Foundations

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What is statistics?

Statistics is the science of learning from data. It includes:

• collecting data
• organizing and summarizing data
• analyzing patterns
• interpreting results to make decisions

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Major branches of statistics

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Descriptive statistics

Descriptive statistics focuses on organizing, summarizing, and presenting data. Examples:

• class average marks
• highest and lowest temperature this week
• bar chart of students by branch

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Inferential statistics

Inferential statistics uses sample data to draw conclusions about a population. Because inference involves uncertainty, probability is essential. Examples:

• surveying 200 voters to estimate support in a city
• testing a sample of bulbs to estimate defect rate in full production

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Population and sample

• Population: the complete set of all units of interest.
• Sample: a subset of the population used for study.

Example:

• Population: all students in your college
• Sample: 120 students chosen for a survey

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Census vs sample survey

• Census: data from every unit in the population
• Sample survey: data from part of the population

Sample surveys are faster and cheaper, but must be designed carefully.

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Parameter vs statistic

• Parameter: numerical summary of a population (usually unknown)
• Statistic: numerical summary from a sample (used to estimate parameter)

Example:

• Population mean height = parameter
• Sample mean height = statistic

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Purpose of statistical analysis

• If the goal is to describe and summarize observed information, the study is descriptive.
• If the goal is to use a sample and make conclusions about a population, the study is inferential.
• A descriptive study can be done on a sample or a population.

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What is data?

Data are facts and figures collected for analysis, presentation, and interpretation. Data can be:

• numbers (exam marks, income)
• labels (department, blood group)
• text or categories (feedback type)

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Why do we collect data?

We collect data to understand characteristics of groups such as people, places, or things. Typical goals:

• comparison (which section performed better)
• prediction (next month sales)
• decision-making (admit/reject, pass/fail)

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Data collection

• Primary data: collected first-hand for current study
• Secondary data: already collected by someone else (reports, government data, publications)

Primary data methods:

• surveys
• experiments
• observations

Secondary data sources:

• census reports
• institutional records
• research articles

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Cases and variables

• Case (observation): a unit from which data are collected.
• Variable: a characteristic or attribute that can vary across cases.

Example (school dataset):

• Cases: each student.
• Variables: name, date of birth, marks, board, etc.

In a data table:

• Rows represent cases.
• Columns represent variables.

Note: a missing value (“not available”) and a value of 0 are not the same. Also note:

• variable names should be clear (attendance_percent, final_marks)
• keep units consistent in one column

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Categorical and numerical variables

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Categorical data (qualitative)

• Represents labels or groups.

Examples:

• gender
• branch (CSE, ECE, ME)
• grade (A, B, C)

Common types:

• Nominal: categories with no natural order (blood group)
• Ordinal: categories with order (poor < average < good)

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Numerical data (quantitative)

• Describes numerical properties of cases.
• Uses measured units.

Examples:

• age, height, weight
• number of siblings
• salary

Common types:

• Discrete: countable values (0, 1, 2, ...)
• Continuous: measurable values on a scale (height, time, temperature)

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Measurement units

The unit gives meaning to numerical values (for example, kilograms for weight, rupees for price, centimeters for height).
Values in a numerical variable should be recorded in a common unit. Bad practice: mixing cm and m in one height column without conversion.

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Scales of measurement (important)

• Nominal: labels only
• Ordinal: rank/order, unequal gaps
• Interval: equal gaps, no true zero (temperature in Celsius)
• Ratio: equal gaps, true zero (height, weight, income)

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Data classification

• Categorical
• Numerical
  • Discrete
  • Continuous

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Cross-sectional and time-series data

• Cross-sectional data: data observed at one point in time across cases.
• Time-series data: data recorded over time for one case or unit.
• Time plot: a graph of time-series values in chronological order.

Examples:

• Cross-sectional: income of 50 households in March 2026
• Time-series: monthly electricity bill of one hostel from Jan-Dec 2025

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Common mistakes to avoid

• confusing population with sample
• treating category labels as numbers for arithmetic
• ignoring units while comparing values
• assuming sample results are exact for population without uncertainty