean.

Step 2: Analyze the given range

In this problem, the values are between 0 and 10. This means that the greatest possible range (difference between the maximum and minimum) is 10. This range imposes a maximum possible spread for the data, and consequently a maximum possible standard deviation.

Step 3: Consider possible standard deviation values

The maximum standard deviation occurs when all data points are at the extremes (for example, half of them at 0 and half at 10), which would theoretically provide the highest spread. However, this is not possible considering there are 15 intervals, not two extreme values fully employed.

Step 4: Calculate the maximal standard deviation

If we consider an evenly distributed set of values across the intervals from 0 to 10, the spread will be maximized theoretically nearer to half of this range. Extreme concentration of values at 0 and 10 would result in much less practical standard deviation than if evenly distributed in practice. Nonetheless, an SD greater than half the total absolute range (5) is not feasible.

Step 5: Determine the impossibility

If we look at options given (1, 2, 4, 6), it becomes quickly apparent that a standard deviation of 6 exceeds the feasible spread within a 0 to 10 range. This is because 6 suggests too much dispersion relative to the actual values' spread.

Key Concepts

Frequency Distribution

A frequency distribution is a way of showing how often each value in a set of data occurs. It's like creating a menu for data values and their frequencies, helping us see which values are common and which are rare. This is particularly useful when dealing with a large number of data points. Imagine you roll a die many times, and you want to know how many times each number appears. By organizing this data into a frequency distribution, you’ll be able to quickly see the frequency of each dice outcome.
Simply put:

• Frequency distribution tables list unique data values alongside their frequencies.
• They help in identifying data trends and patterns by illustrating how the data is spread across different values.

By visualizing how data is distributed, frequency distributions make it easier to calculate other statistical measures, such as the mean or standard deviation.

Range

The range in statistics is the simplest measure of dispersion. It's calculated by subtracting the smallest value from the largest value in a data set. This gives a sense of the spread or extent of the data. For example, if the smallest number in your data set is 2 and the largest is 10, the range is 8.
Considerations when using the range includes:

• It only uses the two extreme values and ignores the rest of the data, making it sensitive to outliers.
• A larger range means more spread out data, while a smaller range indicates data that is closer in value.

Though easy to compute, the range provides only a rough idea of variability, and is better used alongside more robust measures like the standard deviation.

Variation

Variation refers to the overall spread of a data set. It's crucial in statistics as it reflects how much the data points differ from the mean. A high variation means the data points are very spread out, while a low variation indicates they are tightly clustered around the mean.
Variation is often measured using variance or standard deviation. Here's how they help:

• Variance: It averages the squared differences from the mean, offering a detailed view of data spread.
• Standard Deviation: This is the square root of variance, providing an average distance from the mean, in the same units as the data.

Understanding variation aids in determining reliability and consistency within data sets, essential for statistical analysis and decision making.

Dispersion

Dispersion in statistics refers to the way data points are distributed or spread out. It provides insights into the variability within a data set. The more dispersed the values, the less likely they are to cluster around the mean.
Dispersion is crucial for understanding data as it helps identify:

• How reliable an average is, by showing how much variation exists.
• The presence of outliers, which can significantly impact data interpretation.

Tools like the range, variance, and standard deviation are often used to measure dispersion. By capturing how data is spread, dispersion enables researchers to better understand the distribution and predict behavior in similar datasets.