Question

Which one of the following is false?

• A. Mean deviation from the mean and median are always equal.
• B. Coefficient of variation is unit-free.
• C. Coefficient of variation is a relative measure.
• D. Greater CV implies more variability.

Hint:

Mean Deviation Facts. Mean deviation is minimized about the median, not necessarily equal to that about the mean.

Answer 1

The Correct Option is A

The statement that is false is: Mean deviation from the mean and median are always equal.

This is false because:

1. Mean Deviation from Mean: Mean deviation is calculated by taking the average of absolute deviations of each data point from the mean. Mathematically, it is expressed as:
   MD = (1/n) Σ |x_i - μ|
   where n is the number of observations, x_i represents each observation, and μ is the mean.
2. Mean Deviation from Median: Similarly, mean deviation from the median is the average of absolute deviations of each data point from the median. Written as:
   MD_median = (1/n) Σ |x_i - M|
   where M denotes the median.
3. In general, Mean Deviation from the Mean and Mean Deviation from the Median are not always equal because they have different central tendencies (mean versus median).

Coefficiencies:

• Coefficient of Variation (CV): Defined as:
  CV = (Standard Deviation/Mean) × 100%
• It is a unit-free measure because it is a ratio, allowing comparison across different datasets.
• It serves as a relative measure of variability.
• Greater CV implies more variability among the data set.

Answer 2

Step 1: Understand the concepts
Mean deviation is the average of the absolute differences between each data point and a central value, which can be the mean or median.
Mean deviation from the mean and mean deviation from the median are generally different.

Step 2: Compare mean deviation from mean and median
- Mean deviation from the mean is calculated using the mean as the reference point.
- Mean deviation from the median is calculated using the median as the reference point.
Since mean and median can be different, the mean deviations from these two points usually differ.

Step 3: Conclusion about the statement
The statement "Mean deviation from the mean and median are always equal" is false because these two values are generally not equal.

Final answer: Mean deviation from the mean and median are always equal is false.