The question asks us to identify the statistical methods used to measure the 'dispersion' or spread of a dataset from the given options.

Understanding Dispersion in Statistics

Dispersion refers to the extent to which a distribution is stretched or squeezed. In simpler terms, it tells us how spread out or closely clustered the data points are. Key measures of dispersion help us understand this variability.

Analyzing the Options for Dispersion Measures

Let's examine each option provided:

• A. Median: The median is a measure of central tendency. It represents the middle value in a dataset when the data is ordered. It does not measure the spread of the data.
• B. Mean: The mean (or average) is also a measure of central tendency. It indicates the average value of the data points. Like the median, it does not directly measure dispersion.
• C. Mean Deviation: This is a measure of dispersion. It calculates the average absolute difference between each data point and the mean (or median) of the dataset. A lower mean deviation indicates less dispersion. The formula involves absolute values, for example, Mean Deviation $= \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$, where $x_i$ are the data points and $\bar{x}$ is the mean.
• D. Standard Deviation: This is a widely used measure of dispersion. It quantifies the amount of variation or dispersion of a set of values by measuring the square root of the variance. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values. The formula is $\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \mu)^2}{N}}$.
• E. Range: The range is a simple measure of dispersion, calculated as the difference between the maximum and minimum values in a dataset. It provides a quick idea of the spread but is sensitive to outliers. Range $= \text{Maximum Value} - \text{Minimum Value}$.

Conclusion on Dispersion Methods

Based on the analysis, the methods that measure the dispersion or spread of data are Mean Deviation, Standard Deviation, and Range.

Therefore, the correct options are C, D, and E.

Identifying the Correct Option

Comparing our findings with the provided options:

• Option 1: A and E Only (Incorrect, A is central tendency)
• Option 2: A, C and E Only (Incorrect, A is central tendency)
• Option 3: B, C and D Only (Incorrect, B is central tendency)
• Option 4: C, D and E Only (Correct, as C, D, and E are measures of dispersion)