Test: Measures of Dispersion - UGC NET MCQ

- Assertion Analysis: The assertion is true; the range indeed serves as a measure of variability within a dataset, indicating how spread out the values are.

- Reason Analysis: The reason is also true; it accurately describes how the range is computed by finding the difference between the highest and lowest values.

- Explanation Relationship: The reason correctly explains why the assertion is true, as it highlights the method of calculating the range, which directly relates to understanding variability.

The mean is a measure of central tendency, not dispersion. It provides the average value of a dataset but does not indicate how spread out the data points are. In contrast, measures like range, variance, and standard deviation specifically quantify how much the data varies around the mean. For example, a dataset with a high standard deviation indicates that the values are spread out over a wider range, while a low standard deviation suggests that the values are clustered closely around the mean. Understanding these measures is crucial for interpreting the variability in data effectively.

- Statement 1 is correct. The range is indeed calculated by subtracting the minimum value from the maximum value in a dataset, providing a simple measure of the spread.

- Statement 2 is incorrect. The standard deviation is the square root of the variance. Therefore, while the standard deviation can be greater than, less than, or equal to the variance depending on the dataset, it is not true that it is always greater than or equal to the variance.

Overall, only Statement 1 is correct, making the correct answer Option A.

The standard deviation is an absolute measure of dispersion, which quantifies the amount of variation or spread in a set of data values. Unlike relative measures, which are expressed as a percentage of another value, absolute measures such as the standard deviation are presented in the same units as the data itself. For instance, if the data is in meters, the standard deviation will also be in meters. Understanding dispersion is crucial as it provides insights into the variability of data, which can influence decision-making and statistical analysis. An interesting fact is that the standard deviation is widely used in finance to measure market volatility; a higher standard deviation indicates a higher risk due to greater variability in returns.

The Coefficient of Variation is defined as the ratio of the standard deviation to the mean of a data set, typically expressed as a percentage. This measure allows for the comparison of the degree of variation between datasets with different units or scales. A higher coefficient indicates greater relative variability. An interesting fact is that the Coefficient of Variation is particularly useful in fields such as finance, where it helps assess the risk of an investment relative to its expected return.

• The assertion is correct because mean deviation indeed measures the variability of data points in relation to the mean.
• The reason is also correct as mean deviation is calculated by taking the absolute differences of each data point from the mean, confirming the assertion.
• Since the reason accurately explains how the mean deviation is derived, it serves as the correct explanation for the assertion.

- The Assertion (A) is correct because the range is indeed calculated by finding the difference between the largest and smallest values in the dataset.

- The Reason (R) is also correct, as it specifically describes how to calculate the range of a frequency distribution.

- Since the Reason provides a valid explanation for the Assertion, Option A is the correct choice.

- Assertion Analysis: The assertion is true; in open-ended frequency distributions, either the lower limit of the lowest class or the upper limit of the highest class is not specified, making it impossible to calculate a definitive range.

- Reason Analysis: The reason is also true; it correctly identifies the nature of open-ended distributions as lacking defined limits.

- Explanation Relationship: The reason validly explains the assertion as it directly addresses the reason why the range cannot be calculated in such distributions, confirming the assertion's validity.

- Statement 1 is correct because the Coefficient of Dispersion (C.D.) helps in comparing datasets that have different averages or are measured in different units, making it a versatile tool in data analysis.

- Statement 2 is incorrect. Measures of Central Tendency (such as mean, median, mode) provide a central figure for the data, while Measures of Dispersion (like range, variance, standard deviation) describe the spread or variability of the data. Therefore, they serve different purposes in data analysis.

Thus, the correct answer is Option A: 1 Only.

A higher range in a data set indicates greater variability because it measures the difference between the largest and smallest values. This means that when the range is large, the data points are spread out over a wider interval, suggesting diverse values. However, it's important to note that the range only considers the extremes and may not accurately represent the overall distribution of the data. For example, two data sets can have the same range but differ significantly in their other characteristics, such as how the values are distributed within that range. An interesting fact is that while the range provides a quick measure of variability, other statistical measures like the interquartile range or standard deviation give a more comprehensive view of data dispersion.