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Probability & Statistics Calculator

Basics
Descriptive Stats
Probability
Distributions
Inference
Practice
Understanding Probability & Statistics

Key Concepts

  • Data types and measurement scales
  • Descriptive vs. inferential statistics
  • Population vs. sample
  • Random variables and probability
  • Statistical significance

Statistical Foundations

  • Measures of central tendency
  • Measures of dispersion
  • Probability distributions
  • Sampling methods
  • Hypothesis testing framework

Probability Basics

  • Sample space and events
  • Probability axioms
  • Conditional probability
  • Independent events
  • Bayes' theorem

Applications

  • Data science and analytics
  • Research methodology
  • Quality control
  • Risk assessment
  • Decision making under uncertainty

Data & Statistics: The Basics

Statistics is the science of collecting, analyzing, interpreting, and presenting data. It provides tools to make informed decisions in the face of uncertainty. Understanding statistics helps us extract meaningful insights from data and make evidence-based conclusions.

Note: Statistics can be broadly divided into descriptive statistics (summarizing data) and inferential statistics (drawing conclusions from samples). Probability theory provides the mathematical foundation for statistics, especially for modeling random phenomena and uncertainty.
Descriptive Statistics Calculator

Data Input

Enter your dataset below (comma, space, or new-line separated numbers):

Results Summary

Central Tendency

Mean:-
Median:-
Mode:-

Dispersion

Range:-
Variance:-
Std Deviation:-

Position

Minimum:-
Q1 (25%):-
Q3 (75%):-
Maximum:-

Shape

Skewness:-
Kurtosis:-
IQR:-
Count:-

Key Formulas:

Mean: μ = (Σx) / n

Variance: σ² = Σ(x - μ)² / n

Standard Deviation: σ = √σ²

Probability Calculators

Combination & Permutation Calculator

Results

Combinations (nCr)

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C(n,r) = n! / (r! × (n-r)!)

Permutations (nPr)

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P(n,r) = n! / (n-r)!

Conditional Probability Calculator

Bayes' Theorem Results

P(B|A) - Probability of B given A

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P(B|A) = P(A|B) × P(B) / P(A)

Joint Probability P(A∩B)

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P(A∩B) = P(A|B) × P(B)

Probability Distributions

Normal Distribution Calculator

Probability Density
Cumulative Probability
Probability Density at x

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Cumulative Probability

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Binomial Distribution Calculator

Binomial Probability

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P(X = k) = C(n,k) × pk × (1-p)n-k
Statistical Inference Tools

Confidence Interval Calculator

Confidence Interval for Mean

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CI = x̄ ± zα/2 × (s / √n)

Hypothesis Testing Calculator

Z-Test
T-Test
Z-Test Results
Z-statistic:-
P-value:-
Decision:-
T-Test Results
T-statistic:-
Degrees of freedom:-
P-value:-
Decision:-
Practice Problems

Problem 1: Descriptive Statistics

For the dataset [15, 20, 12, 18, 25, 22, 18], what is the mean and median?

(A) Mean: 18.57, Median: 18
(B) Mean: 18, Median: 18.57
(C) Mean: 19.3, Median: 18
(D) Mean: 18.57, Median: 20

Problem 2: Probability Basics

If P(A) = 0.6, P(B) = 0.4, and A and B are independent events, what is P(A ∩ B)?

(A) 0.24
(B) 0.24
(C) 0.36
(D) 1.0

Problem 3: Combinations

How many different 3-person committees can be formed from a group of 8 people?

(A) 24
(B) 336
(C) 56
(D) 512

Problem 4: Normal Distribution

In a normal distribution with mean 70 and standard deviation 5, approximately what percentage of data falls between 65 and 75?

(A) 50%
(B) 95%
(C) 68%
(D) 99.7%

Problem 5: Confidence Intervals

A sample of 49 observations has a mean of 35 and a standard deviation of 7. The 95% confidence interval for the population mean is:

(A) [33.02, 36.98]
(B) [33.02, 36.98]
(C) [32.50, 37.50]
(D) [34.02, 35.98]

Answers:

  1. A. Mean = (15+20+12+18+25+22+18)/7 = 130/7 = 18.57. Median is the middle value when arranged in order: [12, 15, 18, 18, 20, 22, 25], so median = 18.
  2. B. When events are independent, P(A ∩ B) = P(A) × P(B) = 0.6 × 0.4 = 0.24.
  3. C. The formula for combinations is C(n,r) = n!/(r!(n-r)!) = 8!/(3!5!) = 56.
  4. C. In a normal distribution, approximately 68% of data falls within 1 standard deviation of the mean, which is the range 65 to 75 (mean ± 1 SD).
  5. B. The 95% confidence interval is x̄ ± z₀.₀₂₅ × (s/√n) = 35 ± 1.96 × (7/√49) = 35 ± 1.96 × 1 = 35 ± 1.96 = [33.02, 36.98].