Statistics on Arshad Siddiqui

Formula Sheet

Thu, 12 Mar 2026 00:00:00 +0000

Formula Sheet #

This page is a quick reference of definitions + formulas, grouped by the modules.

Notation #

Sample size: \( n \) (sample), \( N \) (population)
Sample mean: \( \bar{x} \) , population mean: \( \mu \)
Sample variance: \( s^2 \) , population variance: \( \sigma^2 \)
Sample SD: \( s \) , population SD: \( \sigma \)
Complement: \( A^c \)
Intersection (“and”): \( A\cap B \) , union (“or”): \( A\cup B \)
Conditional probability: \( P(A\mid B) \)

1. Basic Probability & Statistics #

1.1 Measures of Central Tendency #

Arithmetic mean #

Sample mean (ungrouped):

Stats Formula Sheet

Wed, 25 Feb 2026 00:00:00 +0000

Stats Formula Sheet #

Keep this page as a quick reference of definitions + formulas.

Notation #

Sample size: \( n \) (sample), \( N \) (population)
Mean: \( \bar{x} \) (sample), \( \mu \) (population)
Variance: \( s^2 \) (sample), \( \sigma^2 \) (population)
Standard deviation: \( s \) (sample), \( \sigma \) (population)

Module 1: Basic Statistics #

Measures of Central Tendency #

Sample mean (ungrouped):

Basic Statistics

Mon, 01 Jan 0001 00:00:00 +0000

Basic Statistics #

Statistics: describes data (what you see).
Probability: models uncertainty (what you don’t know yet).

Summarise a dataset using central tendency and variability
Explain core probability ideas using simple examples
Apply the axioms of probability
Distinguish mutually exclusive vs independent events

flowchart TD
 A[Dataset] --> B[Central Tendency]
 A --> C[Variability]
 B --> B1[Mean]
 B --> B2[Median]
 B --> B3[Mode]
 C --> C1[Range]
 C --> C2[Variance]
 C --> C3[Standard Deviation]
 C --> C4[IQR]

Measures of Central Tendency #

Central tendency tells you where the “middle” of the data is. Describes a set of scores with a single number that describes the PERFORMANCE of the group.

Basic Probability

Mon, 01 Jan 0001 00:00:00 +0000

Basic Probability #

Probability models uncertainty: what you don’t know yet, but want to reason about.

Key takeaway: Probability is a number between 0 and 1 that measures how likely an event is. The whole topic is about defining events clearly and applying a few core rules consistently.

Probability quantifies uncertainty: a number between 0 and 1.

0 means: impossible
1 means: certain

Terminology #

Random experiment #

A random experiment is an action whose outcome is not known in advance.

Hypothesis Testing

Mon, 01 Jan 0001 00:00:00 +0000

Hypothesis Testing #

Hypothesis testing is a statistical decision-making method used to decide whether sample evidence is strong enough to reject an initial assumption about a population.

It connects probability, sampling distributions, confidence intervals, significance levels, and decision rules.

Key takeaway:
Hypothesis testing is not about proving something with certainty.

It is about asking:

If the null hypothesis were true, how surprising would this sample result be?

Prediction & Forecasting

Mon, 01 Jan 0001 00:00:00 +0000

Prediction & Forecasting #

Prediction and forecasting use statistical models to estimate unknown or future values.

In this module, the focus is on correlation, regression, and time series forecasting.

Key takeaway:
Prediction estimates a value using a model.

Forecasting is prediction where the order of time matters.

Correlation
Regression
Time series analysis
Components of time series data
Moving average and weighted moving average
AR model
ARMA model
ARIMA model
SARIMA and SARIMAX
VAR and VARMAX
Simple exponential smoothing

Prediction vs Forecasting ☆ #

Concept	Meaning	Example
Prediction	Estimate an unknown output	Predict house price from area and rooms
Forecasting	Predict future values using time order	Forecast sales for next month

All forecasting is prediction, but not all prediction is forecasting.

Overall Workflow #

flowchart LR
 A[Data] --> B[Explore Pattern]
 B --> C[Choose Model]
 C --> D[Train or Fit]
 D --> E[Validate]
 E --> F[Predict or Forecast]
 F --> G[Interpret Error]

 style A fill:#E1F5FE
 style B fill:#C8E6C9
 style C fill:#FFF9C4
 style D fill:#EDE7F6
 style E fill:#C8E6C9
 style F fill:#E1F5FE
 style G fill:#FFF9C4

Correlation ☆ #

Correlation measures the direction and strength of linear relationship between two variables.

Gaussian Mixture Model & Expectation Maximization

Mon, 01 Jan 0001 00:00:00 +0000

Gaussian Mixture Model & Expectation Maximization #

A Gaussian Mixture Model represents data as a weighted combination of multiple Gaussian distributions.

It is commonly used for soft clustering and density estimation.

Key takeaway:
K-means gives hard cluster membership.

GMM gives probabilities of belonging to each cluster.

Gaussian Mixture Model
soft clustering
mixing coefficients
latent variables
likelihood and log-likelihood
Expectation-Maximization algorithm
E-step and M-step
responsibilities
convergence

Motivation ☆ #

Many real datasets are not described well by one Gaussian distribution.