Topic 1

📊 What is Statistics & Why It Matters

The science of collecting, organizing, analyzing, and interpreting data

Introduction

What is it? Statistics is a branch of mathematics that deals with data. It provides methods to make sense of numbers and help us make informed decisions based on evidence rather than guesswork.

Why it matters: From business forecasting to medical research, sports analysis to government policy, statistics powers nearly every decision in our modern world.

When to use it: Whenever you need to understand patterns, test theories, make predictions, or draw conclusions from data.

💡 REAL-WORLD EXAMPLE

Imagine Netflix deciding what shows to produce. They analyze viewing statistics: what genres people watch, when they pause, what they finish. Statistics transforms millions of data points into actionable insights like "Create more thriller series" or "Release episodes on Fridays."

Two Branches of Statistics

Descriptive Statistics

  • Summarizes and describes data
  • Uses charts, graphs, averages
  • Example: "Average class score is 85"

Inferential Statistics

  • Makes predictions and inferences
  • Tests hypotheses
  • Example: "New teaching method improves scores"

Use Cases & Applications

  • Healthcare: Clinical trials testing new drugs, disease outbreak tracking
  • Business: Customer behavior analysis, sales forecasting, A/B testing
  • Government: Census data, economic indicators, policy impact assessment
  • Sports: Player performance metrics, game strategy optimization

🎯 Key Takeaways

  • Statistics transforms raw data into meaningful insights
  • Two main branches: Descriptive (what happened) and Inferential (what will happen)
  • Essential for decision-making across all fields
  • Combines mathematics with real-world problem solving
Topic 2

👥 Population vs Sample

Understanding the difference between the entire group and a subset

Introduction

What is it? A population includes ALL members of a defined group. A sample is a subset selected from that population.

Why it matters: It's usually impossible or impractical to study entire populations. Sampling allows us to make inferences about large groups by studying smaller representative groups.

When to use it: Use populations when you can access all data; use samples when populations are too large, expensive, or time-consuming to study.

💡 REAL-WORLD ANALOGY

Think of tasting soup. You don't need to eat the entire pot (population) to know if it needs salt. A single spoonful (sample) gives you a good idea—as long as you stirred it well first!

Interactive Visualization

Key Differences

Aspect Population Sample
Size Entire group (N) Subset (n)
Symbol N (uppercase) n (lowercase)
Cost High Lower
Time Long Shorter
Accuracy 100% (if measured correctly) Has sampling error
⚠️ COMMON MISTAKE

Biased Sampling: If your sample doesn't represent the population, your conclusions will be wrong. Example: Surveying only morning shoppers at a store will miss evening customer patterns.

✅ PRO TIP

For a sample to be representative, use random sampling. Every member of the population should have an equal chance of being selected.

🎯 Key Takeaways

  • Population (N): All members of a defined group
  • Sample (n): A subset selected from the population
  • Good samples are random and representative
  • Larger samples generally provide better estimates
Topic 3

📈 Parameters vs Statistics

Population measures vs sample measures

Introduction

What is it? A parameter is a numerical characteristic of a population. A statistic is a numerical characteristic of a sample.

Why it matters: We usually can't measure parameters directly (populations are too large), so we estimate them using statistics from samples.

When to use it: Parameters are what we want to know; statistics are what we can calculate.

💡 REAL-WORLD EXAMPLE

You want to know the average height of all students in your country (parameter). You can't measure everyone, so you measure 1,000 students (sample) and calculate their average height (statistic) to estimate the population parameter.

Common Parameters and Statistics

Measure Parameter (Population) Statistic (Sample)
Mean (Average) μ (mu) x̄ (x-bar)
Standard Deviation σ (sigma) s
Variance σ² s²
Proportion p p̂ (p-hat)
Size N n

The Relationship

Key Concept

Statistic → Estimates → Parameter

We use statistics (calculated from samples) to estimate parameters (unknown population values).

📊 EXAMPLE

Scenario: A factory wants to know the average weight of cereal boxes.

  • Population: All cereal boxes produced (millions)
  • Parameter: μ = true average weight of ALL boxes (unknown)
  • Sample: 100 randomly selected boxes
  • Statistic: x̄ = 510 grams (calculated from the 100 boxes)
  • Inference: We estimate μ ≈ 510 grams
⚠️ COMMON MISTAKE

Confusing symbols! Greek letters (μ, σ, ρ) refer to parameters (population). Roman letters (x̄, s, r) refer to statistics (sample).

🎯 Key Takeaways

  • Parameter: Describes a population (usually unknown)
  • Statistic: Describes a sample (calculated from data)
  • Greek letters = population, Roman letters = sample
  • Statistics are used to estimate parameters
Topic 4

🔢 Types of Data

Categorical, Numerical, Discrete, Continuous, Ordinal, Nominal

Introduction

What is it? Data comes in different types, and understanding these types determines which statistical methods you can use.

Why it matters: Using the wrong analysis method for your data type leads to incorrect conclusions. You can't calculate an average of colors!

When to use it: Before any analysis, identify your data type to choose appropriate statistical techniques.

Data Type Hierarchy

DATA
CATEGORICAL
NUMERICAL
Nominal
Ordinal
Discrete
Continuous

Categorical Data

Represents categories or groups (qualitative)

Nominal

Categories with NO order

  • Colors: Red, Blue, Green
  • Gender: Male, Female, Non-binary
  • Country: USA, India, Japan
  • Blood Type: A, B, AB, O

Ordinal

Categories WITH meaningful order

  • Education: High School < Bachelor's < Master's
  • Satisfaction: Poor < Fair < Good < Excellent
  • Medal: Bronze < Silver < Gold
  • Size: Small < Medium < Large

Numerical Data

Represents quantities (quantitative)

Discrete

Countable, specific values only

  • Number of students: 25, 30, 42
  • Number of cars: 0, 1, 2, 3...
  • Dice roll: 1, 2, 3, 4, 5, 6
  • Number of children: 0, 1, 2, 3...

Can't have 2.5 students!

Continuous

Can take any value in a range

  • Height: 165.3 cm, 180.7 cm
  • Weight: 68.5 kg, 72.3 kg
  • Temperature: 23.4°C, 24.7°C
  • Time: 3.25 seconds

Infinite precision possible

💡 QUICK TEST

Ask yourself:

  1. Is it a label/category? → Categorical
  2. Is it a number? → Numerical
  3. Can you count it? → Discrete
  4. Can you measure it? → Continuous
  5. Does order matter? → Ordinal (else Nominal)
📊 EXAMPLES
Data Type Reason
Zip codes Categorical (Nominal) Numbers used as labels, not quantities
Test scores (A, B, C, D, F) Categorical (Ordinal) Categories with clear order
Number of pages in books Numerical (Discrete) Countable whole numbers
Reaction time in milliseconds Numerical (Continuous) Can be measured to any precision
⚠️ COMMON MISTAKE

Just because something is written as a number doesn't make it numerical! Phone numbers, jersey numbers, and zip codes are categorical because they identify categories, not quantities.

🎯 Key Takeaways

  • Categorical: Labels/categories (Nominal: no order, Ordinal: has order)
  • Numerical: Quantities (Discrete: countable, Continuous: measurable)
  • Data type determines which statistical methods to use
  • Always identify data type before analysis
Topic 5

📍 Measures of Central Tendency

Mean, Median, Mode - Finding the center of data

Introduction

What is it? Measures of central tendency are single values that represent the "center" or "typical" value in a dataset.

Why it matters: Instead of looking at hundreds of numbers, one central value summarizes the data. "Average salary" tells you more than listing every employee's salary.

When to use it: When you need to summarize data with a single representative value.

💡 REAL-WORLD ANALOGY

Imagine finding the "center" of a group of people standing on a field. Mean is like finding the balance point where they'd balance on a seesaw. Median is literally the middle person. Mode is where the most people are clustered together.

Mathematical Foundations

Mean (Average)
μ = Σx n

Where:

  • μ (mu) = population mean or x̄ (x-bar) = sample mean
  • Σx = sum of all values
  • n = number of values

Steps:

  1. Add all values together
  2. Divide by the count of values
Median (Middle Value)

If odd number of values: Middle value

If even number of values: Average of two middle values

Steps:

  1. Sort values in ascending order
  2. Find the middle position: (n + 1) / 2
  3. If between two values, average them
Mode (Most Frequent)

The value(s) that appear most frequently

Types:

  • Unimodal: One mode
  • Bimodal: Two modes
  • Multimodal: More than two modes
  • No mode: All values appear equally

Interactive Calculator

Mean: 30
Median: 30
Mode: None
📊 WORKED EXAMPLE

Dataset: Test scores: 65, 70, 75, 80, 85, 90, 95

Mean:

Sum = 65 + 70 + 75 + 80 + 85 + 90 + 95 = 560

Mean = 560 / 7 = 80

Median:

Already sorted. Middle position = (7 + 1) / 2 = 4th value

Median = 80

Mode:

All values appear once. No mode

When to Use Which?

Use Mean

  • Data is symmetrical
  • No extreme outliers
  • Numerical data
  • Need to use all data points

Use Median

  • Data has outliers
  • Data is skewed
  • Ordinal data
  • Need robust measure

Use Mode

  • Categorical data
  • Finding most common value
  • Discrete data
  • Multiple peaks in data
⚠️ COMMON MISTAKE

Mean is affected by outliers! In salary data like $30K, $35K, $40K, $45K, $500K, the mean is $130K (misleading!). The median of $40K better represents typical salary.

✅ PRO TIP

For skewed data (like income, house prices), always report the median along with the mean. If they're very different, your data has outliers or is skewed!

🎯 Key Takeaways

  • Mean: Sum of all values divided by count (affected by outliers)
  • Median: Middle value when sorted (resistant to outliers)
  • Mode: Most frequent value (useful for categorical data)
  • Choose the measure that best represents your data type and distribution
Topic 6

⚡ Outliers

Extreme values that don't fit the pattern

Introduction

What is it? Outliers are data points that are significantly different from other observations in a dataset.

Why it matters: Outliers can indicate data errors, special cases, or important patterns. They can also severely distort statistical analyses.

When to use it: Always check for outliers before analyzing data, especially when calculating means and standard deviations.

💡 REAL-WORLD EXAMPLE

In a salary dataset for entry-level employees: $35K, $38K, $40K, $37K, $250K. The $250K is an outlier—maybe it's a data entry error (someone added an extra zero) or a special case (CEO's child). Either way, it needs investigation!

Detection Methods

IQR Method

Most common approach:

  • Calculate Q1, Q3, and IQR = Q3 - Q1
  • Lower fence = Q1 - 1.5 × IQR
  • Upper fence = Q3 + 1.5 × IQR
  • Outliers fall outside fences

Z-Score Method

For normal distributions:

  • Calculate z-score for each value
  • z = (x - μ) / σ
  • If |z| > 3: definitely outlier
  • If |z| > 2: possible outlier
⚠️ COMMON MISTAKE

Never automatically delete outliers! They might be: (1) Valid extreme values, (2) Data entry errors, (3) Important discoveries. Always investigate before removing.

🎯 Key Takeaways

  • Outliers are extreme values that differ significantly from other data
  • Use IQR method (1.5 × IQR rule) or Z-score method to detect
  • Mean is heavily affected by outliers; median is resistant
  • Always investigate outliers before deciding to keep or remove
Topic 7

📏 Variance & Standard Deviation

Measuring spread and variability in data

Introduction

What is it? Variance measures the average squared deviation from the mean. Standard deviation is the square root of variance.

Why it matters: Shows how spread out data is. Low values mean data is clustered; high values mean data is scattered.

When to use it: Whenever you need to understand data variability—in finance (risk), manufacturing (quality control), or research (reliability).

Mathematical Formulas

Population Variance (σ²)
σ² = Σ(x - μ)² / N

Where N = population size, μ = population mean

Sample Variance (s²)
s² = Σ(x - x̄)² / (n - 1)

Where n = sample size, x̄ = sample mean. We use (n-1) for unbiased estimation.

Standard Deviation
σ = √(variance)

Same units as original data, easier to interpret

📊 WORKED EXAMPLE

Dataset: [4, 8, 6, 5, 3, 7]

Step 1: Mean = (4+8+6+5+3+7)/6 = 5.5

Step 2: Deviations: [-1.5, 2.5, 0.5, -0.5, -2.5, 1.5]

Step 3: Squared: [2.25, 6.25, 0.25, 0.25, 6.25, 2.25]

Step 4: Sum = 17.5

Step 5: Variance = 17.5/(6-1) = 3.5

Step 6: Std Dev = √3.5 = 1.87

🎯 Key Takeaways

  • Variance measures average squared deviation from mean
  • Standard deviation is square root of variance (same units as data)
  • Use (n-1) for sample variance to avoid bias
  • Higher values = more spread; lower values = more clustered
Topic 8

🎯 Quartiles & Percentiles

Dividing data into equal parts

Introduction

What is it? Quartiles divide sorted data into 4 equal parts. Percentiles divide data into 100 equal parts.

Why it matters: Shows relative position in a dataset. "90th percentile" means you scored better than 90% of people.

The Five-Number Summary

  • Minimum: Smallest value
  • Q1 (25th percentile): 25% of data below this
  • Q2 (50th percentile/Median): Middle value
  • Q3 (75th percentile): 75% of data below this
  • Maximum: Largest value
💡 REAL-WORLD EXAMPLE

SAT scores: If you score 1350 and that's the 90th percentile, it means you scored higher than 90% of test-takers. Percentiles are perfect for standardized tests!

🎯 Key Takeaways

  • Q1 = 25th percentile, Q2 = median, Q3 = 75th percentile
  • Percentiles show relative standing in a dataset
  • Five-number summary: Min, Q1, Q2, Q3, Max
  • Useful for understanding data distribution
Topic 9

📦 Interquartile Range (IQR)

Middle 50% of data and outlier detection

Introduction

What is it? IQR = Q3 - Q1. It represents the range of the middle 50% of your data.

Why it matters: IQR is resistant to outliers and is the foundation of the 1.5×IQR rule for outlier detection.

The 1.5 × IQR Rule

Outlier Boundaries
Lower Fence = Q1 - 1.5 × IQR
Upper Fence = Q3 + 1.5 × IQR

Any value outside these fences is considered an outlier

🎯 Key Takeaways

  • IQR = Q3 - Q1 (range of middle 50% of data)
  • Resistant to outliers (unlike standard deviation)
  • 1.5×IQR rule: standard method for outlier detection
  • Box plots visualize IQR and outliers
Topic 10

📉 Skewness

Understanding data distribution shape

Introduction

What is it? Skewness measures the asymmetry of a distribution.

Why it matters: Indicates whether data leans left or right, affecting which statistical methods to use.

Types of Skewness

Negative (Left) Skew

Tail extends to the left

Mean < Median < Mode

Example: Test scores when most students do well

Symmetric (No Skew)

Perfectly balanced

Mean = Median = Mode

Example: Normal distribution

Positive (Right) Skew

Tail extends to the right

Mode < Median < Mean

Example: Income data, house prices

🎯 Key Takeaways

  • Skewness measures asymmetry in distribution
  • Negative skew: tail to left, Mean < Median
  • Positive skew: tail to right, Mean > Median
  • Symmetric: Mean = Median = Mode
Topic 11

🔗 Covariance

How two variables vary together

Introduction

What is it? Covariance measures how two variables change together.

Why it matters: Shows if variables have a positive, negative, or no relationship.

Formula

Sample Covariance
Cov(X,Y) = Σ(xᵢ - x̄)(yᵢ - ȳ) / (n-1)

Interpretation

  • Positive: Variables increase together
  • Negative: One increases as other decreases
  • Zero: No linear relationship
  • Problem: Scale-dependent, hard to interpret magnitude

🎯 Key Takeaways

  • Covariance measures joint variability of two variables
  • Positive: variables move together; Negative: inverse relationship
  • Scale-dependent (unlike correlation)
  • Foundation for correlation calculation
Topic 12

💞 Correlation

Standardized measure of relationship strength

Introduction

What is it? Correlation coefficient (r) is a standardized measure of linear relationship between two variables.

Why it matters: Always between -1 and +1, making it easy to interpret strength and direction of relationships.

Pearson Correlation Formula

Correlation Coefficient (r)
r = Cov(X,Y) / (σₓ × σᵧ)

Covariance divided by product of standard deviations

Interpretation Guide

  • r = +1: Perfect positive correlation
  • r = 0.7 to 0.9: Strong positive
  • r = 0.4 to 0.6: Moderate positive
  • r = 0.1 to 0.3: Weak positive
  • r = 0: No correlation
  • r = -0.1 to -0.3: Weak negative
  • r = -0.4 to -0.6: Moderate negative
  • r = -0.7 to -0.9: Strong negative
  • r = -1: Perfect negative correlation
💡 REAL-WORLD EXAMPLE

Study hours vs exam scores typically show r = 0.7 (strong positive). More study hours correlate with higher scores.

🎯 Key Takeaways

  • r ranges from -1 to +1
  • Measures strength AND direction of linear relationship
  • Scale-independent (unlike covariance)
  • Only measures LINEAR relationships
Topic 13

💪 Interpreting Correlation

Correlation vs causation and common pitfalls

The Golden Rule

⚠️ CORRELATION ≠ CAUSATION

Just because two variables are correlated does NOT mean one causes the other!

Common Scenarios

  • Direct Causation: X causes Y (smoking causes cancer)
  • Reverse Causation: Y causes X (not the direction you thought)
  • Third Variable: Z causes both X and Y (confounding variable)
  • Coincidence: Pure chance with no real relationship
📊 FAMOUS EXAMPLE

Ice cream sales correlate with drowning deaths.

Does ice cream cause drowning? NO! The third variable is summer weather—more people swim in summer (more drownings) and eat ice cream in summer.

🎯 Key Takeaways

  • Correlation shows relationship, NOT causation
  • Always consider third variables (confounders)
  • Need controlled experiments to prove causation
  • Be skeptical of correlation claims in media
Topic 14

🎲 Probability Basics

Foundation of statistical inference

Introduction

What is it? Probability measures the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain).

Why it matters: Foundation for all statistical inference, hypothesis testing, and prediction.

Basic Formula

Probability of Event E
P(E) = Number of favorable outcomes / Total number of possible outcomes

Key Rules

  • Range: 0 ≤ P(E) ≤ 1
  • Complement: P(not E) = 1 - P(E)
  • Addition (OR): P(A or B) = P(A) + P(B) - P(A and B)
  • Multiplication (AND): P(A and B) = P(A) × P(B) [if independent]
📊 EXAMPLE

Rolling a die:

P(rolling a 4) = 1/6 ≈ 0.167

P(rolling even) = 3/6 = 0.5

P(not rolling a 6) = 5/6 ≈ 0.833

🎯 Key Takeaways

  • Probability ranges from 0 to 1
  • P(E) = favorable outcomes / total outcomes
  • Complement rule: P(not E) = 1 - P(E)
  • Foundation for all statistical inference
Topic 15

🔷 Set Theory

Union, intersection, and complement

Introduction

What is it? Set theory provides a mathematical framework for organizing events and calculating probabilities.

Key Concepts

  • Union (A ∪ B): A OR B (either event occurs)
  • Intersection (A ∩ B): A AND B (both events occur)
  • Complement (A'): NOT A (event doesn't occur)
  • Mutually Exclusive: A ∩ B = ∅ (can't both occur)

🎯 Key Takeaways

  • Union (∪): OR operation
  • Intersection (∩): AND operation
  • Complement ('): NOT operation
  • Venn diagrams visualize set relationships
Topic 16

🔀 Conditional Probability

Probability given that something else happened

Introduction

What is it? Conditional probability is the probability of event A occurring given that event B has already occurred.

Formula

Conditional Probability
P(A|B) = P(A and B) / P(B)

Read as: "Probability of A given B"

📊 EXAMPLE

Drawing cards: P(King | Red card) = ?

P(Red card) = 26/52

P(King and Red) = 2/52

P(King | Red) = (2/52) / (26/52) = 2/26 = 1/13

🎯 Key Takeaways

  • P(A|B) = probability of A given B occurred
  • Formula: P(A|B) = P(A and B) / P(B)
  • Critical for Bayes' Theorem
  • Used in machine learning and diagnostics
Topic 17

🎯 Independence

When events don't affect each other

Introduction

What is it? Two events are independent if the occurrence of one doesn't affect the probability of the other.

Test for Independence

Events A and B are independent if:
P(A|B) = P(A)

OR equivalently:

P(A and B) = P(A) × P(B)

Examples

  • Independent: Coin flips, die rolls with replacement
  • Dependent: Drawing cards without replacement, weather on consecutive days

🎯 Key Takeaways

  • Independent events don't affect each other
  • Test: P(A and B) = P(A) × P(B)
  • With replacement → independent
  • Without replacement → dependent
Topic 18

🧮 Bayes' Theorem

Updating probabilities with new evidence

Introduction

What is it? Bayes' Theorem shows how to update probability based on new information.

Why it matters: Used in medical diagnosis, spam filters, machine learning, and countless applications.

The Formula

Bayes' Theorem
P(A|B) = [P(B|A) × P(A)] / P(B)
  • P(A|B) = posterior probability
  • P(B|A) = likelihood
  • P(A) = prior probability
  • P(B) = marginal probability
📊 MEDICAL DIAGNOSIS EXAMPLE

Disease affects 1% of population. Test is 95% accurate.

You test positive. What's probability you have disease?

P(Disease) = 0.01

P(Positive|Disease) = 0.95

P(Positive|No Disease) = 0.05

P(Positive) = 0.01×0.95 + 0.99×0.05 = 0.059

P(Disease|Positive) = (0.95×0.01)/0.059 = 0.161

Only 16.1% chance you have the disease!

🎯 Key Takeaways

  • Updates probability based on new evidence
  • P(A|B) = [P(B|A) × P(A)] / P(B)
  • Critical for medical testing and machine learning
  • Counter-intuitive results common (base rate matters!)
Topic 19

📊 Probability Mass Function (PMF)

Probabilities for discrete random variables

Introduction

What is it? PMF gives the probability that a discrete random variable equals a specific value.

Why it matters: Used for countable outcomes like dice rolls, coin flips, or number of defects.

Properties

  • 0 ≤ P(X = x) ≤ 1 for all x
  • Sum of all probabilities = 1
  • Only defined for discrete variables
  • Visualized with bar charts
📊 EXAMPLE: Die Roll

P(X = 1) = 1/6

P(X = 2) = 1/6

... and so on

Sum = 6 × (1/6) = 1 ✓

🎯 Key Takeaways

  • PMF is for discrete random variables
  • Gives P(X = specific value)
  • All probabilities sum to 1
  • Visualized with bar charts
Topic 20

📈 Probability Density Function (PDF)

Probabilities for continuous random variables

Introduction

What is it? PDF describes probability for continuous random variables. Probability at exact point is 0; we calculate probability over intervals.

Key Differences from PMF

  • For continuous (not discrete) variables
  • P(X = exact value) = 0
  • Calculate P(a < X < b) = area under curve
  • Total area under curve = 1

🎯 Key Takeaways

  • PDF is for continuous random variables
  • Probability = area under curve
  • P(X = exact point) = 0
  • Total area under PDF = 1
Topic 21

📉 Cumulative Distribution Function (CDF)

Probability up to a value

Introduction

What is it? CDF gives the probability that X is less than or equal to a specific value.

Formula: F(x) = P(X ≤ x)

Properties

  • Always non-decreasing
  • F(-∞) = 0
  • F(+∞) = 1
  • P(a < X ≤ b) = F(b) - F(a)

🎯 Key Takeaways

  • CDF: F(x) = P(X ≤ x)
  • Works for both discrete and continuous
  • Always increases from 0 to 1
  • Useful for finding percentiles
Topic 22

🪙 Bernoulli Distribution

Single trial with two outcomes

Introduction

What is it? Models a single trial with two outcomes: success (1) or failure (0).

Examples: Coin flip, pass/fail test, yes/no question

Formula

Bernoulli PMF
P(X = 1) = p
P(X = 0) = 1 - p = q

Mean = p, Variance = p(1-p)

🎯 Key Takeaways

  • Single trial, two outcomes (0 or 1)
  • Parameter: p (probability of success)
  • Mean = p, Variance = p(1-p)
  • Building block for binomial distribution
Topic 23

🎰 Binomial Distribution

Multiple independent Bernoulli trials

Introduction

What is it? Models the number of successes in n independent Bernoulli trials.

Requirements: Fixed n, same p, independent trials, binary outcomes

Formula

Binomial PMF
P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

C(n,k) = n! / (k!(n-k)!)

Mean = np, Variance = np(1-p)

📊 EXAMPLE

Flip coin 10 times. P(exactly 6 heads)?

n=10, k=6, p=0.5

P(X=6) = C(10,6) × 0.5^6 × 0.5^4 = 210 × 0.000977 ≈ 0.205

🎯 Key Takeaways

  • n independent trials, probability p each
  • Counts number of successes
  • Mean = np, Variance = np(1-p)
  • Common in quality control and surveys
Topic 24

🔔 Normal Distribution

The bell curve and 68-95-99.7 rule

Introduction

What is it? The most important continuous probability distribution—symmetric, bell-shaped curve.

Why it matters: Many natural phenomena follow normal distribution. Foundation of inferential statistics.

Properties

  • Symmetric around mean μ
  • Bell-shaped curve
  • Mean = Median = Mode
  • Defined by μ (mean) and σ (standard deviation)
  • Total area under curve = 1

The 68-95-99.7 Rule (Empirical Rule)

  • 68% of data within μ ± 1σ
  • 95% of data within μ ± 2σ
  • 99.7% of data within μ ± 3σ
💡 REAL-WORLD EXAMPLE

IQ scores: μ = 100, σ = 15

68% of people have IQ between 85-115

95% have IQ between 70-130

99.7% have IQ between 55-145

🎯 Key Takeaways

  • Symmetric bell curve, parameters μ and σ
  • 68-95-99.7 rule for standard deviations
  • Foundation for hypothesis testing
  • Central Limit Theorem connects to sampling
Topic 25

⚖️ Hypothesis Testing Introduction

Making decisions from data

Introduction

What is it? Statistical method for testing claims about populations using sample data.

Why it matters: Allows us to make evidence-based decisions and determine if effects are real or due to chance.

The Two Hypotheses

  • Null Hypothesis (H₀): Status quo, no effect, no difference
  • Alternative Hypothesis (H₁ or Hₐ): What we're trying to prove

Decision Process

  1. State hypotheses (H₀ and H₁)
  2. Choose significance level (α)
  3. Collect data and calculate test statistic
  4. Find p-value or critical value
  5. Make decision: Reject H₀ or Fail to reject H₀
📊 EXAMPLE

Claim: New teaching method improves test scores

H₀: μ = 75 (no improvement)

H₁: μ > 75 (scores improved)

🎯 Key Takeaways

  • H₀ = null hypothesis (status quo)
  • H₁ = alternative hypothesis (what we test)
  • We either reject or fail to reject H₀
  • Never "accept" or "prove" anything
Topic 26

🎯 Significance Level (α)

Setting your error tolerance

Introduction

What is it? α (alpha) is the probability of rejecting H₀ when it's actually true (Type I error rate).

Common values: 0.05 (5%), 0.01 (1%), 0.10 (10%)

Interpretation

  • α = 0.05: Willing to be wrong 5% of the time
  • Lower α: More stringent, harder to reject H₀
  • Higher α: More lenient, easier to reject H₀
  • Confidence level: 1 - α (e.g., 0.05 → 95% confidence)

🎯 Key Takeaways

  • α = probability of Type I error
  • Common: α = 0.05 (5% error rate)
  • Set before collecting data
  • Trade-off between Type I and Type II errors
Topic 27

📊 Standard Error

Measuring sampling variability

Introduction

What is it? Standard error (SE) measures how much sample means vary from the true population mean.

Formula

Standard Error of Mean
SE = σ / √n

or estimate: SE = s / √n

Key Points

  • Decreases as sample size increases
  • Measures precision of sample mean
  • Lower SE = better estimate
  • Used in confidence intervals and hypothesis tests

🎯 Key Takeaways

  • SE = σ / √n
  • Measures sampling variability
  • Larger samples → smaller SE
  • Critical for inference
Topic 28

📏 Z-Test

Hypothesis test for large samples with known σ

When to Use Z-Test

  • Sample size n ≥ 30 (large sample)
  • Population standard deviation (σ) known
  • Testing population mean
  • Normal distribution or large n

Formula

Z-Test Statistic
z = (x̄ - μ₀) / (σ / √n)

x̄ = sample mean

μ₀ = hypothesized population mean

σ = population standard deviation

n = sample size

🎯 Key Takeaways

  • Use when n ≥ 30 and σ known
  • z = (x̄ - μ₀) / SE
  • Compare z to critical value or find p-value
  • Large |z| = evidence against H₀
Topic 29

🎚️ Z-Score & Critical Values

Standardization and rejection regions

Z-Score (Standardization)

Z-Score Formula
z = (x - μ) / σ

Converts any normal distribution to standard normal (μ=0, σ=1)

Critical Values

  • α = 0.05 (two-tailed): z = ±1.96
  • α = 0.05 (one-tailed): z = 1.645
  • α = 0.01 (two-tailed): z = ±2.576

🎯 Key Takeaways

  • Z-score standardizes values
  • Critical values define rejection region
  • |z| > critical value → reject H₀
  • Common: ±1.96 for 95% confidence
Topic 30

💯 P-Value Method

Probability of observing data if H₀ is true

Introduction

What is it? P-value is the probability of getting results as extreme as observed, assuming H₀ is true.

Decision Rule

  • If p-value ≤ α: Reject H₀ (statistically significant)
  • If p-value > α: Fail to reject H₀ (not significant)

Interpretation

  • p < 0.01: Very strong evidence against H₀
  • 0.01 ≤ p < 0.05: Strong evidence against H₀
  • 0.05 ≤ p < 0.10: Weak evidence against H₀
  • p ≥ 0.10: Little or no evidence against H₀
⚠️ COMMON MISCONCEPTION

P-value is NOT the probability that H₀ is true! It's the probability of observing your data IF H₀ were true.

🎯 Key Takeaways

  • P-value = P(data | H₀ true)
  • Reject H₀ if p ≤ α
  • Smaller p-value = stronger evidence against H₀
  • Most common approach in modern statistics
Topic 31

↔️ One-Tailed vs Two-Tailed Tests

Directional vs non-directional hypotheses

Two-Tailed Test

  • H₁: μ ≠ μ₀ (different, could be higher or lower)
  • Testing for any difference
  • Rejection regions in both tails
  • More conservative

One-Tailed Test

  • Right-tailed: H₁: μ > μ₀
  • Left-tailed: H₁: μ < μ₀
  • Testing for specific direction
  • Rejection region in one tail
  • More powerful for directional effects

🎯 Key Takeaways

  • Two-tailed: testing for any difference
  • One-tailed: testing for specific direction
  • Choose before collecting data
  • Two-tailed is more conservative
Topic 32

📐 T-Test

Hypothesis test for small samples or unknown σ

When to Use T-Test

  • Small sample (n < 30)
  • Population σ unknown (use sample s)
  • Population approximately normal

Formula

T-Test Statistic
t = (x̄ - μ₀) / (s / √n)

Same as z-test but uses s instead of σ

Follows t-distribution with df = n - 1

🎯 Key Takeaways

  • Use when σ unknown or n < 30
  • t = (x̄ - μ₀) / (s / √n)
  • Follows t-distribution
  • More variable than z-distribution
Topic 33

🔓 Degrees of Freedom

Independent pieces of information

Introduction

What is it? Degrees of freedom (df) is the number of independent values that can vary in analysis.

Common Formulas

  • One-sample t-test: df = n - 1
  • Two-sample t-test: df ≈ n₁ + n₂ - 2
  • Chi-squared: df = (rows-1)(cols-1)

Why It Matters

  • Determines shape of t-distribution
  • Higher df → closer to normal distribution
  • Affects critical values

🎯 Key Takeaways

  • df = number of independent values
  • For t-test: df = n - 1
  • Higher df → distribution closer to normal
  • Critical for finding correct critical values
Topic 34

⚠️ Type I & Type II Errors

False positives and false negatives

The Two Types of Errors

H₀ True H₀ False
Reject H₀ Type I Error (α) Correct!
Fail to Reject H₀ Correct! Type II Error (β)

Definitions

  • Type I Error (α): Rejecting true H₀ (false positive)
  • Type II Error (β): Failing to reject false H₀ (false negative)
  • Power = 1 - β: Probability of correctly rejecting false H₀
📊 MEDICAL ANALOGY

Type I Error: Telling healthy person they're sick (false alarm)

Type II Error: Telling sick person they're healthy (missed diagnosis)

🎯 Key Takeaways

  • Type I: False positive (α)
  • Type II: False negative (β)
  • Trade-off: decreasing one increases the other
  • Power = 1 - β (ability to detect true effect)
Topic 35

χ² Chi-Squared Distribution

Distribution for categorical data analysis

Introduction

What is it? Chi-squared (χ²) distribution is used for testing hypotheses about categorical data.

Properties

  • Always positive (ranges from 0 to ∞)
  • Right-skewed
  • Shape depends on degrees of freedom
  • Higher df → more symmetric

Uses

  • Goodness of fit test
  • Test of independence
  • Testing variance

🎯 Key Takeaways

  • Used for categorical data
  • Always positive, right-skewed
  • Shape depends on df
  • Foundation for chi-squared tests
Topic 36

✓ Goodness of Fit Test

Testing if data follows expected distribution

Introduction

What is it? Tests whether observed frequencies match expected frequencies from a theoretical distribution.

Formula

Chi-Squared Test Statistic
χ² = Σ [(O - E)² / E]

O = observed frequency

E = expected frequency

df = k - 1 (k = number of categories)

📊 EXAMPLE

Testing if die is fair:

Roll 60 times. Expected: 10 per face

Observed: 8, 12, 11, 9, 10, 10

Calculate χ² and compare to critical value

🎯 Key Takeaways

  • Tests if observed matches expected distribution
  • χ² = Σ(O-E)²/E
  • Large χ² = poor fit
  • df = number of categories - 1
Topic 37

🔗 Test of Independence

Testing relationship between categorical variables

Introduction

What is it? Tests whether two categorical variables are independent or associated.

Formula

Chi-Squared for Independence
χ² = Σ [(O - E)² / E]

E = (row total × column total) / grand total

df = (rows - 1)(columns - 1)

📊 EXAMPLE

Are gender and color preference independent?

Create contingency table, calculate expected frequencies, compute χ², and test against critical value.

🎯 Key Takeaways

  • Tests independence of two categorical variables
  • Uses contingency tables
  • df = (r-1)(c-1)
  • Large χ² suggests association
Topic 38

📏 Chi-Squared Variance Test

Testing claims about population variance

Introduction

What is it? Tests hypotheses about population variance or standard deviation.

Formula

Chi-Squared for Variance
χ² = (n-1)s² / σ₀²

n = sample size

s² = sample variance

σ₀² = hypothesized population variance

df = n - 1

🎯 Key Takeaways

  • Tests claims about variance/standard deviation
  • χ² = (n-1)s²/σ₀²
  • Requires normal population
  • Common in quality control
Topic 39

📊 Confidence Intervals

Range of plausible values for parameter

Introduction

What is it? A confidence interval provides a range of values that likely contains the true population parameter.

Why it matters: More informative than point estimates—shows precision and uncertainty.

Formula

Confidence Interval for Mean
CI = x̄ ± (critical value × SE)

For z: CI = x̄ ± z* × (σ/√n)

For t: CI = x̄ ± t* × (s/√n)

Common Confidence Levels

  • 90% CI: z* = 1.645
  • 95% CI: z* = 1.96
  • 99% CI: z* = 2.576
📊 EXAMPLE

Sample: n=100, x̄=50, s=10

95% CI = 50 ± 1.96(10/√100)

95% CI = 50 ± 1.96 = (48.04, 51.96)

🎯 Key Takeaways

  • CI = point estimate ± margin of error
  • 95% CI most common
  • Wider CI = more uncertainty
  • Larger sample = narrower CI
Topic 40

± Margin of Error

Measuring estimate precision

Introduction

What is it? Margin of error (MOE) is the ± part of a confidence interval, showing the precision of an estimate.

Formula

Margin of Error
MOE = (critical value) × SE

MOE = z* × (σ/√n) or t* × (s/√n)

Factors Affecting MOE

  • Sample size: Larger n → smaller MOE
  • Confidence level: Higher confidence → larger MOE
  • Variability: Higher σ → larger MOE

🎯 Key Takeaways

  • MOE = critical value × SE
  • Indicates precision of estimate
  • Inversely related to sample size
  • Trade-off between confidence and precision
Topic 41

🔍 Interpreting Confidence Intervals

Common misconceptions and proper interpretation

Correct Interpretation

"We are 95% confident that the true population parameter lies within this interval."

This means: If we repeated this process many times, 95% of the intervals would contain the true parameter.

⚠️ COMMON MISCONCEPTIONS
  • WRONG: "There's a 95% probability the parameter is in this interval."
  • WRONG: "95% of the data falls in this interval."
  • WRONG: "We are 95% sure our sample mean is in this interval."

Using CIs for Hypothesis Testing

  • If hypothesized value is INSIDE CI → fail to reject H₀
  • If hypothesized value is OUTSIDE CI → reject H₀
  • 95% CI corresponds to α = 0.05 test
✅ PRO TIP

Report confidence intervals instead of just p-values! CIs provide more information: effect size AND statistical significance.

🎯 Key Takeaways

  • Correct interpretation: confidence in the method, not the specific interval
  • 95% refers to long-run success rate
  • Can use CIs for hypothesis testing
  • More informative than p-values alone