Stats1_Ultimate_Pattern_Guide

Stats 1: Ultimate Pattern Recognition Guide 🎯

IMPORTANT

Purpose: Complete hierarchical map of every Stats 1 question pattern. Organized by difficulty with quick recognition strategies.

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📊 Pattern Hierarchy Overview

Week	Pattern Families	Total Patterns
1: Data Collection	Sampling, Scales, Data Types	6
2: Descriptive Stats	Charts, Summaries	7
3: Central Tendency	Mean, Median, Mode	8
4: Dispersion	Variance, SD, Correlation	9
5: Counting	Permutations, Circular, Constraints	10
6: Combinations	Selection, Binomial, Constraint Events	8
7: Probability	Axioms, Union/Intersection, Complement	9
8: Conditional	Bayes, Independence, Trees	10

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Week 1: Data Collection

Pattern Family 1.1: Sampling Methods

🟢 Level 1: Sample vs Population

Spot it Fast: “Sample” vs “Census”

Key Distinction:

• Population: ENTIRE group
• Sample: SUBSET of population

Common Traps:

• ❌ Confusing parameter (population) with statistic (sample)

🟡 Level 2: Sampling Techniques

Spot it Fast: “Random”, “Stratified”, “Systematic”

Types:

1. Simple Random: Every member equal chance
2. Stratified: Divide into groups, sample from each
3. Systematic: Every $k$-th member
4. Convenience: Easiest to reach (biased!)

🔴 Level 3: Bias Identification

Spot it Fast: “What type of bias?”

Types:

• Selection bias: Non-random sampling
• Response bias: Question wording influences answer
• Non-response bias: Certain groups don’t respond

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Pattern Family 1.2: Scales of Measurement

🟢 Level 1: Identify Scale

Spot it Fast: “What scale is ‘temperature’?”

Hierarchy (least to most):

1. Nominal: Categories (Gender, Color)
2. Ordinal: Ordered categories (Ratings 1-5)
3. Interval: Numeric, no true zero (Temperature °C)
4. Ratio: Numeric, true zero (Height, Weight)

Quick Test:

• Can you order? → Ordinal+
• Can you subtract? → Interval+
• Can you take ratio? → Ratio

🟡 Level 2: Appropriate Operations

Spot it Fast: “Can you find mean of ratings 1-5?”

Rules:

• Nominal: Mode only
• Ordinal: Mode, Median
• Interval/Ratio: All (Mean, Median, Mode)

Common Traps:

• ❌ Taking mean of nominal data (e.g., “average gender”)

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Week 2: Descriptive Statistics

Pattern Family 2.1: Chart Selection

🟢 Level 1: Categorical Data

Spot it Fast: “Best chart for favorite colors?”

Options:

• Bar chart: Different categories
• Pie chart: Parts of whole (percentages)

Rule: Use pie ONLY if showing proportions of 100%

🟡 Level 2: Numerical Data

Spot it Fast: “Distribution of test scores”

Options:

• Histogram: Continuous data grouped into bins
• Box plot: Shows quartiles, outliers
• Scatter plot: Relationship between two variables

🔴 Level 3: Misleading Charts

Spot it Fast: “What’s wrong with this chart?”

Common Issues:

• Truncated y-axis exaggerates differences
• 3D effects distort proportions
• Different bin widths in histogram

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Pattern Family 2.2: Frequency Distributions

🟢 Level 1: Absolute vs Relative

Spot it Fast: “Frequency” vs “Relative frequency”

Formulas:

• Absolute: Raw count
• Relative: $\frac{\text{count}}{\text{total}}$
• Percentage: Relative × 100%

🟡 Level 2: Cumulative Frequency

Spot it Fast: “How many scored ≤ 70?”

Method: Add frequencies up to that value

Example: Scores [60, 70, 80] with frequencies [5, 10, 8]

• Cumulative at 70: 5 + 10 = 15

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Week 3: Central Tendency

Pattern Family 3.1: Mean Calculations

🟢 Level 1: Simple Mean

Spot it Fast: “Average of 2, 4, 6, 8”

Formula: $\bar{x}=\frac{\sum x_i}{n}$

🟡 Level 2: Weighted Mean

Spot it Fast: “Frequency table” or “Grouped data”

Formula: $\bar{x}=\frac{\sum f_i{x}_i}{\sum f_i}$

Example: Values [2, 6, 11] with frequencies [10, 5, 3]

• $\bar{x}=\frac{10(2)+5(6)+3(11)}{18}=\frac{83}{18}=4.61$

🔴 Level 3: Find Unknown from Mean

Spot it Fast: “Mean is 5.63, find $x$”

Mental Algorithm:

1. Setup: $\frac{\text{sum of terms}}{n}=\text{mean}$
2. Express sum using $x$
3. Solve equation

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Pattern Family 3.2: Median & Mode

🟢 Level 1: Median (Odd vs Even)

Spot it Fast: “Middle value”

Method:

• Odd $n$: Middle position = $\frac{n+1}{2}$
• Even $n$: Average of positions $\frac{n}{2}$ and $\frac{n}{2}+1$

Example: [1, 3, 5, 7, 9]

• $n=5$, position = 3
• Median = 5

🟡 Level 2: Mode

Spot it Fast: “Most frequent value”

Cases:

• Unimodal: One mode
• Bimodal: Two modes
• No mode: All unique

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Pattern Family 3.3: Combined/Corrected Mean

🟢 Level 1: Correcting Mean

Spot it Fast: “Value 7 wrongly recorded as 11”

Mental Algorithm:

1. Find total: $\text{old mean}\times n$
2. Correct: Subtract wrong, add right
3. New mean: New total $÷n$

🟡 Level 2: Combined Mean

Spot it Fast: “Two groups with different means”

Formula: ${\bar{x}}_{combined}=\frac{n_1{\bar{x}}_1+n_2{\bar{x}}_2}{n_1+n_2}$

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[Patterns continue for Weeks 4-8…]

Quick Reference: Stats Pattern Spotting

Week 1-2: Data & Descriptions

• “Sample” vs “Population” → Different formulas for SD
• Nominal data → Mode ONLY (no mean!)
• Pie vs Bar → Pie for parts of whole

Week 3: Central Tendency

• “Wrongly recorded” → Correct the total first
• Frequency table → Weighted mean

Week 4: Dispersion

• Add constant → SD unchanged
• Multiply by constant → SD × |constant|
• Linear transformation $Y=aX+b$ → $r_{XY}=1$ if $a>0$

Week 5-6: Counting

• Circular arrangement → $(n-1)!$
• “At least one” → Total - None
• With constraints → Use complement OR casework

Week 7-8: Probability

• “At least” → Complement method
• Base rate fallacy → Use Bayes carefully
• Independence → $P(A\cap B)=P(A)P(B)$