Theory_Consolidated

Stats 1: Complete Theory Notes

IIT Madras BS DS - Foundational Level All 12 Weeks Covered

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Week 1: Introduction to Data

1.1 Population vs Sample

Term	Definition
Population	Entire group of interest
Sample	Subset of population actually studied
Parameter	Numerical measure of population
Statistic	Numerical measure of sample

1.2 Types of Statistics

Type	Purpose	Example
Descriptive	Summarize data	”Average score is 75”
Inferential	Make predictions about population	”95% of all students pass”

1.3 Types of Variables

Categorical (Qualitative):

• Nominal: Names only (Gender, Color)
• Ordinal: Ordered categories (Grades: A > B > C)

Numerical (Quantitative):

• Discrete: Countable values (Number of students)
• Continuous: Any value in range (Height, Weight)

1.4 Scales of Measurement

Scale	Properties	Examples	Operations
Nominal	Labels only	Gender, City	Mode
Ordinal	Order exists	Ratings, Grades	Mode, Median
Interval	Equal intervals, no true zero	Temperature (°C)	Mean, SD
Ratio	True zero exists	Height, Weight, Money	All operations

Key Test: Can you say ”$x$ is twice $y$“? Only valid for Ratio.

1.5 Data Types

Type	Description
Cross-sectional	Data at single point in time
Time-series	Data over time
Structured	Organized in rows/columns
Unstructured	No predefined format

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Week 2: Describing Categorical Data

2.1 Frequency Tables

Term	Formula
Frequency	Count of occurrences
Relative Frequency	$\frac{\text{Frequency}}{\text{Total}}$
Cumulative Frequency	Running total

2.2 Visualizations

Chart	Best For
Bar Chart	Comparing categories
Pie Chart	Showing proportions (parts of whole)
Pareto Chart	Bar chart with cumulative line

2.3 Mode for Categorical Data

Mode = Most frequent category

In pie chart: Widest slice = Mode

Note: Mean and Median are NOT defined for nominal data!

2.4 Association (Contingency Tables)

Two variables are associated if the distribution of one changes depending on the value of the other.

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Week 3: Measures of Central Tendency & Dispersion

3.1 Central Tendency

Mean (Average)

$\bar{x}=\frac{{\sum }_{i=1}^n{x}_i}{n}$

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

Median

• Middle value when sorted
• If $n$ is even: Average of two middle values
• Robust to outliers

Mode

• Most frequent value
• Can have multiple modes (bimodal, multimodal)

3.2 Dispersion

Range

$\text{Range}=\text{Maximum}-\text{Minimum}$

Variance

Population: ${\sigma }^2=\frac{\sum (x_i-\mu {)}^2}{N}$

Sample: $s^2=\frac{\sum (x_i-\bar{x}{)}^2}{n-1}$

Shortcut Formula: ${\sigma }^2=\frac{\sum x_i^2}{N}-{\mu }^2$

Standard Deviation

$\sigma =\sqrt{{\sigma }^2}$

Interquartile Range (IQR)

$\text{IQR}=Q_3-Q_1$

Where:

• $Q_1$ = 25th percentile
• $Q_2$ = Median = 50th percentile
• $Q_3$ = 75th percentile

3.3 Linear Transformation

If $Y=aX+b$: $\text{Mean}(Y)=a\cdot \text{Mean}(X)+b$ $\text{Var}(Y)=a^2\cdot \text{Var}(X)$ $\text{SD}(Y)=|a|\cdot \text{SD}(X)$

Note: Adding constant $b$ does NOT change variance!

3.4 Outlier Detection

$\text{Lower Bound}=Q_1-1.5\times \text{IQR}$ $\text{Upper Bound}=Q_3+1.5\times \text{IQR}$

Values outside these bounds are outliers.

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Week 4: Correlation & Regression

4.1 Covariance

$\text{Cov}(X,Y)=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{N}$

• Positive: Both increase together
• Negative: One increases, other decreases
• Zero: No linear relationship

4.2 Correlation Coefficient (Pearson’s r)

$r=\frac{\text{Cov}(X,Y)}{S{D}_X\cdot S{D}_Y}$

Value of $r$	Interpretation
$r=1$	Perfect positive linear
$r=-1$	Perfect negative linear
$0.7 \leq	r
$0.3 \leq	r
$	r

Properties:

• $r\in [-1,1]$
• Unitless
• Symmetric: $r_{XY}=r_{YX}$

4.3 Linear Regression

$\hat{y}=a+bx$

Slope: $b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}=r\cdot \frac{S{D}_Y}{S{D}_X}$

Intercept: $a=\bar{y}-b\bar{x}$

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Week 5: Permutations & Combinations

5.1 Factorial

$n!=n\times (n-1)\times \cdots \times 2\times 1$ $0!=1$

5.2 Permutations (Order Matters)

$P(n,r)=\frac{n!}{(n-r)!}$

With Repetition (Identical Objects): $\frac{n!}{n_1!\cdot n_2!\cdots n_k!}$

5.3 Combinations (Order Doesn’t Matter)

$C(n,r)=\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$

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Week 6-8: Probability

6.1 Basic Probability

$P(A)=\frac{\text{Favorable outcomes}}{\text{Total outcomes}}$

Properties:

• $0\le P(A)\le 1$
• $P(\text{Sure event})=1$
• $P(\text{Impossible event})=0$

6.2 Addition Rule

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

For mutually exclusive events: $P(A\cup B)=P(A)+P(B)$

6.3 Multiplication Rule

$P(A\cap B)=P(A)\cdot P(B|A)$

For independent events: $P(A\cap B)=P(A)\cdot P(B)$

6.4 Conditional Probability

$P(A|B)=\frac{P(A\cap B)}{P(B)}$

6.5 Bayes’ Theorem

$P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$

Total Probability: $P(B)={\sum }_{i}P(B|A_i)\cdot P(A_i)$

6.6 Independence vs Mutual Exclusivity

Independent	Mutually Exclusive
$P(A\cap B)=P(A)\cdot P(B)$	$P(A\cap B)=0$
Both can occur	Cannot occur together

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Week 9-10: Discrete Random Variables

9.1 Probability Mass Function (PMF)

$P(X=x)$ for each value $x$

Requirements:

1. $P(X=x)\ge 0$
2. $\sum P(X=x)=1$

9.2 Expected Value

$E[X]=\sum x\cdot P(X=x)$

Properties:

• $E[aX+b]=aE[X]+b$
• $E[X+Y]=E[X]+E[Y]$

9.3 Variance

$\text{Var}(X)=E[X^2]-(E[X]{)}^2$

Properties:

• $\text{Var}(aX+b)=a^2\text{Var}(X)$
• For independent $X,Y$: $\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)$

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Week 11-12: Special Distributions

11.1 Binomial Distribution

$X\sim \text{Bin}(n,p)$: $n$ trials, probability $p$ of success

$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$

$E[X]=np$ $\text{Var}(X)=np(1-p)$

11.2 Poisson Distribution

$X\sim \text{Poisson}(\lambda )$: Average rate $\lambda$

$P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$

$E[X]=\text{Var}(X)=\lambda$

12.1 Uniform Distribution

$X\sim \text{Uniform}(a,b)$

$f(x)=\frac{1}{b-a}\text{ for }a\le x\le b$

$P(c<X<d)=\frac{d-c}{b-a}$

$E[X]=\frac{a+b}{2}$

12.2 Exponential Distribution

$X\sim \text{Exp}(\lambda )$

$f(x)=\lambda e^{-\lambda x}\text{ for }x\ge 0$

$P(X>x)=e^{-\lambda x}$

$E[X]=\frac{1}{\lambda }$

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End of Stats 1 Theory. Good luck!