Among these options, which one is NOT an example of relative measure of dispersion?
Variance
Understanding Measures of Dispersion in Statistics
In statistics, measures of dispersion help us understand how spread out a set of data is. There are two main types of measures of dispersion:
- Absolute measures of dispersion: These measures express the variation in the same units as the data itself. Examples include Range, Quartile Deviation, Mean Deviation, and Standard Deviation, and Variance.
- Relative measures of dispersion: These measures express the variation as a proportion or percentage of an average (like the mean or median). They are unitless and are useful for comparing the dispersion of two different datasets, even if they have different units or scales. These are typically coefficients calculated from the absolute measures.
Identifying Relative vs. Absolute Measures
The question asks to identify which option is NOT an example of a relative measure of dispersion. Let's examine each option:
Variance as a Measure of Dispersion
Variance is an absolute measure of dispersion. It is calculated as the average of the squared differences from the mean. The unit of variance is the square of the unit of the data. For example, if data is in meters, variance is in square meters. Because it has a unit related to the square of the data unit, it is not a relative measure.
Coefficient of Variation (CV)
The Coefficient of Variation is a relative measure of dispersion. It is calculated as the ratio of the standard deviation ($\sigma$) to the mean ($\mu$), usually expressed as a percentage: $\text{CV} = \left( \frac{\sigma}{\mu} \right) \times 100\%$ It is a dimensionless quantity, making it suitable for comparing variability between datasets with different means or scales.
Coefficient of Standard Deviation
The Coefficient of Standard Deviation is also a relative measure of dispersion. It is simply the ratio of the standard deviation ($\sigma$) to the mean ($\mu$): $\text{Coefficient of Standard Deviation} = \frac{\sigma}{\mu}$ This is essentially the Coefficient of Variation before being multiplied by 100 to express it as a percentage. It is unitless.
Coefficient of Quartile Deviation (CQD)
The Coefficient of Quartile Deviation is a relative measure of dispersion based on quartiles. It is calculated as the ratio of the Quartile Deviation (QD) to the average of the third quartile ($Q_3$) and the first quartile ($Q_1$): $\text{CQD} = \frac{Q_3 - Q_1}{Q_3 + Q_1}$ Since it is a ratio of values in the same units, the units cancel out, making it a unitless, relative measure. Quartile Deviation ($QD = \frac{Q_3 - Q_1}{2}$) itself is an absolute measure.
Comparing the Options
Let's summarize the nature of each measure:
| Measure of Dispersion | Type | Description / Formula Basis |
|---|---|---|
| Variance | Absolute | Average of squared deviations from the mean ($\sigma^2$). Unit is data unit squared. |
| Coefficient of Variation | Relative | Ratio of Standard Deviation to Mean ($\frac{\sigma}{\mu}$). Unitless. |
| Coefficient of Standard Deviation | Relative | Ratio of Standard Deviation to Mean ($\frac{\sigma}{\mu}$). Unitless. |
| Coefficient of Quartile Deviation | Relative | Ratio based on quartiles ($\frac{Q_3 - Q_1}{Q_3 + Q_1}$). Unitless. |
Based on this comparison, Variance is the only measure among the options that is an absolute measure, not a relative measure of dispersion. The coefficients (Coefficient of variation, Coefficient of standard deviation, Coefficient of quartile deviation) are specifically designed to be relative measures, allowing for comparison across different datasets.
Conclusion
The question asks for the option that is NOT an example of a relative measure of dispersion. As analyzed, Variance is an absolute measure, while Coefficient of variation, Coefficient of standard deviation, and Coefficient of quartile deviation are all relative measures.
Therefore, Variance is the correct answer.
Revision Table: Measures of Dispersion
| Type of Measure | Examples | Key Characteristic | Use Case |
|---|---|---|---|
| Absolute Measures | Range, Quartile Deviation, Mean Deviation, Standard Deviation, Variance | Expressed in the same units as the data. | Describing dispersion within a single dataset. |
| Relative Measures | Coefficient of Variation, Coefficient of Standard Deviation, Coefficient of Quartile Deviation, Coefficient of Mean Deviation | Expressed as a ratio or percentage, unitless. | Comparing dispersion across datasets with different units or scales. |
Additional Information on Dispersion
Measures of dispersion are crucial in statistics because they provide context to measures of central tendency (like mean, median, mode). A dataset with a small measure of dispersion is tightly clustered around the average, while a dataset with a large measure of dispersion is widely spread out.
For example, two classes might have the same average test score (mean), but one class could have scores very close to the mean (low dispersion, e.g., everyone scored between 70 and 80 if the mean is 75), while the other class could have scores ranging widely (high dispersion, e.g., scores from 40 to 100 if the mean is 75). Measures of dispersion help highlight this difference in data distribution.
Relative measures are particularly useful when comparing variability. For instance, comparing the variability in the heights of adult humans (measured in cm) and the variability in the weights of adult humans (measured in kg). Using standard deviation directly wouldn't be meaningful as the units are different. However, the coefficient of variation can be used to see which attribute (height or weight) shows more relative variability.
Important Questions from Variance and Standard Deviation
The sum of deviations of n numbers from 10 and 20 are p and q respectively. If (p - q)2 = 10000, then what is the value of n?
The mean and the variance of 10 observations are given to be 4 and 2 respectively. If every observation is multiplied by 2, the mean and the variance of the new series will be respectively.
If the data are moderately non-symmetrical, then which one of the following empirical relationships is correct?
If the total number of observations is 20, ∑ x i= 1000 and \(\sum {\rm{x}}_{\rm{i}}^2 = 84000\) , then what is the variance of the distribution?
The standard deviation of the first 10 natural numbers is 3.028. What will be the standard deviation of the first 20 natural numbers?
