STATISTICS & PROBABILITY — COMPLETE VISUAL STUDY GUIDE
STATISTICS & PROBABILITY
Complete Visual Study Guide
For University Exams · Data Science · AI · GRE · GMAT · AP Stats
■ 12 Core Topics ■ 50+ Formulas ■ Exam Tips Included ■ Used Worldwide
What's Inside This Guide:
1. Types of Data & Levels of Measurement
2. Descriptive Statistics — Mean, Median, Mode, Variance, SD
3. Probability Fundamentals — Rules, Trees, Venn Diagrams
4. Conditional Probability & Bayes' Theorem
5. Random Variables & Probability Distributions
6. Binomial Distribution
7. Normal Distribution & Z-Scores
8. Central Limit Theorem
9. Confidence Intervals
10. Hypothesis Testing — Step-by-Step
11. Chi-Square & ANOVA
12. Correlation & Regression
■ Best used alongside lectures. Highlight key formulas. Practice every example!
Page 1 | Statistics & Probability — Complete Visual Study Guide
, STATISTICS & PROBABILITY — COMPLETE VISUAL STUDY GUIDE
1. TYPES OF DATA & MEASUREMENT LEVELS
Understanding data types is the first step in any statistical analysis.
Data Type Description Examples Operations Allowed
Nominal Categories, no order Gender, Colors, Country Mode, Frequency
Ordinal Ordered categories Ratings (1-5), Grades Median, Mode
Interval Equal spacing, no true zero Temperature (°C), IQ Mean, SD, +/−
Ratio Equal spacing + true zero Height, Weight, Age All operations
■ Exam Tip: Nominal = Name only. Ordinal = Order matters. Interval = no zero. Ratio = everything!
2. DESCRIPTIVE STATISTICS
Descriptive statistics summarize and describe the main features of a dataset.
Measures of Central Tendency
Mean (x■) x■ = (Σx) / n Sum of all values divided by count
Median Middle value when sorted If even: average of two middle values
Mode Most frequent value Can have multiple modes or none
Used when values have different
Weighted Mean x■w = Σ(w·x) / Σw importance
Measures of Spread / Dispersion
Range = Max − Min
Variance (Population): σ2 = Σ(x − µ)2 / N
Variance (Sample): s2 = Σ(x − x■)2 / (n−1)
Standard Deviation: σ = √(σ2) | s = √(s2)
IQR = Q3 − Q1 (Interquartile Range)
■ Why (n−1) in sample variance? It corrects for bias — this is called Bessel's Correction. Always use (n−1)
for SAMPLES and N for the full POPULATION.
THE EMPIRICAL RULE (for Normal Distributions)
µ ± 1σ → ~68% of data falls here
µ ± 2σ → ~95% of data falls here
µ ± 3σ → ~99.7% of data falls here
Page 2 | Statistics & Probability — Complete Visual Study Guide
STATISTICS & PROBABILITY
Complete Visual Study Guide
For University Exams · Data Science · AI · GRE · GMAT · AP Stats
■ 12 Core Topics ■ 50+ Formulas ■ Exam Tips Included ■ Used Worldwide
What's Inside This Guide:
1. Types of Data & Levels of Measurement
2. Descriptive Statistics — Mean, Median, Mode, Variance, SD
3. Probability Fundamentals — Rules, Trees, Venn Diagrams
4. Conditional Probability & Bayes' Theorem
5. Random Variables & Probability Distributions
6. Binomial Distribution
7. Normal Distribution & Z-Scores
8. Central Limit Theorem
9. Confidence Intervals
10. Hypothesis Testing — Step-by-Step
11. Chi-Square & ANOVA
12. Correlation & Regression
■ Best used alongside lectures. Highlight key formulas. Practice every example!
Page 1 | Statistics & Probability — Complete Visual Study Guide
, STATISTICS & PROBABILITY — COMPLETE VISUAL STUDY GUIDE
1. TYPES OF DATA & MEASUREMENT LEVELS
Understanding data types is the first step in any statistical analysis.
Data Type Description Examples Operations Allowed
Nominal Categories, no order Gender, Colors, Country Mode, Frequency
Ordinal Ordered categories Ratings (1-5), Grades Median, Mode
Interval Equal spacing, no true zero Temperature (°C), IQ Mean, SD, +/−
Ratio Equal spacing + true zero Height, Weight, Age All operations
■ Exam Tip: Nominal = Name only. Ordinal = Order matters. Interval = no zero. Ratio = everything!
2. DESCRIPTIVE STATISTICS
Descriptive statistics summarize and describe the main features of a dataset.
Measures of Central Tendency
Mean (x■) x■ = (Σx) / n Sum of all values divided by count
Median Middle value when sorted If even: average of two middle values
Mode Most frequent value Can have multiple modes or none
Used when values have different
Weighted Mean x■w = Σ(w·x) / Σw importance
Measures of Spread / Dispersion
Range = Max − Min
Variance (Population): σ2 = Σ(x − µ)2 / N
Variance (Sample): s2 = Σ(x − x■)2 / (n−1)
Standard Deviation: σ = √(σ2) | s = √(s2)
IQR = Q3 − Q1 (Interquartile Range)
■ Why (n−1) in sample variance? It corrects for bias — this is called Bessel's Correction. Always use (n−1)
for SAMPLES and N for the full POPULATION.
THE EMPIRICAL RULE (for Normal Distributions)
µ ± 1σ → ~68% of data falls here
µ ± 2σ → ~95% of data falls here
µ ± 3σ → ~99.7% of data falls here
Page 2 | Statistics & Probability — Complete Visual Study Guide
