Skewness and Kurtosis

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Skewness: Measuring Asymmetry of Distribution

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Definition and Concept

Skewness is a statistical measure that describes the degree of **asymmetry** in the shape of a probability distribution or a dataset. It indicates whether the data is concentrated more on one side of the central value than the other. In simpler terms, it tells us if the distribution is lopsided.

• A distribution is considered **symmetric** if it looks the same on both sides of a central point (like folding it in half). A perfectly symmetric distribution has zero skewness. For a unimodal (single-peaked) symmetric distribution, the mean, median, and mode are all located at the same point.
• An distribution is **asymmetric** or **skewed** if one side (or "tail") is longer or extends further than the other.

Skewness helps in understanding the shape of the distribution and how it deviates from the ideal symmetric shape.

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Types of Skewness and Relationship with Averages

There are three main types of skewness, characterized by the direction of the longer tail and the typical relationship between the mean, median, and mode:

1. Zero Skewness (Symmetric Distribution):

   In a perfectly symmetric distribution, the shape is identical on both sides of the center. There is no skewness.

   • The mean, median, and mode are all equal and coincide at the center of the distribution.
   • Examples include the Normal Distribution (bell curve), uniform distribution, and symmetric bimodal distributions.

2. Positive Skewness (Skewed to the Right):

   A distribution is positively skewed if the longer tail extends to the right side, towards higher values. This happens when there are some relatively high values that are far from the majority of the data.

   • The bulk of the data is concentrated on the left side of the distribution, around lower values.
   • The presence of extreme high values in the right tail pulls the mean towards the right more than the median or mode.
   • Typically, for a unimodal positively skewed distribution, the relationship between the measures of central tendency is: Mean > Median > Mode. The mode is at the peak, the median is to the right of the mode, and the mean is further to the right.
   • Examples: Income distribution (most people earn less, a few earn significantly more), scores on a difficult test (most students score lower, a few score very high).

3. Negative Skewness (Skewed to the Left):

   A distribution is negatively skewed if the longer tail extends to the left side, towards lower values. This happens when there are some relatively low values that are far from the majority of the data.

   • The bulk of the data is concentrated on the right side of the distribution, around higher values.
   • The presence of extreme low values in the left tail pulls the mean towards the left more than the median or mode.
   • Typically, for a unimodal negatively skewed distribution, the relationship between the measures of central tendency is: Mode > Median > Mean. The mode is at the peak, the median is to the left of the mode, and the mean is further to the left.
   • Examples: Scores on an easy test (most students score high, a few score very low), age of death (most people die at older ages, some die at younger ages due to illness or accident).

Understanding the type of skewness helps in choosing the most appropriate measure of central tendency (median is often preferred over mean for skewed data) and in interpreting the data distribution correctly.

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Methods of Measuring Skewness (Karl Pearson's, Bowley's)

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Quantitative Measurement of Skewness

While visual inspection of a histogram or frequency polygon provides a qualitative idea of skewness, quantitative measures are needed to precisely determine the degree and direction of asymmetry and to compare the skewness of different distributions. These quantitative measures are called coefficients of skewness.

Several methods exist for calculating coefficients of skewness. The most common ones encountered in introductory statistics are based on the relationship between the mean, median, and mode, or based on quartiles.

1. Karl Pearson's Coefficient of Skewness ($Sk_p$)

This measure relates the difference between the mean and the mode (or median) to the standard deviation. It indicates how many standard deviations the mean is away from the mode (or median).

Formula using Mode:

This formula is based on the definition of skewness as the difference between the mean and the mode, standardized by the standard deviation to make it a relative measure.

This formula is meaningful and accurate when the mode is clearly defined and the distribution is unimodal.

Formula using Median (based on the empirical relationship):

When the mode is not well-defined (e.g., for multimodal distributions or grouped data where the modal class is ambiguous) or when the empirical relationship (Mode $\approx$ 3 Median - 2 Mean) is assumed to hold, a modified formula using the median is often used. From the empirical relationship, Mean - Mode $\approx$ 3(Mean - Median).

This median-based formula is generally preferred when the distribution is moderately skewed and unimodal, as the median is less affected by extreme values than the mode.

Interpretation of Karl Pearson's Coefficient of Skewness ($Sk_p$):

• If $Sk_p = 0$, the distribution is symmetric.
• If $Sk_p > 0$, the distribution is positively skewed (Mean > Mode or Mean > Median). A larger positive value indicates greater positive skewness.
• If $Sk_p < 0$, the distribution is negatively skewed (Mean < Mode or Mean < Median). A larger negative value (smaller in algebraic sense) indicates greater negative skewness.
• Using the median-based formula (Formula 2), the value of $Sk_p$ typically lies between -1 and +1 for most moderately skewed distributions. Values outside this range suggest a more extreme degree of skewness. Values between -0.5 and +0.5 are often considered to indicate roughly symmetric distributions.

2. Bowley's Coefficient of Skewness ($Sk_b$)

This measure, also known as the Quartile Coefficient of Skewness, is based on the quartiles of the distribution. It measures the asymmetry of the central 50% of the data.

Recall that $Q_1$ is the first quartile (25th percentile), $Q_2$ is the second quartile (the Median, 50th percentile), and $Q_3$ is the third quartile (75th percentile).

The formula compares the distances of the median from the first and third quartiles: $(Q_3 - Q_2)$ is the distance from the median to the third quartile, and $(Q_2 - Q_1)$ is the distance from the first quartile to the median. In a symmetric distribution, these distances are equal.

Formula:

This can be simplified to:

Interpretation of Bowley's Coefficient of Skewness ($Sk_b$):

• If $Sk_b = 0$, the median is equidistant from $Q_1$ and $Q_3$, indicating symmetry in the central part of the distribution.
• If $Sk_b > 0$, $(Q_3 - Q_2) > (Q_2 - Q_1)$, meaning the distance from Median to $Q_3$ is greater than the distance from $Q_1$ to Median. This suggests positive skewness in the central part.
• If $Sk_b < 0$, $(Q_3 - Q_2) < (Q_2 - Q_1)$, meaning the distance from Median to $Q_3$ is less than the distance from $Q_1$ to Median. This suggests negative skewness in the central part.
• Bowley's coefficient always falls between -1 and +1 ($ -1 \le Sk_b \le +1$).

Bowley's coefficient is less sensitive to extreme values (outliers) than Pearson's coefficient because it only uses values at the quartiles, not the mean or standard deviation which are affected by all values. It is particularly useful when dealing with distributions that have extreme values or when the mean and standard deviation are not appropriate (e.g., with open-ended classes).

3. Moment-Based Coefficient of Skewness ($\gamma_1$)

A more general and mathematically rigorous measure of skewness is based on the third central moment ($m_3$).

Formula:

Where $m_3 = \frac{\sum (x_i - \bar{x})^3}{n}$ for ungrouped data or $m_3 = \frac{\sum f_i (x_i - \bar{x})^3}{N}$ for frequency distributions.

Interpretation:

• If $\gamma_1 = 0$, the distribution is symmetric.
• If $\gamma_1 > 0$, the distribution is positively skewed.
• If $\gamma_1 < 0$, the distribution is negatively skewed.

This measure is used in more advanced statistical theory and applications. It captures the asymmetry using the cubed deviations, which gives more weight to large deviations and their direction (positive or negative).

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Kurtosis: Measuring Peakedness of Distribution

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Definition and Concept

Kurtosis is a statistical measure that describes the **shape** of a probability distribution, specifically its degree of "peakedness" or "flatness" compared to a standard reference distribution, usually the normal distribution. It also gives an indication of the heaviness or thickness of the tails of the distribution.

Kurtosis does not measure the steepness of the slopes of the distribution. Instead, it measures the concentration of data in the peak and tails. High kurtosis means more data in the tails (outliers) and often a sharper peak around the mean. Low kurtosis means less data in the tails and a flatter distribution with data spread more evenly.

It is a measure of how outlier-prone a distribution is. Distributions with high kurtosis have a greater propensity for extreme values than distributions with low kurtosis.

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Types of Kurtosis

Kurtosis is typically assessed by comparing it to the kurtosis of the normal distribution, which serves as a benchmark.

1. Mesokurtic:

   A distribution that has a kurtosis similar to that of the normal distribution. The peak and tail thickness are considered "normal".

   • The kurtosis value (using the moment-based definition $\beta_2$) is equal to 3.
   • The excess kurtosis ($\gamma_2 = \beta_2 - 3$) is equal to 0.

   The normal distribution itself is the prime example of a mesokurtic distribution.

2. Leptokurtic:

   A distribution that has a higher peak and heavier/fatter tails than a mesokurtic (normal) distribution. "Lepto" means thin, referring to the thin waist of the distribution around the center, not the tails.

   • More data is concentrated around the mean (sharper peak).
   • There is a higher probability of extreme values (outliers), contributing to the heavier tails.
   • The kurtosis value ($\beta_2$) is greater than 3.
   • The excess kurtosis ($\gamma_2$) is greater than 0.

   Examples: T-distribution with low degrees of freedom, Laplace distribution.

3. Platykurtic:

   A distribution that has a flatter peak and lighter/thinner tails than a mesokurtic (normal) distribution. "Platy" means broad or flat.

   • Data is more spread out across the range, with less concentration around the mean (flatter peak).
   • There is a lower probability of extreme values (outliers).
   • The kurtosis value ($\beta_2$) is less than 3.
   • The excess kurtosis ($\gamma_2$) is less than 0.

   Example: Uniform distribution.

Understanding kurtosis helps in selecting appropriate statistical models and tests, as many methods assume a normal distribution (mesokurtic).

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Measurement (Moment-Based)

Kurtosis is quantitatively measured using the fourth central moment ($m_4$). The coefficient of kurtosis is derived from the fourth central moment and the variance.

Coefficient of Kurtosis ($\beta_2$):

Where $m_4 = \frac{\sum (x_i - \bar{x})^4}{n}$ for ungrouped data or $m_4 = \frac{\sum f_i (x_i - \bar{x})^4}{N}$ for frequency distributions.

Alternatively, the concept of **Excess Kurtosis** ($\gamma_2$) is often used, which compares the distribution's kurtosis directly to that of the normal distribution ($\beta_2 = 3$).

Coefficient of Excess Kurtosis ($\gamma_2$):

Interpretation based on Excess Kurtosis ($\gamma_2$):

• If $\gamma_2 = 0$, the distribution is Mesokurtic (like the normal distribution).
• If $\gamma_2 > 0$, the distribution is Leptokurtic (sharper peak, fatter tails than normal).
• If $\gamma_2 < 0$, the distribution is Platykurtic (flatter peak, thinner tails than normal).

Calculating moments, especially the third and fourth, can be computationally intensive, requiring calculating $(x_i - \bar{x})^3$ and $(x_i - \bar{x})^4$ for each observation or class mark and summing them up, weighted by frequency. While the calculation is involved, understanding the interpretation of skewness and kurtosis values provides valuable insight into the shape of the data distribution.