Skewness and Distribution Shapes

Skewness describes the asymmetry of a distribution. If the data pile up on one side and the tail stretches longer on the other side, the distribution is skewed.
The transcript emphasizes left-tailed and negatively skewed distributions: "Go to your peak long tail to the left. Right. Left tailed. Negatively skewed." This means the long tail is on the left side.
It’s also noted that there will be examples where distributions are negatively or positively skewed; skewness is a property of the data, not a defect.

Left-tailed distribution = negatively skewed distribution where the tail extends further to the left (lower values).
Right-tailed distribution = positively skewed distribution where the tail extends further to the right (higher values).
In left-skewed data, most observations are concentrated on the right (higher values) with a tail toward smaller values.
In right-skewed data, most observations are concentrated on the left (lower values) with a tail toward larger values.

Characteristics:
- Tail is longer on the left side; bulk of the data is toward the right.
- Common intuitive statement: mean is pulled toward the tail; thus the mean is typically less than the median.
Typical relationship among measures of central tendency:
\text{Mode} > \text{Median} > \text{Mean}
Significance:
- The mean may not be the best representative of the center when data are highly skewed.
- Median often provides a better sense of a "typical" value in skewed distributions.
Real-world intuition: skewness can arise naturally in data where a floor bound exists (e.g., nonnegative values) and there are a few very small values pulling the tail left.

The transcript notes that skewness appears in various datasets; some contexts favor negative skewness, others positive.
Examples (typical in practice):
- Right-skewed (positive skew): income distributions, waiting times, time-to-failure data, stock returns in certain regimes.
- Left-skewed (negative skew): very easy tests where most people score high with a few low performers, certain completion time scenarios where most finish quickly but a few take much longer.
Practical interpretation:
- Skewness informs which summary statistics to report (mean vs median) and which statistical methods are appropriate.
- Skewness affects the suitability of methods that assume normality.

In skewed distributions, the order of mean, median, and mode shifts depending on the direction of skewness:
- Left-skewed (negatively skewed):
  \text{Mode} > \text{Median} > \text{Mean}.
- Right-skewed (positively skewed):
  \text{Mean} > \text{Median} > \text{Mode}.
Implications:
- The mean is sensitive to outliers and the tail; it may not represent the "typical" observation in skewed data.
- The median is more robust to extreme values and often better represents central tendency in skewed distributions.

Normality assumptions: Many parametric tests assume approximately normal distributions; skewness can violate these assumptions.
Methods to address skewness:
- Data transformation to reduce skewness (e.g., logarithmic, square root, Box-Cox transformations).
- Use non-parametric tests when data remain skewed or when transforming is inappropriate.
- Report both mean and median when skewness is present to provide a fuller picture.
Contextual decision-making:
- Whether skewness is "bad" depends on the data and the analysis goal; skewness is a natural feature of many real-world datasets.

Skewness indicates asymmetry; the long tail direction determines whether a distribution is negatively (left) or positively (right) skewed.
There is nothing inherently wrong with skewness; it depends on the data and the questions being asked.
Always consider which measure of central tendency and which statistical methods are appropriate given the skewness observed in the data.
Recognize that skewness informs practical decisions: transformation choices, reporting practices, and the robustness of conclusions.