Types+of+Distribution+and+Skewness lecture

Unit 4: Types of Distribution and Skewness lecture

1. Unit 4: Introduction to Distribution Shapes
Types of Distribution and Skewness
Understanding Data Shapes, Symmetry, and T ailedness
Symmetrical Positively Skewed Negatively Skewed

2. Introduction to Distribution Shapes
What is a Data Distribution?
A data distribution illustrates how frequently differ ent
values appear within a dataset, providing insight into
its underlying shape and structure.
By visualizing a distribution, we can understand the
spread, central tendency, and potential outliers of
the data, which are crucial for statistical analysis and
interpretation.
Why Distribution Shapes
Matter
Reveal patterns and underlying structures in data
Provide insights into central tendency and variability
Identify potential outliers and anomalies
Inform appropriate statistical methods and models
Support decision-making and hypothesis generation
Common Distribution Shapes
The shape of a data distribution reveals important characteristics about the
data-generating process and can significantly impact the choice of
appropriate statistical methods for analysis.
Symmetrical
Equal spread on both
sides
Positively Skewed
Tail extends to the right
Negatively
Skewed
Tail extends to the left

3. Symmetrical/Normal Distribution
The Normal Distribution
The normal distribution, often called the Gaussian
distribution or bell curve, is a fundamental concept in
statistics due to its symmetrical shape and
prevalence in natural phenomena.
Symmetry
Perfectly symmetrical
around its central point. If
folded in half, both sides
would be identical.
Central
Tendency
Mean, median, and mode
are all equal and located at
the center of the
distribution.
Parameters
Defined by mean (μ) and
standard deviation (σ). μ
determines the center, σ
the spread.
Probability
The total area under the
curve equals 1 (100%),
representing total
probability.
The Empirical Rule
68%
Within 1 standard
deviation (μ ± σ)
95%
Within 2 standard
deviations (μ ± 2σ)
99.7%
Within 3 standard
deviations (μ ± 3σ)
Practical Applications
Biological measurements (height, weight,
IQ)
Quality control in manufacturing
Financial markets (option pricing, risk
assessment)
Statistical testing (t-tests, ANOVA)

4. Positively Skewed Distribution
Definition
A positively skewed distribution, also known as a right-skewed
distribution, is characterized by a longer tail extending towards
the right side. This implies that the majority of data values are
clustered around the lower end, while a few larger, extreme
values pull the mean towards the right.
Key Characteristics
Tail Direction: The "tail" extends to the right.
Central Tendency Relationship:
Mean > Median > Mode
Peak Location: The peak of the distribution is towards the left.
Impact on Measures: The mean is pulled towards the higher
values in the longer right tail.
Common Examples
Income Distribution:
Most individuals earn lower
to moderate incomes, with
a smaller number earning
very high incomes.
Exam Scores: On a
challenging exam, most
students score low, with
only a few achieving very
high scores.
Implications
For data interpretation, the
median is often a better
measure of central
tendency than the mean.
Outliers have a greater
impact on the mean than
on other measures.
Right-Skewed Distribution
Lower values Higher values

5. Negatively Skewed Distribution
What is a Negatively Skewed
Distribution?
A negatively skewed distribution, also known as a left-skewed
distribution, has a longer tail extending towards the left side. This
indicates that most values are clustered towards the higher end,
with a few smaller, extreme values pulling the mean towards the
left.
Visual Representation
Higher values (more frequent) Lower values (fewer frequent)
Key Properties
Tail Direction: The "tail" extends to the left side of the
distribution.
Central Tendency Relationship: The mean is typically less
than the median, which is less than the mode (Mean < Median <
Mode).
Peak Location: The peak of the distribution is towards the right
side.
Data Clustering: Data points are concentrated towards the
higher end of the range.
Real-World Examples
Age of Death
In many populations, most
people live to an older age
(e.g., 70-80 years), with fewer
dying at very young ages,
resulting in a left-skewed
distribution.
Easy Exam Scores
On an easy exam, most
students score very high, with
only a few scoring much lower
than average.

6. Coefficient of Sk ewness
What is Skewness?
Skewness is a fundamental statistical measure that quantifies the
asymmetry of a data distribution. Unlike symmetrical
distributions where data is evenly spread around the center,
skewed distributions have a longer "tail" on one side.
The coefficient of sk ewness provides a standardized way to
measure and compare this asymmetry across differ ent datasets.
Pearson's Coefficient of Sk ewness
Formula: 3 × (Mean - Median) / Standard
Deviation
This method is often preferred when:
The mode is ill-defined
The median is a more robust measure than the mode
Interpreting Skewness Coefficients
Negative
Value
Indicates negative
skewness (left-
skewed), meaning
the tail is longer
on the left side.
Zero Value
Indicates a
perfectly
symmetrical
distribution (e.g.,
normal
distribution).
Positive Value
Indicates positive
skewness (right-
skewed), meaning
the tail is longer
on the right side.
The coefficient of sk ewness provides a numerical measure of the
direction and degree of skew, which is crucial for understanding
the shape of a distribution and its implications for statistical
analysis.
Strong
Negative
Moderate
Negative
Symmetrical
(Normal)
Moderate
Positive
Strong
Positive
Skewness Coefficient

7. Kurtosis: Measuring Tailedness
What is Kurtosis?
Kurtosis is a measure of the "tailedness" of a
distribution, indicating how often outliers occur. It
describes the shape of the distribution's tails relative
to the tails of a normal distribution.
While sometimes described as "peakedness," its
primary focus is on the weight of the tails, which
helps identify the presence of outliers.
Why Kurtosis Matters
Identifies the pr esence and frequency of outliers
Evaluates the "tailedness" compared to normal
distribution
Important in financial analysis for risk management
Impacts statistical modeling and analysis selection
Kurtosis Visualization
Common Calculation Note
Statistical software often reports "excess kurtosis," which is (Kurtosis - 3),
making the normal distribution have an excess kurtosis of 0. This
standardization helps in comparing tailedness across differ ent
distributions.
Mesokurtic
Kurtosis ≈ 3
Normal distribution
Leptokurtic
Kurtosis > 3
Fatter tails, more outliers
Platykurtic
Kurtosis < 3
Thinner tails, fewer outliers

8. Types of Kurtosis
Kurtosis measures the "tailedness" of a distribution, indicating how often outliers occur relative to a normal distribution (Kurtosis ≈ 3).
Mesokurtic
Kurtosis: ≈ 3
Excess Kurtosis: 0
Characteristics:
Moderate tailedness
Similar to normal distribution
Outliers are neither frequent nor infrequent
Platykurtic
Kurtosis: < 3
Excess Kurtosis: Negative
Characteristics:
Thin tails
Flatter and wider at the peak
Outliers are less frequent than normal
Leptokurtic
Kurtosis: > 3
Excess Kurtosis: Positive
Characteristics:
Fat tails
Higher frequency of outliers
More peaked around the mean
Implications for Data Analysis
Statistical Testing
Kurtosis helps identify distributions where standard
statistical procedures might be invalid, leading to
incorrect conclusions.
Finance & Risk
Leptokurtic distributions (fat tails) imply higher
probability of extreme events than normal
distributions, crucial for risk management.
Model Selection
Understanding kurtosis helps select appropriate
statistical models and interpret results correctly
based on data characteristics.

9. Distribution Shapes in Practice
Understanding distribution shapes is crucial across various fields, as it pr ovides insights into data patterns and guides appropriate analytical
approaches.
Financial Analysis
Normal distributions are foundational in financial
modeling, though real data often exhibits skewness and
heavy tails.
Option pricing and risk assessment
Portfolio optimization under normal market conditions
Quality Control
Manufacturing processes often approximate normal
distributions, enabling statistical process control.
Setting upper and lower control limits
Identifying outliers as defects
Research Methodology
Many statistical tests assume normally distributed data,
making distribution shape a critical consideration.
Transformations to meet test assumptions
Non-parametric alternatives for non-normal data
Additional Applications
Distribution knowledge applies across diverse fields
where data patterns matter.
Biological Measurements
Height, weight, blood pressure
Environmental Data
Temperature, precipitation
Key Insight: Recognizing distribution types is essential for appropriate statistical modeling and interpretation.

10. Identifying Distribution Types
Visual Methods
Histograms
Bar charts showing frequency or density that can reveal symmetry,
skewness, and modality.
Density Plots
Smooth curves showing data distribution that help identify shape and
outliers.
Box Plots
Displays median, quartiles, and outliers that reveal symmetry and
skewness.
Q-Q Plots
Compares data to theoretical distributions, showing departures from
expected patterns.
Numerical Measures
Coefficient of Sk ewness
Quantifies asymmetry : 3 × (Mean - Median) / Standard Deviation
Negatively Skewed Symmetrical Positively Skewed
Kurtosis
Measures tailedness relative to normal distribution (excess kurtosis).
Platykurtic: Flatter tails, less outliers
Mesokurtic: Similar to normal distribution
Leptokurtic:Sharper peaks, more outliers
Common Applications
Statistical software for automatic identification
Data preprocessing for machine learning
Financial risk analysis (especially for fat tails)

11. Impact on Statistical Analysis
How Distribution Shapes Matter
Statistical Method Selection
Different distribution shapes require differ ent statistical
approaches. For example, normal distributions work well with
parametric tests like t-tests and ANOVA, while non-normal
distributions may require non-parametric alternatives.
Assumption Validity
Many statistical tests assume normally distributed data. Significant
skewness or kurtosis can invalidate these assumptions, leading to
incorrect conclusions.
Result Interpretation
Distribution shape affects how we interpr et statistical results. For
instance, in a leptokurtic distribution, extreme values are more
likely than in a normal distribution, which impacts how we view
outliers.
Practical Implications
Normality-Based
Methods
t-tests, ANOVA, linear
regression
Assumes symmetrical
distribution with no significant
outliers
Non-Parametric Tests
Mann-Whitney, Kruskal-Wallis,
sign test
More robust to skewness and
outliers
Key Considerations:
System performance analysis often involves non-normal distributions
(e.g., response times with long tails)
Financial data typically shows leptokurtic patterns with fat tails,
requiring differ ent modeling approaches
Understanding distribution shapes guides appropriate data
transformation and modeling strategies
Remember: The goal is not to "fix" non-nor mal data, but to
select appropriate methods that account for its characteristics.

12. Summary: Distribution Characteristics
Understanding the shape of a data distribution provides crucial insights into its characteristics and implications for analysis.
Feature / Distribution Type Symmetrical (Normal) Positively Skewed (Right-Skewed) Negatively Skewed (Left-Skewed)
Shape Bell-shaped Tail to the right Tail to the left
Mean, Median, Mode Mean = Median = Mode Mean > Median > Mode Mean < Median < Mode
Tail Length Equal on both sides Longer right tail Longer left tail
Outliers Infrequent, balanced More frequent high values More frequent low values
Skewness Coefficient 0 Positive Negative
Key Insights
Distribution shape matters for statistical analysis
Symmetry vs. skewness indicates data imbalance
Remember
Central tendency relationship indicates skew direction
Recognizing distribution types guides appropriate analysis