Skewness

• Positive Skewness (Right-Skewed): Occurs when the tail on the right side of the distribution is longer or fatter than the left side. In this scenario, the mean is typically greater than the median, which is greater than the mode ($Mean > Median > Mode$).

• Negative Skewness (Left-Skewed): Occurs when the tail on the left side is longer or fatter. This indicates a concentration of high values with a few extremely low values pulling the mean down, resulting in $Mean < Median < Mode$.

• Zero Skewness: Represents a perfectly symmetrical distribution, such as the Normal Distribution, where the mean, median, and mode are equal.

Pearson's Coefficients

• First Coefficient: Based on the difference between the mean and the mode: $S_p = \frac{Mean - Mode}{\sigma}$ where $\sigma$ is the standard deviation. This is used when a clear mode exists.

• Second Coefficient: Used when the mode is ill-defined, utilizing the relationship between mean and median: $S_p = \frac{3(Mean - Median)}{\sigma}$

Bowley's Coefficient (Quartile Skewness)

• This measure relies on the position of quartiles and is less affected by extreme outliers: $S_b = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1}$ where $Q_2$ is the median. It ranges from $-1$ to $+1$.

Moment-Based Skewness

• Calculated using the third central moment ($\mu_3$), which measures the cubical deviations from the mean: $\gamma_1 = \frac{\mu_3}{\sigma^3}$ A positive $\mu_3$ indicates positive skewness, while a negative $\mu_3$ indicates negative skewness.

• Identify the Tail: Always look at the direction of the long, thin tail to determine the type of skewness. Students often mistakenly look at where the 'hump' is; if the hump is on the left, the skew is positive (right).

• Formula Selection: Use Pearson's first coefficient if the mode is distinct. If the distribution is multi-modal or the mode is not clearly defined, default to the median-based formula.

• Sanity Check: If you calculate a positive skewness value, verify that your $Mean$ is indeed larger than your $Median$. If the math doesn't align with the conceptual relationship, re-check your calculations.

• Range Awareness: Remember that Bowley's coefficient is bounded between $-1$ and $+1$, whereas Pearson's coefficients can occasionally exceed these bounds depending on the data distribution.

• The 'Direction' Confusion: A common error is labeling a distribution based on the location of the peak. A peak on the left side of the graph indicates Positive Skewness because the tail extends to the right.

• Mean Sensitivity: Students often forget that the mean is the most sensitive to outliers. In a skewed distribution, the mean is 'pulled' toward the tail more than the median or mode.

• Symmetry vs. Normality: Not all symmetric distributions are 'Normal'. A distribution can have zero skewness but still be non-normal due to its kurtosis (peakedness).