Kurtosis and Skewness 

Kurtosis and skewness are statistical measures used to describe the shape of a data distribution. Skewness indicates the asymmetry of the distribution, showing whether data points are more spread on one side of the mean. Kurtosis measures the "tailedness" or the presence of outliers in the distribution. Together, they help in understanding the nature, behavior, and spread of data, playing a crucial role in data analysis, quality control, finance, and decision-making.

1.0What is Skewness in Statistics?

Skewness in statistics measures the asymmetry of a probability distribution. It shows whether the data leans more toward the left or the right of the mean.

Types of Skewness:

1. Positive Skewness (Right Skew):

• The tail is stretched on the right side.
• Most values lie to the left of the mean.
• Mean > Median > Mode.
• Common in income distributions or time to failure data.

2. Negative Skewness (Left Skew):

• The tail is stretched on the left side.
• Most values lie to the right of the mean.
• Mean < Median < Mode.

3. Zero Skewness:

Perfectly symmetrical distribution (e.g., normal distribution).

2.0Calculating Skewness

You can compute skewness using:

Formula for Skewness:

$\text{Skewness}=\frac{n}{(n-1)(n-2)}\sum {\left(\frac{x_i-\bar{x}}{s}\right)}^3$

Where:

n = number of data points

$\bar{x}=\text{sample mean}$

s = sample standard deviation

$x_i$ = individual data values

Many statistical software and calculators also offer built-in functions for calculating skewness.

3.0What is Kurtosis in Statistics?

Kurtosis in statistics measures the "tailedness" of the distribution — how heavy or light the tails are compared to a normal distribution. It provides insights into the likelihood of outliers.

Types of Kurtosis

1. Mesokurtic:

• • Normal distribution.
  • Moderate tails (Kurtosis = 3).

2. Leptokurtic:

• • Heavy tails, sharp peak.
  • High probability of outliers.
  • Kurtosis > 3.

3. Platykurtic:

• • Light tails, flat peak.
  • Few or no outliers.
  • Kurtosis < 3.

4.0Kurtosis Formula

The kurtosis formula is:

$\text{Kurtosis}=\frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum {\left(\frac{x_i-\bar{x}}{s}\right)}^4-\frac{3(n-1{)}^2}{(n-2)(n-3)}$

This is the excess kurtosis, where:

• A result of 0 indicates a normal distribution.
• Positive → Leptokurtic.
• Negative → Platykurtic.

5.0Skewness and Kurtosis in Statistics: Why They Matter

Understanding skewness and kurtosis in statistics helps:

• Detects outliers and data irregularities.
• Validate assumptions for statistical models.
• Inform the choice of further analysis methods.

For example:

• A dataset with positive skewness might need transformation before applying linear regression.
• High kurtosis could signal the presence of rare but extreme events (important in risk management and finance).

6.0Solved Examples on Kurtosis and Skewness

Example 1: The marks of 5 students are: 20, 25, 30, 35, 80. Determine the skewness.

Solution:

• Mean = (20 + 25 + 30 + 35 + 80) / 5 = 38
• Median = 30
• Since Mean > Median, the distribution is positively skewed.

Example 2: Given a dataset: 2, 3, 5, 7, 10. Find skewness.

Solution:

$$\begin{gathered} \bar{x}=\frac{2+3+5+7+10}{5}=5.4 \\ \text{Mean} \\ \text{Standard deviation (s)} \\ (\approx 2.87) \\ \text{Skewness formula (simplified for small data):} \\ \text{Skewness}\approx \frac{n}{(n-1)(n-2)}\sum (\frac{x_i-\bar{x}}{s}{)}^3 \\ \text{Compute each value and sum:} \\ \text{Final Skewness}(\approx 0.46)\text{(Moderate positive skew)} \end{gathered}$$ 

Example 3: A dataset has the following values: 4, 4, 5, 5, 50. Is the kurtosis high?

Solution:

• Large outlier (50) suggests heavy tails
• Kurtosis will be high → Leptokurtic distribution

Example 4: A normal distribution has skewness = 0 and kurtosis = 3. What does it indicate?

Answer:

• Skewness = 0 → symmetrical
• Kurtosis = 3 → mesokurtic
• Thus, it is a normal distribution

Example 5: A histogram has a long tail to the left. What is the skewness?

Answer:

• Tail on left → Negative skewness

7.0Practice Questions on Kurtosis and Skewness

1. Identify the skewness: Data = 12, 15, 18, 20, 100
2. Calculate skewness: Data = 3, 6, 6, 7, 8
3. Interpret kurtosis: If a dataset has frequent extreme values, is it leptokurtic or platykurtic?
4. Multiple choice:

A distribution with kurtosis < 3 is called:

1. Mesokurtic
2. Leptokurtic
3. Platykurtic
4. None of these
5. True or False: If a distribution is symmetrical, then its skewness is always 0.