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📊 Standard Deviation Calculator

Calculate standard deviation, variance, mean, and other statistics

Supports comma, space, and newline separators
Sample Std Dev (s)
6.1473
Divided by n-1
Population Std Dev (σ)
5.8318
Divided by n
Statistics Summary
Count (n)10
Sum (Σx)867
Mean (x̄)86.7
Median88.5
Sample Variance (s²)37.7889
Population Variance (σ²)34.01
Coefficient of Variation (CV)6.73%
Data Range
Minimum76
Q1 (25%)83
Median (50%)88.5
Q3 (75%)91
Maximum95
Range19

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Standard Deviation Calculator Guide

The Standard Deviation Calculator is a professional statistical analysis tool. Enter a set of data (supports comma, space, or newline separators) to automatically calculate key statistics including sample standard deviation (s), population standard deviation (σ), mean, variance, median, quartiles, range, and coefficient of variation (CV).

Standard deviation is the most important measure of data dispersion. A small standard deviation indicates data is concentrated around the mean, while a large standard deviation means data is more spread out. This tool provides both sample and population standard deviation calculations to meet different statistical analysis needs.

Core Statistical Formulas

【Mean】x̄ = Σxᵢ / n

【Population Variance】σ² = Σ(xᵢ - x̄)² / n
【Population Std Dev】σ = √(Σ(xᵢ - x̄)² / n)

【Sample Variance】s² = Σ(xᵢ - x̄)² / (n-1)  (Bessel's correction)
【Sample Std Dev】s = √(Σ(xᵢ - x̄)² / (n-1))

【Coefficient of Variation】CV = (σ / |x̄|) × 100%

【Median】The middle value after sorting (average of two middle values for even count)

Practical Examples

📋
【Example 1】Student grade analysis Data: 85, 90, 78, 92, 88, 76, 95, 83, 89, 91 Count n=10, Mean x̄=86.7 Population std dev σ ≈ 5.66 Sample std dev s ≈ 5.97 CV ≈ 6.53% Interpretation: Grades are relatively concentrated; CV<10% indicates good data consistency 【Example 2】Comparing two datasets A: 50, 50, 50, 50, 50 → Mean=50, Std Dev=0 (no variation) B: 0, 25, 50, 75, 100 → Mean=50, Std Dev≈39.53 (high dispersion) Same mean but vastly different distributions — this is what standard deviation reveals

Important Notes

💡
- Sample standard deviation divides by (n-1) instead of n; this is Bessel's correction for unbiased estimation - Use population std dev (σ) when data represents the entire population; use sample std dev (s) for sampled data - At least 2 data points are needed to calculate standard deviation - Coefficient of variation (CV) is used to compare dispersion across data with different units - Outliers significantly affect standard deviation; consider using median and IQR for a more robust assessment

Application Scenarios

  • - Education: Analyze grade distribution, evaluate teaching quality consistency
  • - Quality control: Monitor product dimension deviations, determine process stability
  • - Financial risk: Calculate portfolio volatility, assess asset return stability
  • - Scientific experiments: Measurement error analysis, experiment reproducibility testing
  • - Market research: User rating distribution, satisfaction survey data analysis

Standard Deviation vs Other Dispersion Measures

There are multiple ways to measure data dispersion: (1) Range = Maximum - Minimum, simplest but sensitive to outliers; (2) Interquartile Range (IQR) = Q3 - Q1, more robust against outliers; (3) Mean Absolute Deviation (MAD) = average of |xᵢ-x̄|, intuitive but mathematically less convenient than standard deviation.

Standard deviation is the most commonly used dispersion measure because of its special role in the normal distribution: approximately 68% of data falls within μ±1σ, 95% within μ±2σ, and 99.7% within μ±3σ (the 68-95-99.7 rule). This property makes standard deviation irreplaceable in hypothesis testing and confidence interval construction.

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