Dinuka Himsara

Probability & Statistics

Interactive visual guide to understanding statistical concepts with live demonstrations

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Descriptive Statistics

Summarizing and describing data through measures of central tendency, variability, and distribution shape.

Measures of Central Tendency

Mean (average), Median (middle value), Mode (most frequent)

Mean = Σx / n
Median = middle value when ordered
Mode = most frequent value

Measures of Variability

Standard Deviation, Variance, Range, Interquartile Range

Variance = Σ(x - μ)² / n
Standard Deviation = √Variance

Distribution Shape

Skewness, Kurtosis, Symmetry

Click a button to generate sample statistics

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Probability

Mathematical framework for quantifying uncertainty and likelihood of events.

Basic Rules

Addition rule, Multiplication rule, Complement rule

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A ∩ B) = P(A) × P(B|A)
P(A') = 1 - P(A)

Conditional Probability

Probability of an event given another event has occurred

P(A|B) = P(A ∩ B) / P(B)

Bayes' Theorem

Updating probabilities based on new evidence

P(A|B) = P(B|A) × P(A) / P(B)

Interactive probability demonstrations

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Probability Distributions

Mathematical functions describing the likelihood of different outcomes.

Discrete Distributions

For countable outcomes

Binomial: Fixed trials, success/failure

P(X=k) = C(n,k) × p^k × (1-p)^(n-k)

Poisson: Rare events over time/space

P(X=k) = (λ^k × e^(-λ)) / k!

Continuous Distributions

For continuous values

Normal: Bell-shaped, symmetric

f(x) = (1/σ√(2π)) × e^(-½((x-μ)/σ)²)

Exponential: Time between events

f(x) = λe^(-λx)

Click to visualize different distributions

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Sampling

Methods for selecting representative subsets from populations to make inferences.

Population vs Sample

Population: Complete group of interest
Sample: Subset used for analysis

Population Parameter: μ, σ, p
Sample Statistic: x̄, s, p̂

Sampling Methods

Random, Systematic, Stratified, Cluster sampling

• Random: Every member has equal chance
• Stratified: Divide into groups, sample from each
• Cluster: Sample entire groups
• Systematic: Every nth member

Sampling Bias

Selection bias, Non-response bias, Convenience sampling issues

Compare different sampling methods

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Sampling Distributions

Distribution of sample statistics across multiple samples, foundation for statistical inference.

Central Limit Theorem

Sample means approach normal distribution as sample size increases

X̄ ~ N(μ, σ/√n)
Standard Error = σ/√n

Distribution of Sample Mean

Mean = population mean, Standard deviation = σ/√n

E[X̄] = μ
Var(X̄) = σ²/n
SE(X̄) = σ/√n

Applications

Confidence intervals, Hypothesis testing, Statistical inference

Explore sampling distribution concepts