Introduction to Statistics

Lecture 01 — Basic Concepts & Foundations

Abdullah Al Mahmud

Statistics: What & How

Topics

• What is Statistics?
• How Statistics Works
• Probability and Statistics
• Applications of Statistics
• Example Problems

What is Statistics?

Three Meanings of “Statistics”

1. Plural of statistic — numerical facts (e.g., “the statistics show 60% approval”)
2. Table of data — organized numerical information
3. Methodology — the science of collecting, organizing, analyzing, and interpreting data

Definition (Croxton & Cowden): Statistics is the science of collection, presentation, analysis, and interpretation of numerical data.

How Statistics Works

The Statistical Process

1. Population → Define the group of interest
2. Sample → Select a representative subset
3. Analyze → Apply statistical methods
4. Infer → Draw conclusions about the population

There are many sampling techniques: simple random, stratified, cluster, systematic, etc.

Probability vs. Statistics

Aspect	Probability	Statistics
Direction	Population → Sample (deductive)	Sample → Population (inductive)
Known	Population parameters	Sample data
Unknown	Sample outcomes	Population parameters
Question	“Given the model, what data?”	“Given the data, what model?”

Applications of Statistics

• Identification of unwanted spam messages in e-mail
• Segmentation of customer behavior for targeted advertising
• Forecasts of weather and long-term climate changes
• Prediction of election outcomes
• Algorithms for auto-piloting drones & self-driving cars
• Optimization of energy use in buildings
• Crime hotspot projection
• Discovery of genetic sequences linked to diseases

Chapter 01: Basic Concepts

Chapter Overview

• Definition & Mechanism of Statistics
• Population and Sample
• Variable Types
• Scales of Measurement
• Summation Notation & Theorems
• Mathematical Problems

Mechanism of Statistics

The statistical workflow involves five stages:

1. Data Collection — Gathering raw information
2. Organization — Tabulating, grouping, coding
3. Presentation — Tables, charts, graphs
4. Analysis — Descriptive & inferential methods
5. Interpretation — Drawing meaningful conclusions

Population and Sample

Population — The complete set of items/events of interest Sample — Any subset of the population

Population Types

Type	Description	Examples
Finite	Countable, limited size	Students in a classroom, books in a library, employees in a company
Infinite	Uncountable, unlimited	Stars in the sky, outcomes of infinite coin flips, water molecules in the ocean

Variable and Constant

Concept	Description	Examples
Variable	Characteristic that varies across units	Income, height, GPA, die roll result
Random Variable	Variable whose value depends on chance	Result of a die throw, number of heads in 10 flips
Constant	Fixed, unchanging value	\(\pi\), \(e\), speed of light in vacuum

Note: “Name of students” is a label (qualitative variable), not a constant.

Types of Variables

By Nature of Data

Type	Also Called	Values	Examples
Qualitative (Categorical)	Non-numeric	Categories, labels	Gender, religion, blood type, color
Quantitative (Numerical)	Numeric	Numbers with magnitude	Income, temperature, height, GPA

By Continuity (Quantitative Only)

Type	Description	Examples
Discrete	Countable, gaps between values	Number of children, die roll, defect count
Continuous	Any value in an interval	Height, weight, temperature, time

By Role

• Univariate — One variable studied
• Bivariate — Two variables studied
• Multivariate — Multiple variables studied

Scales of Measurement

Describes the nature of information within the values assigned to a variable.

Scale	Properties	Operations	Examples
Nominal	Categories only, no order	\(=, \neq\)	Gender, religion, blood type, street #, room #
Ordinal	Categories with order	\(=, \neq, <, >\)	Rating (poor–excellent), income group, education level
Interval	Order + equal intervals, no true zero	\(=, \neq, <, >, +, -\)	Temperature (°C/°F), dates (AD), IQ scores, Cartesian coordinates
Ratio	Order + equal intervals + true zero	All arithmetic	Height, weight, income, distance, radius, diameter

Key distinction: Interval scales lack a true zero (0°C ≠ “no temperature”); ratio scales have meaningful zero (0 kg = no mass).

Scale Examples — Practice

Classify each variable:

Variable	Scale	Reason
Gender	Nominal	Categories, no order
Religion	Nominal	Categories, no order
Temperature (°C)	Interval	Equal intervals, no true zero
Income group (Lower, Middle, High)	Ordinal	Ordered categories
Income ($)	Ratio	True zero, all operations valid
Distance of stars	Ratio	True zero, measurable
Radius of screws	Ratio	True zero, measurable
Diameter of trees	Ratio	True zero, measurable
Room number	Nominal	Label only, no magnitude

Another Example — Match the Scale

Data	Correct Scale	Why
Poor, Bad, Good, Excellent	Ordinal	Ordered categories
Scale of −10 to 10: −10, −2, 0, 5, 10	Interval	Equal intervals, arbitrary zero
Awesome, Amazing, Mind-blowing, Stunning	Ordinal	Ordered by intensity
Scale of 0 to 10: 0, 5, 8, 10	Ratio	True zero, equal intervals

⚠️ Common mistake: “Poor–Excellent” is ordinal (not ratio); “Awesome–Stunning” is ordinal (not nominal).

Interval Scale — Special Properties

Multiplication/division are not meaningful on interval scales:

• Temperature: 20°C is not “twice as hot” as 10°C
• Dates: Year 2000 is not “twice” Year 1000
• Cartesian coordinates: (2,2) is not “twice” (1,1)
• Direction in degrees: 90° is not “twice” 45°

For more: Wikipedia — Level of measurement · Stat Mania discussion

Operations Allowed by Scale

Origin and Scale Shifts

Given data \(x_1, x_2, \dots, x_n\):

Transformation	Formula	Effect
Origin shift	\(y_i = x_i \pm a\)	Adds/subtracts constant
Scale shift	\(y_i = b \cdot x_i\) or \(x_i / b\)	Multiplies/divides by constant
Both (standardization)	\(y_i = \frac{x_i - a}{b}\)	Center & scale (e.g., z-scores)

Why? To compare variables on different scales, simplify calculations, or meet model assumptions.

Summation Notation

\[\sum_{i=1}^n x_i = x_1 + x_2 + \cdots + x_n\]

Examples: - \(\sum_{i=1}^4 x_i = x_1 + x_2 + x_3 + x_4\) - \(\sum_{i=1}^{10} x_i = x_1 + x_2 + \cdots + x_{10}\) - \(\sum_{i=1}^n a = n a\) (summing a constant \(n\) times)

Summation Theorems

Theorem 1: Constant Multiple

\[\sum_{i=1}^n b x_i = b \sum_{i=1}^n x_i\]

Theorem 2: Linear Transformation

\[\sum_{i=1}^n (a x_i - b) = a \sum_{i=1}^n x_i - n b\]

Theorem 3: Quadratic Form

\[\sum_{i=1}^n (a x_i^2 - b x_i + c) = a \sum_{i=1}^n x_i^2 - b \sum_{i=1}^n x_i + n c\]

Theorem 4: Two Variables

\[\sum_{i=1}^n (a x_i - b y_i) = a \sum_{i=1}^n x_i - b \sum_{i=1}^n y_i\]

Theorem 5: Squared Linear Form

\[\sum_{i=1}^n (a x_i - b)^2 = a^2 \sum_{i=1}^n x_i^2 - 2ab \sum_{i=1}^n x_i + n b^2\]

Theorem 6: Square of Sum vs Sum of Squares

\[\left(\sum_{i=1}^n x_i\right)^2 = \sum_{i=1}^n x_i^2 + 2\sum_{i < j} x_i x_j\] > Key: \((\sum x_i)^2 \neq \sum x_i^2\) (cross-product terms!)

Product Notation

\[\prod_{i=1}^k x_i = x_1 \times x_2 \times \cdots \times x_k\]

Theorem 7: Product of Products

\[\prod_{i=1}^k (x_i y_i) = \left(\prod_{i=1}^k x_i\right)\left(\prod_{i=1}^k y_i\right)\]

Theorem 8: Constant Product

\[\prod_{i=1}^n c = c^n\]

Double Summation

Given \(X = \{20, 25, 15\}\), \(Y = \{15, 30, 20\}\):

\[\sum_{i=1}^3 \sum_{j=1}^3 x_i y_j = (20+25+15)(15+30+20) = 60 \times 65 = 3900\]

Row-by-row expansion: - Row 1 (\(x_1=20\)): \(20(15+30+20) = 1300\) - Row 2 (\(x_2=25\)): \(25(15+30+20) = 1625\) - Row 3 (\(x_3=15\)): \(15(15+30+20) = 975\) - Total: \(3900\)

Double Summation Theorems

Theorem 9: \(\displaystyle \sum_{i=1}^m \sum_{j=1}^n (x_i + y_j) = n\sum_{i=1}^m x_i + m\sum_{j=1}^n y_j\)

Theorem 10: \(\displaystyle \sum_{i=1}^m \sum_{j=1}^n x_i y_j = \left(\sum_{i=1}^m x_i\right)\left(\sum_{j=1}^n y_j\right)\)

Practice Problems

Problem 1

Given \(f_1=2, f_2=4, f_3=6\) and \(x_1=-3, x_2=7, x_3=4\), find: - i. \(\sum f_i x_i\) - ii. \(\sum f_i x_i^2\) - iii. \(\sum f_i (x_i - 5)^2\)

Problem 2

Find \(\sum_{i=1}^{10} (x_i - 4)\) given \(\sum_{i=1}^{10} x_i = 20\).

Problem 3

Prove: \(\sum_{i=1}^k ab x_i = ab \sum_{i=1}^k x_i\)

Problem 4

Prove: \(\prod_{i=1}^n c = c^n\)

Problem 5

Given \(\sum_{i=1}^{10} x_i = 20\) and \(\sum_{i=1}^{10} x_i^2 = 400\), find \(\sum_{i=1}^{10} (x_i^2 + 5x_i + 10)\).

Problem 6

With \(X = \{20, 25\}\), \(Y = \{15, 30\}\): - i. Show \(\sum X_i^2 \neq (\sum X_i)^2\) - ii. Show \(\sum X_i Y_i \neq \sum\sum X_i Y_j\)

Creative Questions

See the question bank: STAT1 Creative Questions

Creative Question 2

Income (\(x\))	120	130	88	150	175	144	180	200	160	155
Expense (\(y\))	80	120	70	100	160	114	170	195	140	131

Prove: 1. \(\displaystyle \sum_{i=1}^{10}\sum_{j=1}^{10} x_i y_j = \left(\sum_{i=1}^{10} x_i\right)\left(\sum_{j=1}^{10} y_j\right)\) 2. \(\displaystyle \sum_{i=1}^{10}\sum_{j=1}^{10} (x_i - y_j) = 10\sum_{i=1}^{10} x_i - 10\sum_{j=1}^{10} y_j\) 3. \(\displaystyle \sum_{i=1}^{10} x_i y_i \neq \left(\sum_{i=1}^{10} x_i\right)\left(\sum_{i=1}^{10} y_i\right)\)

End of Chapter 01

Thank You! 🎓

Questions?