Stats Patterns

Consolidated Question Patterns & Abstractions: Statistics I (Weeks 1-4)

This document synthesizes the core problem types and mental algorithms for the first four weeks of Statistics I. Use this to rapidly identify what a question is asking and which tools you need to solve it.

📚 Table of Contents

1. Week 1: Introduction to Statistics & Data
2. Week 2: Describing Categorical Data
3. Week 3: Describing Numerical Data
4. Week 4: Association Between Two Variables

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Week 1: Introduction to Statistics & Data

• Core Idea: Learning the fundamental vocabulary to classify and understand the nature of data.

Pattern #	Pattern Name	Frequency	Difficulty	Core Skill & Abstraction
1.1	Population vs. Sample Identification	High	Easy	Abstract: Is it the entire group of interest (Population) or the subset you have data for (Sample)? Keywords: “all”, “every” vs. “selected”, “a group of”.
1.2	Inferential vs. Descriptive Logic	High	Easy	Abstract: Is the statement just describing the sample, or is it inferring a conclusion about the whole population? Keywords: “the sample had…” vs. “we conclude that all…“.
1.3	Classifying Variable Types	High	Medium	Abstract: Apply the tests: 1. Math? (Can I average it? Yes=Numerical, No=Categorical). 2. Gaps? (Can it be a decimal? Yes=Continuous, No=Discrete).
1.4	Identifying the Scale of Measurement	High	Medium	Abstract: Use the NOIR framework: Nominal (names), Ordinal (order), Interval (equal intervals, no true zero), Ratio (true zero).

🧠 Week 1 Mental Algorithm: The Classification Checklist

When asked to classify a variable (e.g., “Education Level”):

1. Triage: It’s a classification problem.
2. Abstract & Act:
   • Math Test: Can I average “High School” and “PhD”? No. $\to$ It’s Categorical.
   • Order Test: Is there a natural ranking? Yes, PhD > High School. $\to$ It’s Ordinal.
   • Final Answer: Categorical, Ordinal Scale.

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Week 2: Describing Categorical Data

• Core Idea: Summarizing and visualizing data that falls into non-numerical groups using counts, proportions, and charts.

Pattern #	Pattern Name	Frequency	Difficulty	Core Skill & Abstraction
2.1	Calculating Frequencies & Proportions	High	Easy	Abstract: Use the core relationship: Part = Whole × Percentage. Find the piece you’re missing.
2.2	Identifying Measures of Central Tendency	High	Easy	Abstract: For nominal data, only the Mode (most frequent) is defined. Mean and Median require order/numbers and are not applicable.
2.3	Choosing the Appropriate Graph	High	Medium	Abstract: Bar Chart to compare counts. Pie Chart to show percentages of a whole. Pareto Chart to identify the most important categories (a sorted bar chart).
2.4	Interpreting Graphical Representations	Medium	Easy	Abstract: Read the values directly from the chart. For pie charts, convert percentages to counts if a total is given. For bar charts, read the axis labels carefully.

🧠 Week 2 Mental Algorithm: The Categorical Toolkit

When you see categorical data (e.g., a list of academies and the number of players in each):

1. Triage: It’s a categorical description problem.
2. Abstract & Act:
   • “What is the most common?” $\to$ Find the Mode. Look for the highest bar in a bar chart or the biggest slice in a pie chart.
   • “What share/proportion…?” $\to$ Calculate Relative Frequency: (Frequency of Category) / (Total).
   • “How to best visualize…?” $\to$ If comparing counts, use a Bar Chart. If showing parts of a whole, a Pie Chart is an option. If prioritizing, a Pareto Chart.

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Week 3: Describing Numerical Data

• Core Idea: Calculating statistics that measure the “center” (central tendency) and “spread” (dispersion) of numerical datasets.

Pattern #	Pattern Name	Frequency	Difficulty	Core Skill & Abstraction
3.1	Calculating Center & Spread	High	Medium	Abstract: Calculate Mean (average), Median (sorted middle), Mode (most frequent), Sample Variance ($s^2$), and Sample Standard Deviation ($s$).
3.2	Correcting Mean and Variance	High	Medium	Abstract: Work backward from the wrong statistic to find the wrong Sum or Sum of Squares. Correct the sum (Sum_correct = Sum_wrong - wrong_val + correct_val), then recalculate.
3.3	Calculating Percentiles and Quartiles	High	Medium	Abstract: 1. Sort the data. 2. Find the median (Q2). 3. Find the median of the lower half (Q1). 4. Find the median of the upper half (Q3). 5. Calculate IQR = Q3 - Q1.
3.4	Identifying Outliers	Medium	Medium	Abstract: Calculate the fences: Lower = Q1 - 1.5*IQR and Upper = Q3 + 1.5*IQR. Any data point outside this range is an outlier.

🧠 Week 3 Mental Algorithm: The Numerical Description Flow

When given a list of numbers:

1. Triage: It’s a numerical description problem.
2. Abstract & Act:
   • First step is always to SORT the data.
   • “Find the center”: Calculate Mean and Median. If they are very different, it hints at skewness or outliers.
   • “Find the spread”: Calculate IQR (resistant to outliers) and Standard Deviation (sensitive to outliers).
   • “Check for outliers”: Use the IQR and the 1.5*IQR fence rule.

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Week 4: Association Between Two Variables

• Core Idea: Moving from describing one variable at a time to describing the relationship between two variables.

Pattern #	Pattern Name	Frequency	Difficulty	Core Skill & Abstraction
4.1	Calculating Covariance and Correlation	High	Medium	Abstract: A procedural calculation, best done with a table. Find means, then deviations from the mean for both variables, then products of deviations. Sum these products to find covariance, then standardize to find correlation r.
4.2	Interpreting the Correlation Coefficient r	High	Easy	Abstract: Look at the sign for direction (positive/negative) and the magnitude for strength (close to 1 or -1 is strong; close to 0 is weak).
4.3	Analyzing Contingency Tables	High	Medium	Abstract: Differentiate between Marginal (uses grand totals in the denominator) and Conditional (uses a row or column total in the denominator) proportions. Read the question carefully to find the correct “whole”.
4.4	Conceptual Understanding of Correlation	Medium	Easy	Abstract: Remember the key rules: Correlation ≠ Causation. Correlation only measures linear relationships. A perfect linear relationship means $r=1$ or $r=-1$.

🧠 Week 4 Mental Algorithm: The Relationship Analysis Flow

When given a dataset with two variables, X and Y:

1. Triage: It’s a relationship/association problem.
2. Abstract & Act:
   • Are X and Y both categorical? $\to$ Build a Contingency Table. Analyze it by calculating conditional proportions.
   • Are X and Y both numerical? $\to$ Your goal is to find the Correlation Coefficient r.
     1. Visualize with a Scatterplot in your mind. Does it seem to go up or down?
     2. Calculate Covariance. The sign will confirm your visual check.
     3. Calculate Standard Deviations for both X and Y.
     4. Calculate $r=cov(x,y)/(sx\ast sy)$.
     5. Interpret r: State the direction (positive/negative) and strength (weak/moderate/strong).