📐 Math ResourcesLast updated March 29, 2026

Statistics & Data Analysis Guide: Standard Deviation, Probability & More

Master statistical concepts with visual explanations and calculators

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Key Takeaways

Standard deviation measures data spread: σ = √[Σ(xi – μ)² / N] for population, s = √[Σ(xi – x̄)² / (N-1)] for sample.
Mean is the average, median is the middle value, mode is the most frequent — each tells a different story.
Probability ranges from 0 (impossible) to 1 (certain): P(A) = favorable outcomes / total outcomes.
The 68-95-99.7 rule: 68% of data falls within 1 SD of the mean, 95% within 2, 99.7% within 3.

Statistics transforms raw data into actionable insights. Whether you're a student preparing for exams, a researcher analyzing results, or a professional making data-driven decisions, this guide covers essential statistical concepts — aligned with Khan Academy and UC Berkeley Statistics curriculum — with our free statistics calculators.

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Measures of Central Tendency

Mean (average), median (middle value), and mode (most frequent) are the three measures of central tendency. Each has strengths: mean uses all data points, median resists outliers, and mode identifies the most common value. Calculate all three with our mean median mode calculator.

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Measures of Dispersion

Standard deviation, variance, and range measure how spread out data is. Variance = SD². Range = max – min. A larger standard deviation indicates more variability. Use our standard deviation calculator to analyze your dataset.

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Probability Fundamentals

Probability quantifies likelihood on a scale of 0 to 1. Key concepts: independent vs. dependent events, combinations vs. permutations, conditional probability (Bayes' theorem), and expected value. Try our probability calculator for common scenarios.

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The Normal Distribution

The bell curve (normal distribution) is fundamental to statistics. Key properties: symmetrical around the mean, described by mean (μ) and standard deviation (σ), and follows the 68-95-99.7 rule. Z-scores measure how many standard deviations a value is from the mean: z = (x – μ) / σ.

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Related Tools & Calculators

4 free tools linked to this guide

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Standard Deviation Calculator

Calculate standard deviation and variance for any dataset.

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Mean Median Mode Calculator

Find mean, median, and mode instantly.

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Probability Calculator

Calculate probabilities for events and combinations.

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Prime Number Calculator

Check prime numbers and find prime factorizations.

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Frequently Asked Questions

What is standard deviation?

Standard deviation measures how spread out numbers are from the mean. A low SD means data points cluster near the average; a high SD means they're more dispersed. Calculate it with our standard deviation calculator.

When should I use mean vs. median?

Use mean for symmetrically distributed data. Use median when data is skewed or has outliers (e.g., income data where a few very high values inflate the mean). Median is more robust to outliers.

How do I calculate probability?

Basic probability: P(event) = number of favorable outcomes / total possible outcomes. For combined events: P(A and B) = P(A) × P(B) if independent; P(A or B) = P(A) + P(B) – P(A and B).

What is a normal distribution?

A normal (Gaussian) distribution is a bell-shaped curve where data is symmetrically distributed around the mean. About 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3. Many natural phenomena follow this pattern.

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