Section 6 - Chapter 2 - Introduction to Statistics Part II

Chapter 2 - Introduction to Statistics Part II
Section 6 – Statistics Analysis
Presented By :
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Agenda
 Introduction to Statistics Part 2
 Data Visualization
 Correlation
 Linear Regression
 Putting It All Together
 Microsoft Excel Functions Used

Anscombe’s Quartet
Presented By :

Key Facts:
1. Definition: Anscombe’s Quartet consists of four datasets that have nearly identical
statistical properties but display distinct distributions when graphed.
2. Creator: Francis Anscombe (1973).
3. Purpose: Demonstrates the importance of data visualization in statistical analysis.
4. Statistical Properties (for all four datasets):
o Mean of XXX: 9
o Mean of YYY: 7.5
o Variance of XXX: 11
o Variance of YYY: 4.12
o Correlation coefficient (rrr): 0.816
o Linear regression equation: y=3+0.5xy = 3 + 0.5xy=3+0.5x

Interpretation of Each Dataset:
1. Dataset I:
o Appears to follow a standard linear relationship.
o The regression model accurately fits the data.
2. Dataset II:
o Clearly follows a non-linear (quadratic) relationship.
o A straight-line regression is not appropriate.
3. Dataset III:
o Mostly linear but includes an outlier.
o The outlier distorts the regression results.
4. Dataset IV:
o Strong vertical outlier that drives correlation.
o The regression line is misleading due to a single influential point.

Comparison of the Four Datasets
Feature Dataset I Dataset II Dataset III Dataset IV
Linear Fit Good Poor (curved)
Affected by an
outlier
Distorted by an
extreme point
Outliers None None One influential point One extreme outlier
Suitable for Linear
Regression?
Yes No Somewhat No
Visualization
Needed?
Less critical Very important Very important Extremely important

Cheat Sheet:
Key Takeaways:
• Summary statistics alone can be misleading.
• Always visualize your data before making conclusions.
• Outliers and non-linearity significantly impact regression results.
• Anscombe’s Quartet warns against blind reliance on summary statistics and automated analyses.
Dataset Visual Pattern Key Insight
I Linear, with minimal scatter
Well-behaved dataset,
regression valid
II Non-linear (quadratic) Linear regression is misleading
III Has an outlier
One point strongly affects
regression
IV Vertical outlier One point distorts correlation

Histogram in Data Visualization
Presented By :

Key Facts:
1. Definition: A histogram is a graphical representation of the distribution of numerical
data using bars.
2. Purpose: Helps visualize the frequency of data points within specified intervals (bins).
3. X-axis: Represents the range of values (bins).
4. Y-axis: Represents the frequency (count) of occurrences in each bin.
5. Key Features:
o Shows the shape of the distribution (e.g., normal, skewed, bimodal).
o Helps identify outliers, trends, and data spread.
o Useful for understanding distributions before applying statistical models.

Cheat Sheet:
Concept Description
Bins Intervals that group data points
Skewness Asymmetry in data distribution (left/right skewed)
Kurtosis Measures how heavy/light the tails of a distribution are
Uniform Distribution Bars of roughly equal height
Normal Distribution Bell-shaped, symmetric around the mean
Bimodal Distribution Two peaks, indicating two different groups in the data
Right-Skewed Tail extends to the right (high values less frequent)
Left-Skewed Tail extends to the left (low values less frequent)

Interpretation of Histograms:
1. Symmetric (Normal Distribution)
o The highest frequency is in the center.
o Data is evenly distributed around the mean.
2. Right-Skewed Distribution (Positively Skewed)
o The tail extends to the right (higher values are rare).
o Example: Income distribution in a population.
3. Left-Skewed Distribution (Negatively Skewed)
o The tail extends to the left (lower values are rare).
o Example: Age of retirement (most retire later, few retire very young).

Interpretation of Histograms:
4. Bimodal Distribution
o Two peaks indicate two dominant groups.
o Example: Exam scores of two distinct student groups.
5. Uniform Distribution
o All bins have similar heights, meaning data is evenly spread.
o Example: Rolling a fair die multiple times.

Comparison of Histograms vs. Other Charts:
Feature Histogram Bar Chart Box Plot Density Plot
Data Type Numerical Categorical Numerical Numerical
X-Axis Ranges/Bins Categories Single Variable Continuous Values
Shows Distribution? Yes No Yes Yes
Shows Outliers? Limited No Yes Yes
Best For?
Frequency
Distribution
Comparing
Categories
Data Spread &
Outliers
Smoother Data
Distribution
Key Takeaways:
✅ Histograms help understand the shape, spread, and patterns of data.
✅ Choice of bin size affects histogram appearance (too few = oversimplified, too many = noisy).
✅ Great for checking normality, skewness, and trends before deeper statistical analysis.
✅ Not ideal for categorical data—use a bar chart instead.

Box-and-Whisker Plot (Boxplot)
Presented By :

Key Facts:
1. Definition: A boxplot is a graphical summary of data distribution showing median, quartiles, and
potential outliers.
2. Purpose: Helps visualize spread, central tendency, and variability of numerical data.
3. Components:
o Box: Represents the interquartile range (IQR: Q1 to Q3).
o Whiskers: Extend from the box to show variability outside the quartiles.
o Median Line: Inside the box, marks the middle value.
o Outliers: Individual points beyond whiskers (potential anomalies).
4. Used For:
o Identifying skewness and spread of data.
o Comparing multiple distributions.
o Spotting outliers. This Content is Copyright Reserved Rights Copyright 2025@PTAIndia

Cheat Sheet:
Term Description
Minimum (Lower Whisker) Smallest data point within 1.5 × IQR from Q1
Q1 (First Quartile, 25%) 25% of data falls below this value
Median (Q2, 50%) Middle value of the dataset
Q3 (Third Quartile, 75%) 75% of data falls below this value
Maximum (Upper Whisker) Largest data point within 1.5 × IQR from Q3
Interquartile Range (IQR) Q3 − Q1 (middle 50% of data)
Outliers Points beyond 1.5 × IQR from Q1 or Q3

Interpretation of Boxplots:
1. Symmetric Distribution:
o Median is centered in the box.
o Whiskers are of roughly equal length.
o Suggests a normal or balanced distribution.
2. Right-Skewed Distribution (Positively Skewed):
o Median is closer to Q1.
o Upper whisker is longer.
o Suggests more high-value outliers.

Interpretation of Boxplots:
3. Left-Skewed Distribution (Negatively Skewed):
o Median is closer to Q3.
o Lower whisker is longer.
o Suggests more low-value outliers.
4. Presence of Outliers:
o Individual dots outside the whiskers indicate extreme values.
o Could suggest measurement errors or significant variations in data.
Key Takeaways:
✅ Boxplots are great for comparing multiple distributions side by side.
✅ They quickly reveal outliers, skewness, and spread in data.
✅ Unlike histograms, boxplots do not show exact frequency distribution.
✅ Violin plots offer a more detailed alternative to boxplots by displaying density.

Comparison: Boxplot vs. Other Charts
Feature Boxplot Histogram Violin Plot Bar Chart
Data Type Numerical Numerical Numerical Categorical
Shows Distribution? Yes Yes Yes No
Shows Outliers? Yes Limited Yes No
Shows Exact
Frequency?
No Yes No Yes
Best For?
Comparing
distributions,
spotting outliers
Understanding
frequency
distribution
Detailed shape of
distribution
Comparing
categorical data

Scatterplot
Presented By :

Scatterplot
Key Facts:
1. Definition: A scatterplot is a graph that displays individual data points using dots to
show relationships between two numerical variables.
2. Purpose: Used to identify patterns, trends, correlations, and potential outliers in data.
3. Axes:
o X-axis: Independent variable.
o Y-axis: Dependent variable.
4. Key Features:
o Reveals correlations (positive, negative, or none).
o Helps detect outliers.
o Shows clusters or gaps in data.

Scatterplot

Scatterplot
Cheat Sheet:
Feature Description
Positive Correlation As X increases, Y increases (upward trend).
Negative Correlation As X increases, Y decreases (downward trend).
No Correlation No clear pattern in the data points.
Strong Correlation Points are closely clustered around a trend.
Weak Correlation Points are widely spread but still follow a trend.
Outliers Points that are far from the general trend.
Clusters Groups of points indicating subgroups in the data.

Scatterplot
Interpretation of Scatterplots:
1. Strong Positive Correlation:
o Points form a tight upward trend.
o Example: Height vs. weight.
2. Strong Negative Correlation:
o Points form a tight downward trend.
o Example: Age of a car vs. resale value.
3. Weak Correlation (Positive or Negative):
o Points loosely follow a trend but are widely spread.
o Example: Studying hours vs. test scores (if other factors influence performance).

Scatterplot
Interpretation of Scatterplots:
4. No Correlation:
o Points are randomly scattered with no trend.
o Example: Shoe size vs. intelligence.
5. Outliers Present:
o Some points are far away from the general pattern.
o Example: A single student with extremely high or low test scores.
6. Clusters in Data:
o Indicates subgroups or different categories in the dataset.
o Example: Income vs. age showing different groups for students, professionals, and
retirees.

Scatterplot
Comparison: Scatterplot vs. Other Charts
Feature Scatterplot Line Chart Bar Chart Bubble Chart
Data Type
Numerical vs.
Numerical
Numerical vs.
Numerical (Trend
Over Time)
Categorical vs.
Numerical
Numerical vs.
Numerical (with extra
variable)
Shows Relationship? Yes Yes (over time) No Yes
Shows Trends? Yes Yes No Yes
Shows Outliers? Yes Limited No Yes
Best For?
Exploring
relationships &
correlations
Time-series trends
Comparing
categories
Comparing
relationships with an
extra dimension

Correlation in Financial Markets
Presented By :

Key Facts
1. Definition: Correlation measures the statistical relationship between two financial
assets, indicating how they move relative to each other.
2. Correlation Coefficient (rrr):
o Ranges from -1 to +1.
o +1: Perfect positive correlation (assets move in the same direction).
o 0: No correlation (assets move independently).
o -1: Perfect negative correlation (assets move in opposite directions).

Key Facts
3. Types of Correlation in Finance:
o Positive Correlation: Stocks in the same sector (e.g., Apple & Microsoft).
o Negative Correlation: Stocks vs. safe-haven assets (e.g., S&P 500 & Gold).
o Zero Correlation: Unrelated assets (e.g., Bitcoin & oil prices).
4. Importance in Investing:
o Helps with portfolio diversification.
o Identifies hedging opportunities.
o Assists in risk management.
5. Commonly Used in:
o Stocks vs. Bonds: Typically negatively correlated.
o Cryptocurrency & Stocks: Often weak correlation but varies in crises.
o Commodities vs. Equities: Gold often negatively correlates with equities.

Cheat Sheet
Correlation Type rrr Value Range Meaning Example
Perfect Positive R =1.0 Move in the same direction Nasdaq & S&P 500
Strong Positive R = 0.70 to 1.0 Mostly move together Oil & Energy Stocks
Moderate Positive R= 0.40 to 0.70 Some relationship USD & U.S. Treasury Bonds
Weak Positive R = 0.10 to 0.40 Limited connection Real Estate & Stocks
No Correlation R = 0 No consistent relationship Bitcoin & Natural Gas
Weak Negative R = - 0.10 to - 0.40 Limited inverse relationship Tech Stocks & Gold
Moderate Negative R = - 0.40 to - 0.70 Often move opposite Stocks & Bonds
Strong Negative R = - 0.70 to – 1.0 Almost always inverse USD & Emerging Markets
Perfect Negative R = 1.0
Always move in opposite
directions
VIX (Volatility Index) & S&P
500

Interpretation of Correlation in Financial Markets
1. High Positive Correlation (r>0.7):
o Assets move together; not good for diversification.
o Example: Tech stocks (Apple & Google).
2. Moderate Positive Correlation (R = 0.40 to 0.70):
o Partial dependence; still some diversification benefits.
o Example: Crude oil & energy sector stocks.
3. Near Zero Correlation (R = 0):
o No predictable relationship; good for diversification.
o Example: Bitcoin & S&P 500 (historically, but fluctuates over time).

Interpretation of Correlation in Financial Markets
4. Moderate Negative Correlation (R = −0.4 to −0.7):
o Helps hedge against losses.
o Example: Stocks & Bonds in a normal market.
5. Strong Negative Correlation (R Below – 0.70):
o Ideal for risk management and hedging strategies.
o Example: VIX (Volatility Index) & Stock Market—VIX rises when stocks fall.

Comparison: Correlation vs. Other Financial Metrics.
Metric Measures Range Best For
Correlation (rrr)
Relationship between asset
movements
-1 to +1 Portfolio Diversification
Beta (β) Sensitivity to the overall market Any value Risk & Volatility
Volatility Price fluctuations over time 0 to ∞ Risk Management
Sharpe Ratio Risk-adjusted returns Any value Portfolio Efficiency
Covariance
Direction of movement, not
strength
Any value Initial Relationship Analysis

Key Takeaways
✅ Correlation helps investors balance portfolios by combining assets that
behave differently.
✅ Negative correlation assets reduce risk (e.g., bonds & stocks).
✅ High correlation limits diversification, increasing vulnerability to market
downturns.
✅ Correlation changes over time, especially during financial crises.
✅ Understanding correlation is essential for risk management and asset
allocation.

Correlation vs. Causation
Presented By :

Key Facts
1. Correlation: Measures the statistical relationship between two variables (how they move
together).
2. Causation: Indicates that one variable directly affects the other.
3. Key Difference: Correlation does not imply causation. Just because two variables move
together does not mean one causes the other.
4. Examples in Finance:
o Correlation: Stock prices and interest rates may move together but are influenced by external
factors.
o Causation: A central bank’s interest rate hike directly affects loan costs, causing businesses to
borrow less.
5. Spurious Correlation: When two variables appear related but are actually influenced by an
unrelated third factor.

Interpretation: How to Distinguish Correlation from Causation
1. Observe the Data Relationship:
o Strong correlation does not automatically imply one variable is driving the other.
2. Look for a Logical Explanation:
o Does a clear mechanism explain why one variable influences the other?
3. Check for Confounding Variables:
o Is there a third variable affecting both?
o Example: A rise in stock market & luxury car sales—driven by economic growth, not direct causation.
4. Use Time-Series Data:
o If changes in A always precede changes in B, causation is more likely.
5. Conduct Controlled Experiments:
o In non-financial fields (medicine, science), controlled experiments confirm causality.

Cheat Sheet: Correlation vs. Causation
Aspect Correlation Causation
Definition
Measures relationship between
two variables
One variable directly causes the
other to change
Direction of Influence
No direction (A → B or B → A or
both)
One variable influences the other
(A → B)
Proven Relationship? No, just association Yes, direct cause-effect
Example in Finance
Stock market & oil prices moving
together
Interest rate hike causing lower
loan demand
Example in Health
People who exercise more tend
to weigh less
Eating excess calories causes
weight gain
Can Be Spurious? Yes No
Proven By?
Statistical analysis (correlation
coefficient)
Experiments, controlled studies,
logical reasoning

Comparison: Correlation vs. Causation vs. Coincidence
Feature Correlation Causation Coincidence
Relationship Type Statistical association Direct cause-effect Random occurrence
Example
Stock market & GDP
growth move together
Interest rate cuts lead to
more borrowing
Number of movies
featuring cats & stock
market returns both rise
Proof Needed?
Statistical correlation
coefficient
Logical explanation,
experiments
No pattern or link
Common Mistake?
Assuming one causes
the other
Ignoring correlation
Believing unrelated
events are connected

Linear Regression
Presented By :

Linear Regression
Key Facts
1. Definition: Linear regression is a statistical method for modeling the relationship between
a dependent variable (YYY) and one or more independent variables (XXX).
2. Equation of Simple Linear Regression:
3. Types of Linear Regression:
o Simple Linear Regression: One independent variable.
o Multiple Linear Regression: Multiple independent variables.

Linear Regression
4. Assumptions:
o Linearity: Relationship between XXX and YYY is linear.
o Independence: Data points are independent.
o Homoscedasticity: Variance of residuals is constant.
o No Multicollinearity: Independent variables in multiple regression are not highly correlated.
o Normality of Residuals: Errors follow a normal distribution.
5. Applications:
o Finance: Stock price prediction, risk modeling.
o Economics: Demand forecasting, GDP estimation.
o Marketing: Sales forecasting, customer behavior analysis.
o Healthcare: Disease progression modeling.

Linear Regression

Linear Regression
Cheat Sheet
Term Description
Intercept Value of YYY when X=0X = 0X=0
Slope How much YYY changes per unit increase in XXX
R-Squared Goodness of fit (how well the model explains variance in YYY)
Adjusted Adjusted for number of predictors in multiple regression
P-value Tests statistical significance of predictors (typically < 0.05)
Residuals Differences between actual and predicted values
Multicollinearity
High correlation among independent variables (causes
instability)
Overfitting
Model learns noise instead of real trends (happens with too
many predictors)

Linear Regression
Comparison: Linear Regression vs. Other Models
Feature Linear Regression Logistic Regression Decision Tree Neural Network
Output Type
Continuous
(numeric)
Binary/Categorical
Discrete or
continuous
Discrete or
continuous
Relationship Linear Non-linear Non-linear Complex patterns
Interpretability High Moderate Low Very Low
Computational Cost Low Low Medium High
Handles Outliers
Well?
No No Yes Yes
Handles
Multicollinearity
Well?
No Yes Yes Yes

Linear Regression
Interpretation of Linear Regression Outputs
1. Slope
o If positive, YYY increases as XXX increases.
o If negative, YYY decreases as XXX increases.
2. Intercept
o The expected value of YYY when X=0X = 0X=0.
o Sometimes not meaningful (e.g., predicting salary when years of experience = 0).
3. Coefficient of Determination – R Square
o Measures how well the independent variable(s) explain the variance in YYY.
o R2=1R^2 = 1R2=1 → Perfect fit (rare in real-world data).
o R2=0R^2 = 0R2=0 → No relationship.

Linear Regression
Interpretation of Linear Regression Outputs
o Rule of thumb:
 R2>0.7 → Strong fit.
 R2 = 0.30 to 0.70. → Moderate fit.
 R2<0.3 → Weak fit.
4. P-Value
o If p < 0.05, the predictor is statistically significant.
o If p > 0.05, the predictor may not be meaningful.
5. Residuals & Homoscedasticity
o Residuals should be randomly distributed.
o A funnel shape suggests heteroscedasticity (violates assumptions).

Linear Regression
Key Takeaways
✅ Linear regression is a powerful, interpretable model for predicting
numerical values.
✅ Best used when variables have a linear relationship and
assumptions hold.
✅ Multiple regression extends it to multiple predictors but requires
checking for multicollinearity.
✅ Compared to non-linear models, it is computationally efficient but
may not capture complex relationships.

Next Chapter 3 - Introduction to Probability
Section 6 – Statistics Analysis
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Section 6 - Chapter 2 - Introduction to Statistics Part II

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