In the realm of statistics, skewness or asymmetry is a measure that can be identified by examining the relationships between the mean, median, and mode of a distribution.
For symmetric, unimodal distributions, which have only one peak, the mean, median, and mode all align perfectly. However, when a distribution is skewed, this symmetry is disrupted, causing the mean and median to diverge from their typical positions relative to the mode.
The presence or absence of symmetry in a distribution gives rise to either positive or negative skewness. This measure helps in determining the degree of asymmetry in a probability distribution of a random variable. A bell-shaped plot or histogram can visually represent this asymmetry.
Using the mode as a reference point, the type of skewness is determined by the dispersion of data on either side, often referred to as “tails.”
- Positive Skewness: In a positively skewed distribution, the tail on the curve’s right side is longer than the left side. Here, both the mean and median are greater than the mode, with the mean typically exceeding the median.
- Negative Skewness: In a negatively skewed distribution, the tail on the curve’s left side is longer than the right side. Here, both the mean and median are less than the mode, and the mean is usually smaller than the median.
Methods to Calculate Skewness
While visual inspection can provide an idea of skewness, precise measurements offer a more accurate assessment. Several methods exist for this purpose:
- Pearson’s First Skewness Coefficient: This measure, primarily used in unimodal distributions, involves subtracting the mode from the mean and dividing the result by the data’s standard deviation.
- Second Pearson Skewness Coefficient: This involves subtracting the median from the mode, multiplying the outcome by three, and then dividing by the standard deviation.
- Fisher’s Coefficient of Asymmetry: This method, slightly more intricate, is grounded in the deviations observed values have from the mean. It’s computed by dividing the third moment about the mean by the cube of the standard deviation.
- Bowley-Yule Skewness Coefficient: This coefficient is based on the median and quartiles. If the distribution is symmetric, the result will be 0. Positive and negative skewness will yield values greater or less than 0, respectively.
Bibliography:
- Martínez Pineda, A. (2018, April 4). Distribution types. RStudio.
- Spiegel, M. Probability and statistics. (2014). Spain. McGraw-Hill Interamericana.
- Mullor Ibáñez, R. Basic Statistics: I. Introduction to Statistics. (2017). Publications of the University of Alicante.
