Distribution Analysis

📊 Distribution Analysis in R: Your Data Detective Toolkit

Imagine you’re a detective. Your job? To understand the story hidden inside numbers. Today, we’ll learn three super-powers that help us crack the case: Quantiles, Correlation, and Scaling!

🎯 The Big Picture

Think of your data like a classroom of kids standing in a line from shortest to tallest.

• Quantiles help you find who’s in the middle, who’s really tall, and who’s really short.
• Correlation tells you if tall kids also tend to have big feet (do two things go together?).
• Scaling is like converting everyone’s height to the same measuring tape so we can compare fairly.

📏 Part 1: Quantile Functions — Finding the Landmarks

What Are Quantiles?

Imagine 100 kids standing in a line from shortest to tallest. Quantiles are like signposts telling you where you are in the line.

• 25th percentile (Q1): 25 kids are shorter than you
• 50th percentile (Median): You’re right in the middle!
• 75th percentile (Q3): 75 kids are shorter than you

🧪 R Example: Finding Quantiles

# Heights of 10 students (in cm)
heights <- c(120, 125, 130, 135, 140,
             145, 150, 155, 160, 165)

# Find the median (50th percentile)
median(heights)
# Result: 142.5

# Find Q1, Median, Q3
quantile(heights, c(0.25, 0.50, 0.75))
#   25%   50%   75%
# 128.75 142.5 156.25

📦 The quantile() Function

quantile(x, probs)

• x = your data
• probs = which percentiles you want (0 to 1)

🎁 Special Quantiles You’ll Use Often

🔍 Quick Summary with fivenum()

fivenum(heights)
# [1] 120 128.75 142.5 156.25 165

This gives you: Min, Q1, Median, Q3, Max — all at once!

🔗 Part 2: Correlation and Covariance — Do Things Move Together?

The Story

Imagine two friends: Height and Shoe Size. When one friend gets bigger, does the other friend also get bigger? That’s what correlation measures!

Three Types of Relationships

graph TD
    A["Two Variables"] --> B["Positive Correlation"]
    A --> C["Negative Correlation"]
    A --> D["No Correlation"]
    B --> E["📈 Both go up together"]
    C --> F["📉 One up, other down"]
    D --> G["🎲 No pattern"]

🧪 R Example: Correlation

# Heights and shoe sizes of 5 people
height <- c(150, 160, 170, 180, 190)
shoe_size <- c(6, 7, 8, 9, 10)

# Calculate correlation
cor(height, shoe_size)
# Result: 1 (perfect positive!)

Understanding Correlation Values

🎯 Covariance: Correlation’s Cousin

Covariance also measures if things move together, but it’s in the original units (harder to interpret).

# Calculate covariance
cov(height, shoe_size)
# Result: 50

# Correlation is easier to understand!
# It's always between -1 and +1

When to Use Which?

• Correlation (cor()): When you want to know HOW STRONG the relationship is (always -1 to +1)
• Covariance (cov()): When you need the actual units (used in advanced math)

📊 Correlation Matrix: Many Variables at Once

# Three variables
age <- c(25, 30, 35, 40, 45)
income <- c(30, 45, 60, 75, 90)
savings <- c(5, 12, 20, 30, 42)

# Create a data frame
data <- data.frame(age, income, savings)

# See all correlations at once!
cor(data)
#           age   income  savings
# age      1.00    1.00    1.00
# income   1.00    1.00    1.00
# savings  1.00    1.00    1.00

⚖️ Part 3: Scaling Data — Making Fair Comparisons

The Problem

Imagine comparing:

• A test score: 85 out of 100
• A race time: 12 seconds
• A weight: 50 kg

How do you compare these? They’re all in different units! Scaling puts everything on the same measuring stick.

Two Popular Scaling Methods

graph TD
    A["Scaling Methods"] --> B["Z-Score / Standardization"]
    A --> C["Min-Max Normalization"]
    B --> D["Mean = 0, SD = 1"]
    C --> E["Range 0 to 1"]

🧪 Z-Score Scaling with scale()

The Z-score tells you: “How many steps away from average are you?”

# Test scores
scores <- c(70, 80, 90, 100, 110)

# Scale the data
scaled_scores <- scale(scores)
print(scaled_scores)
#            [,1]
# [1,] -1.2649111
# [2,] -0.6324555
# [3,]  0.0000000
# [4,]  0.6324555
# [5,]  1.2649111

Understanding Z-Scores

🎯 Min-Max Scaling: 0 to 1

This squishes all values between 0 and 1.

# Custom min-max function
min_max <- function(x) {
  (x - min(x)) / (max(x) - min(x))
}

# Apply it
scores <- c(70, 80, 90, 100, 110)
min_max(scores)
# [1] 0.00 0.25 0.50 0.75 1.00

When to Use Each?

🎯 Quick Reference: All Functions

# QUANTILES
quantile(x, 0.5)    # Get median
quantile(x, c(0.25, 0.75))  # Get Q1 and Q3
fivenum(x)          # Min, Q1, Median, Q3, Max

# CORRELATION & COVARIANCE
cor(x, y)           # Correlation (-1 to +1)
cov(x, y)           # Covariance (original units)
cor(data_frame)     # Correlation matrix

# SCALING
scale(x)            # Z-score standardization
# Custom min-max: (x - min) / (max - min)

🏆 You Did It!

You just learned three powerful detective tools:

1. Quantiles: Find where values stand in the lineup
2. Correlation: Discover if things move together
3. Scaling: Put everything on the same measuring stick

Now you can analyze any dataset like a pro! 🎉

🧠 Remember This Story

A teacher wanted to understand her class better. She used quantiles to find who scored in the top 25%. She checked correlation to see if study hours predicted test scores. Finally, she used scaling to fairly compare math and art scores. The end!