Range

Range of a Distribution

Definition:

• The range of a distribution measures the spread or width of data values in a dataset.
• Formula: Range=L−S, where L is the largest (maximum) value and S is the smallest (minimum) value in the dataset.

Properties:

• Units: The range has the same units as the data points.
• Transformation: If a random variable X is transformed to Y=aX+b, then Range Y​=∣a∣×RangeX​. The shape of the distribution remains unchanged with origin shifts, only scaling matters.
• Grouped Data: For grouped data, the range is defined between the extreme class boundaries.
• Coefficient of Range: This is a percentage measure of range relative to the total range of the data, given by Coefficient of Range = $\frac{L-S}{L+S}\times 100$ It allows comparison of dispersion across different distributions.

Limitations:

• Absolute Measure: Provides only endpoints information, neglecting distribution shape.
• Dispersion: Absolute measure of dispersion, not considering the distribution's internal structure.

Advantages:

• Simplicity: Easy to calculate and interpret, making it useful for quick assessments of data spread.
• Intuitive: Provides a basic understanding of how widely spread data values are.

Applications:

• Preliminary Analysis: Often used initially to get a quick sense of variability in data.
• Comparative Studies: Useful in comparing the spread of data between different datasets or groups.

Conclusion: 

The range serves as a fundamental measure of dispersion, indicating the spread of data values from the smallest to the largest. While simple and intuitive, it is limited by its focus on only the endpoints of the distribution. For more comprehensive insights into data variability and distribution shape, other measures of dispersion such as variance, standard deviation, or interquartile range are often preferred.