Understanding Key Statistical Measures
In statistics, understanding the shape and variability of data is crucial. Today, we explore two fundamental concepts: Skewness and the Coefficient of Variation (C.V.).
The Problem
Given values:
- Mean ($\bar{X}$) = 56.80
- Median ($M_d$) = 59.50
- Standard Deviation ($S.D. or \sigma$) = 12.40
Find:
- Skewness (Karl Pearson's coefficient)
- Coefficient of Variation (C.V.)
1. Karl Pearson's Coefficient of Skewness
Skewness measures the asymmetry of a distribution. Karl Pearson's coefficient of skewness ($S_k$) is calculated as:
$$S_k = \frac{3(\bar{X} - M_d)}{\sigma}$$
Calculation: $$S_k = \frac{3(56.80 - 59.50)}{12.40}$$ $$S_k = \frac{3(-2.70)}{12.40}$$ $$S_k = \frac{-8.10}{12.40} \approx -0.653$$
Interpretation: Since the value is negative, the distribution is negatively skewed (left-skewed).
2. Coefficient of Variation (C.V.)
The C.V. measures relative variability or the spread of the data relative to the mean. It is expressed as a percentage.
$$C.V. = \left( \frac{\sigma}{\bar{X}} \right) \times 100$$
Calculation: $$C.V. = \left( \frac{12.40}{56.80} \right) \times 100$$ $$C.V. \approx 0.2183 \times 100 \approx 21.83\%$$
Interpretation: The standard deviation is approximately 21.83% of the mean, indicating the relative volatility or dispersion of the dataset.