🧠 IIT Madras Notes

Stats W1 to W4 Triage

12 min read

Ultimate Statistics Problem-Solver’s Arsenal (Weeks 1-4)

This guide is the culmination of analyzing all provided assignments. It identifies every unique question pattern in your Statistics course and provides a detailed, solved example for each one. For each pattern, we will focus on the Abstraction—the core idea you need to recognize—and the Execution, the step-by-step process to solve it.

Week 1: Introduction to Statistics & Data

  • TL;DR Concepts: Week 1 is the dictionary of statistics. We learn to speak the language. Population is everyone, Sample is a few. Descriptive just describes the few; Inferential makes a guess about everyone. Data is either Categorical (labels) or Numerical (numbers). We classify it with the NOIR scale (Nominal, Ordinal, Interval, Ratio).

Identified Patterns & Solved Examples

Pattern 1: Population vs. Sample Distinction

  • Abstraction: The Population is the entire, vast group you are curious about. The Sample is the small, manageable group you actually collect data from.
  • Example (from PYQ): An analyst studies placement for B.Tech students in India by collecting data from four randomly selected IITs. Identify the population and sample.
Click for Solution 1. **Identify the Broad Goal:** The study is about "B.Tech students in India". This is the whole group of interest. 2. **Identify the Data Source:** The data comes from "four randomly selected IITs". This is the specific subset that was observed. 3. **Conclusion:** * **Population:** All engineering institutes of India. * **Sample:** The four randomly selected IITs.

Pattern 2: Inferential vs. Descriptive Statements

  • Abstraction: A Descriptive statement is a fact about your sample. An Inferential statement is a conclusion or prediction about the population that you infer from your sample.
  • Example (from PYQ): Based on the sample of four IITs, the report states, “The campus placement of B.Tech students is 95% in different engineering institutes of India.” Is this descriptive or inferential?
Click for Solution 1. **Analyze the Scope:** The statement makes a claim about "engineering institutes of India" (the population). 2. **Compare to Data Source:** The data was only from four IITs (the sample). 3. **Conclusion:** Since the statement generalizes from the small sample to the large population, it is making an inference. It is **Inferential Statistics**. (A descriptive statement would be: "In the four IITs we sampled, the average placement was 95%.")

Pattern 3: Classifying Variables & Scales (The NOIR Framework)

  • Abstraction: Use a checklist. 1. Math? (No → Categorical, Yes → Numerical). 2. Order? (No → Nominal, Yes → Ordinal). 3. True Zero? (No → Interval, Yes → Ratio).
  • Example (from PYQ): Classify the variable “Soccer positions (i.e. Defender, Midfielder, Forward)“.
Click for Solution 1. **Math Test:** Can you average a Defender and a Forward? No. It's **Categorical**. 2. **Order Test:** Is there a natural order or rank? Yes, there's a progression up the field from Defender to Midfielder to Forward. The order has a tactical meaning. 3. **Conclusion:** Because the order matters, it is on an **Ordinal Scale**.

Week 2: Describing Categorical Data

  • TL;DR Concepts: Week 2 is about counting and proportions. Since we can’t use ‘mean’, our main tools are Frequency Tables and Relative Frequencies. The most typical category is the Mode. We visualize with Bar Charts (for comparison), Pie Charts (for parts of a whole), and Pareto Charts (for prioritization).

Identified Patterns & Solved Examples

Pattern 4: Calculating Frequencies and Proportions

  • Abstraction: Proportions and percentages are just fractions of a total. Use the core relationship: Part = Whole × Percentage.
  • Example (from PYQ): A pie chart shows the distribution of 500 total marks. Physics is 25%, Maths is 20%, Biology is 18%. What is the aggregate mark total for these three?
Click for Solution 1. **Find the Total Percentage (The "Part"):** * Total % = 25% + 20% + 18% = 63% or 0.63. 2. **Apply the Formula:** * Aggregate Marks = Total Marks (The "Whole") × Total Percentage * Aggregate Marks = 500 × 0.63 = 315.

Pattern 5: Central Tendency for Categorical Data

  • Abstraction: For nominal categorical data, Mean and Median are meaningless. The only measure of “center” is the Mode, which is simply the category that appears most often.
  • Example (from PYQ): The number of players in academies are: A(30), B(40), C(60), D(20), E(90). Find the mode and determine if the median is defined.
Click for Solution 1. **Find the Mode:** The mode is the category with the highest frequency. The highest count is 90. * **Mode:** Academy E. 2. **Check for Median:** The categories (A, B, C...) are just names (nominal). There's no inherent order. You cannot sort them to find a "middle" value. * **Median:** Not defined for this data.

Week 3: Describing Numerical Data

  • TL;DR Concepts: Week 3 is about the center and spread of numerical data. Mean is the balance point, Median is the middle person in line. Standard Deviation is the average distance from the mean, while IQR is the range of the middle 50% of your data. Outliers are the weirdos far away from everyone else.

Identified Patterns & Solved Examples

Pattern 6: Correcting Mean and Variance After an Error

  • Abstraction: You can reverse-engineer a statistic. An incorrect mean can give you the incorrect sum. You can then fix the sum and recalculate the correct mean.
  • Example (from PYQ): The mean of 6 observations is 19. An observation of 11 was wrongly recorded as 7. What is the correct mean?
Click for Solution 1. **Find the Incorrect Sum:** * Sum_incorrect = Mean_incorrect × n = 19 × 6 = 114. 2. **Fix the Sum:** * Sum_correct = Sum_incorrect - (Wrong Value) + (Correct Value) * Sum_correct = 114 - 7 + 11 = 118. 3. **Calculate the Correct Mean:** * Mean_correct = Sum_correct / n = 118 / 6 ≈ 19.67.

Pattern 7: Quartiles, IQR, and Outliers

  • Abstraction: This is a fixed procedure: 1. Sort. 2. Find Median (Q2). 3. Find Median of lower half (Q1). 4. Find Median of upper half (Q3). 5. Calculate IQR. 6. Check fences.
  • Example (from PYQ): For the data {30, 39, 44, 46, 73, 89, 91, 96, 112, 115}, find the IQR and the number of outliers.
Click for Solution 1. **Data is Sorted (n=10):** 2. **Find Quartiles:** * **Median (Q2):** Average of 5th and 6th values: (73+89)/2 = 81. * **Lower Half:** {30, 39, 44, 46, 73}. **Q1** is the median of this half, which is the 3rd value: 44. * **Upper Half:** {89, 91, 96, 112, 115}. **Q3** is the median of this half, which is the 3rd value: 96. 3. **Calculate IQR:** * IQR = Q3 - Q1 = 96 - 44 = 52. 4. **Calculate Outlier Fences:** * Lower Fence = Q1 - 1.5 * IQR = 44 - 1.5 * 52 = 44 - 78 = -34. * Upper Fence = Q3 + 1.5 * IQR = 96 + 1.5 * 52 = 96 + 78 = 174. 5. **Identify Outliers:** The valid range is [-34, 174]. All data points are within this range. There are **0 outliers**.

Week 4: Association Between Two Variables

  • TL;DR Concepts: Week 4 is about relationships. For categorical data, use Contingency Tables and conditional proportions. For numerical data, visualize with a Scatterplot and measure the linear relationship with Covariance (direction) and the Correlation Coefficient r (direction and strength, from -1 to +1). Remember: Correlation is not Causation!

Identified Patterns & Solved Examples

Pattern 8: Calculating Covariance and Correlation

  • Abstraction: This is a procedural calculation. The key is to create a table to systematically compute all the components needed for the formulas: deviations from the mean for both variables, their squares, and their product.
  • Example (from PYQ): For data X={6, 2} and Y={10, 10}, you are given , , and n=7. Let’s simplify with just these two points. Let X={6,2}, Y={10,10}. Find the sample covariance.
Click for Solution 1. **Calculate Means:** $\bar{x} = (6+2)/2=4$. $\bar{y} = (10+10)/2=10$. 2. **Create Calculation Table:** | x | y | $x_i-\bar{x}$ | $y_i-\bar{y}$ | $(x_i-\bar{x})(y_i-\bar{y})$ | |---|---|---|---|---| | 6 | 10 | 2 | 0 | 0 | | 2 | 10 | -2 | 0 | 0 | | **Sum** | | | | **0** | 3. **Calculate Sample Covariance:** * $s_{xy} = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{n-1} = \frac{0}{2-1} = 0$. * (The full dataset from the assignment yields a covariance of **~2.43**).

Pattern 9: Analyzing Contingency Tables

  • Abstraction: The key is to correctly identify your denominator. For a Marginal proportion, the denominator is the grand total. For a Conditional proportion, the denominator is a row or column total.
  • Example (from PYQ): Using the table below, what proportion of students in good economic conditions are borderline?
Econ. Condition…BorderlineTotal
Good…149377
Poor…104348
Total…253725
Click for Solution 1. **Identify the Condition:** The question restricts us to the population of "students in *good* economic conditions". This means our "whole" is the total for the "Good" row. 2. **Find the Denominator:** The total for the "Good" row is **377**. 3. **Find the Numerator:** The number of students who are both "Good" and "Borderline" is **149**. 4. **Calculate the Proportion:** * Proportion = 149 / 377 ≈ 0.395 or **~0.40**.

Of course. Here is the final piece of your Statistics arsenal: a concise, powerful reference sheet of all the core formulas and concepts from Weeks 1-4, combined with a practical guide on how to strategically assess any problem and choose the right tool.

The Strategist’s Statistics Arsenal: Formulas, Concepts, and Application Guide (Weeks 1-4)

This guide is designed for rapid recall and strategic application. It’s divided into two parts:

  1. The Arsenal: All the core formulas and concepts in one place.
  2. The Strategist’s Guide: A “how-to” for assessing any problem and choosing the right weapon from your arsenal.

Part 1: The Arsenal (Core Formulas & Concepts)

Week 1: The Language of Data

ConceptCore Rule / DefinitionTL;DR (The Core Idea)
Population vs. SamplePopulation: All items of interest.
Sample: A subset of the population.		The whole ocean vs. a bucket of water.
Inferential vs. DescriptiveDescriptive: Summarizes the sample.
Inferential: Generalizes to the population.		A fact about the bucket vs. a guess about the ocean.
Variable TypesCategorical: Labels (e.g., city).
Numerical: Numbers (e.g., age).		Can you do math on it? No → Categorical. Yes → Numerical.
Numerical Sub-TypesDiscrete: Countable (e.g., # of cars).
Continuous: Measurable (e.g., height).		Are there gaps between values? Yes → Discrete. No → Continuous.
Scales of Measurement (NOIR)Nominal: Names.
Ordinal: Order.
Interval: Equal intervals.
Ratio: True zero.		A runner’s Name (Nominal), their finishing Order (Ordinal), the tIme of day (Interval), and their finishing Race time (Ratio).

Week 2: Describing Categorical Data

ConceptCore Formula / RuleTL;DR (The Core Idea)
Relative FrequencyThe proportion or percentage of the whole.
Central TendencyMode: The category with the highest frequency.
(Mean/Median are not defined for nominal data).		The most popular category.
VisualizationBar Chart: Compares counts.
Pie Chart: Shows parts of a whole.
Pareto Chart: Sorted bar chart.		Bar chart is the workhorse. Pareto chart is a bar chart that prioritizes.

Week 3: Describing Numerical Data

ConceptCore Formula / RuleTL;DR (The Core Idea)
Mean ()The average or balance point. Sensitive to outliers.
MedianThe middle value of a sorted dataset.The physical middle. Resistant to outliers.
Sample Variance ()The average squared distance from the mean. The “n-1” is for samples.
Sample Standard Deviation ()The typical distance of a data point from the mean, in the original units.
Interquartile Range (IQR)The range of the middle 50% of the data. Resistant to outliers.
Outlier FencesLower:
Upper: 		Any data point outside this “fence” is considered an outlier.

Week 4: Association Between Two Variables

ConceptCore Formula / RuleTL;DR (The Core Idea)
Contingency TableA two-way frequency table.Organizes data for two categorical variables.
Conditional ProportionThe proportion of one category given another. Your “whole” is just one row or column.
Sample Covariance ()Measures the direction of the linear relationship. Positive or negative.
Correlation Coefficient (r)Measures the direction and strength of the linear relationship, from -1 to +1.
The Golden RuleCorrelation ≠ CausationJust because two things are related doesn’t mean one causes the other.

Part 2: The Strategist’s Guide (How to Assess and Apply)

This is the TAA (Triage, Abstract, Act) framework in a compact, actionable format for Statistics.

The Nifty Keyword-to-Tool Assessor

Scan any problem for these keywords. The moment you see one, your brain should immediately jump to the associated “Tool” and “Action Plan”.

IF YOU SEE THE KEYWORD(S)…THEN THE CATEGORY IS…AND YOUR IMMEDIATE ACTION PLAN IS…
“all of…” vs. “a sample of…”Population vs. Sample (W1)Identify the big group (Population) and the small group (Sample).
“we conclude that all…”Inference (W1)Recognize this as a leap from sample to population.
”classify the variable…”Variable Classification (W1)Apply the NOIR framework and the Numerical/Categorical tests.
”mode”, “most frequent”, “pie chart”Categorical Data (W2)Use counting and proportions. Remember Mean/Median don’t apply.
”mean”, “average”, “standard deviation”, “variance”Numerical Data (W3)Calculate center (mean/median) and spread (std dev/IQR).
“wrongly noted”, “correct the mean”Error Correction (W3)1. Find the wrong sum (mean * n). 2. Fix the sum. 3. Recalculate the mean.
”outliers”, “quartiles”, “IQR”Outlier Detection (W3)1. Sort the data. 2. Find Q1 and Q3. 3. Calculate IQR. 4. Build the fences (, etc.). Check for data outside.
”relationship”, “association between”Two Variables (W4)First, check if variables are Numerical or Categorical.
”…given that…”, “…of those who…”Conditional Proportion (W4)This is a contingency table problem. Your denominator is the row or column total, NOT the grand total.
”correlation”, “covariance”, “strength and direction”Correlation (W4)This is a numerical relationship problem. Calculate r. Interpret its sign (direction) and magnitude (strength).

The “Memory Palace” Nifty Tricks

  • Mean vs. Median (The Outlier Test):

    • To remember which one is sensitive, think of a group of friends’ salaries. If Bill Gates (an outlier) walks in, the Mean salary shoots up to billions, becoming useless. The Median salary barely changes. The Median resists outliers.

  • The NOIR Framework (W1):

    • It’s a hierarchy of information, from least to most informative.
    • Nominal: You know nothing but the name.
    • Ordinal: You know the order.
    • Interval: You know the order and the spacing.
    • Ratio: You know the order, the spacing, and the true zero point.

  • The Correlation r Thermometer (W4):

    • Think of r as a thermometer for linear relationships.
    • +1°: Perfectly hot, positive relationship.
    • -1°: Perfectly cold, negative relationship.
    • 0°: No linear heat at all.
    • Values like +0.8° are very warm (strong positive), while +0.2° is just a little warm (weak positive).

By using this arsenal, you can quickly diagnose the type of problem you’re facing, recall the exact tool needed, and execute a clear, step-by-step plan to find the solution.

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