Analyzing Statistical Measures: Dispersion and Central Tendency

The question asks us to evaluate the correctness of three statements related to statistical measures: Coefficient of Variation, Range, and Mean Deviation. Let's examine each statement carefully to determine its validity in the context of statistics.

Statement 1: Coefficient of variation depends on the unit of measurement of the variable.

The coefficient of variation (CV) is a standardized measure of dispersion of a probability distribution or frequency distribution. It is defined as the ratio of the standard deviation \( \sigma \) to the mean \( \mu \):

\( \text{CV} = \frac{\sigma}{\mu} \times 100\% \)

Let's consider the units of measurement. If a variable \( X \) is measured in units, its mean \( \mu \) will have the same units, and its standard deviation \( \sigma \) will also have the same units. When we calculate the ratio \( \frac{\sigma}{\mu} \), the units in the numerator and the denominator cancel each other out. Therefore, the coefficient of variation is a unitless measure or a dimensionless number.

For example, if we measure height in centimeters or inches, the mean and standard deviation will change values but will retain their respective units. However, the ratio \( \frac{\sigma}{\mu} \) will remain the same regardless of whether centimeters or inches are used, as long as the conversion is consistent. Thus, the coefficient of variation is independent of the unit of measurement.

Based on this analysis, statement 1 is incorrect.

Statement 2: Range is measure of dispersion.

Dispersion in statistics refers to the extent to which a distribution is stretched or squeezed. It describes the variability or spread of the data points in a dataset.

The Range is one of the simplest measures of dispersion. It is calculated as the difference between the maximum value and the minimum value in a dataset:

\( \text{Range} = \text{Maximum Value} - \text{Minimum Value} \)

A larger range indicates greater variability or spread in the data, while a smaller range indicates less variability. Because it quantifies the spread, the range is indeed a measure of dispersion.

Therefore, statement 2 is correct.

Statement 3: Mean deviation is least when measured about median.

The mean deviation is the average of the absolute differences between each data point and a central point (like the mean, median, or mode). The formula for mean deviation about a central value 'a' is:

\( \text{Mean Deviation about 'a'} = \frac{1}{n} \sum_{i=1}^{n} |x_i - a| \)

Here, \( x_i \) are the data points, 'a' is the central value, and 'n' is the number of data points.

A fundamental property in statistics is that the sum of the absolute deviations, \( \sum_{i=1}^{n} |x_i - a| \), is minimized when 'a' is the median of the dataset. Since the mean deviation is simply this sum divided by the number of observations (n), the mean deviation will also be minimized when calculated about the median.

Thus, the mean deviation is least when measured about the median.

Therefore, statement 3 is correct.

Summary of Statement Analysis

• Statement 1: Coefficient of variation depends on the unit of measurement of the variable - Incorrect
• Statement 2: Range is measure of dispersion - Correct
• Statement 3: Mean deviation is least when measured about median - Correct

Based on our analysis, statements 2 and 3 are correct, while statement 1 is incorrect.

Conclusion on Correct Statistical Statements

Evaluating the given statements, we find that only statements 2 and 3 are correct regarding statistical measures of dispersion and central tendency. Statement 1 incorrectly asserts that the coefficient of variation is unit-dependent.

Revision Table: Key Statistical Concepts

Additional Information: Measures of Dispersion

Measures of dispersion quantify the spread or variability of data. Understanding dispersion is crucial in data analysis to know how spread out the data points are from the central value. Common measures of dispersion include:

• Range: Simple difference between max and min values. Sensitive to outliers.
• Interquartile Range (IQR): Difference between the third quartile (Q3) and the first quartile (Q1). Represents the range of the middle 50% of the data. Less sensitive to outliers than the range.
• Mean Deviation: Average of the absolute deviations from the mean, median, or mode. Usually calculated about the mean or median. \( MD = \frac{\sum |x_i - a|}{n} \)
• Variance: The average of the squared differences from the mean. \( \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \) (for population) or \( s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \) (for sample).
• Standard Deviation: The square root of the variance. It is the most commonly used measure of dispersion and has the same unit as the original data. \( \sigma = \sqrt{\sigma^2} \) or \( s = \sqrt{s^2} \).
• Coefficient of Variation (CV): A relative measure of dispersion, expressed as a percentage of the mean. Useful for comparing the variability of datasets with different units or vastly different means. \( \text{CV} = (\sigma/\mu) \times 100\% \)

These measures help describe the characteristics of a dataset beyond just its central tendency.