Standard Deviation Calculator
Online calculator for calculating the standard deviation (spread measure) of a data series
Standard Deviation Calculator
Standard Deviation
Standard deviation is a measure of spread of values around the mean. It is the square root of variance.
Standard Deviation Concept
Standard deviation shows how much values spread.
Small σ: Values close to mean. Large σ: Values widely distributed.
● Small Standard Deviation ● Large Standard Deviation
What is Standard Deviation?
Standard deviation is a fundamental spread measure in descriptive statistics:
- Definition: Square root of variance, measures average deviation from mean
- Notation: σ (sigma) for population, s for sample
- Unit: Same unit as the original data
- Property: Always non-negative (σ ≥ 0)
- Application: Spread analysis, quality control, risk measure
- Interpretation: The larger, the more the data spreads
Sample vs. Population
Depending on whether you have the entire population or just a sample, the formulas differ:
Sample (s)
Usage: When you have only a part of the population.
Formula: Division by (n-1) - Bessel's correction
Purpose: Unbiased estimator for σ
Example: 100 randomly selected students from a school
Population (σ)
Usage: When you have the entire population.
Formula: Division by n
Purpose: Exact value for the population
Example: All students in a class with 25 students
Applications of Standard Deviation
Standard deviation is used in many fields:
Quality Control
- Monitor process spread (Six Sigma)
- Set tolerance limits (µ ± 3σ)
- Evaluate machine capability (Cpk value)
- Minimize defect rate
Finance and Investment
- Measure stock price volatility
- Quantify portfolio risk
- Calculate Value-at-Risk (VaR)
- Sharpe ratio for risk-adjusted return
Standard Deviation Formulas
Sample Standard Deviation (s)
With Bessel's correction (n-1) for unbiased estimation
Population Standard Deviation (σ)
Exact value for the entire population
Relationship to Variance
Standard deviation = square root of variance
Empirical Rule (68-95-99.7)
For normal distribution:
• 68% of values in [µ-σ, µ+σ]
• 95% of values in [µ-2σ, µ+2σ]
• 99.7% of values in [µ-3σ, µ+3σ]
Symbol Explanations
| \(s\) | Sample standard deviation |
| \(\sigma\) | Population standard deviation |
| \(x_i\) | Individual data value |
| \(\overline{x}\) | Sample mean |
| \(\mu\) | Population mean |
| \(n\) | Number of values |
Example Calculations for Standard Deviation
Example 1: Sample Standard Deviation
Calculate: Standard deviation of the sample
1. Calculate Mean
2. Squared Deviations
| (3-5.75)² = | 7.5625 |
| (5-5.75)² = | 0.5625 |
| (7-5.75)² = | 1.5625 |
| (8-5.75)² = | 5.0625 |
| Sum: | 14.75 |
3. Standard Deviation
Example 2: Comparing Two Datasets
Both have mean = 12, but different spread
Dataset A - Low Spread
| Mean: | 12 |
| Variance: | 2 |
| Standard Deviation: | 1.41 |
Interpretation: Values are close together (10-14), low variability.
Dataset B - High Spread
| Mean: | 12 |
| Variance: | 22 |
| Standard Deviation: | 4.69 |
Interpretation: Values widely distributed (5-19), high variability.
Key Insight
Dataset A has σ = 1.41, Dataset B has σ = 4.69.
Although both have the same mean, the three times higher standard deviation of B
shows that values spread much more. This is important for risk analysis, quality control,
and assessing the reliability of measurements.
Example 3: Empirical Rule with Normal Distribution
Application of the 68-95-99.7 rule
68% of Values
µ ± 1σ
[100-15, 100+15]
[85, 115]
68% have IQ between 85 and 115
95% of Values
µ ± 2σ
[100-30, 100+30]
[70, 130]
95% have IQ between 70 and 130
99.7% of Values
µ ± 3σ
[100-45, 100+45]
[55, 145]
99.7% have IQ between 55 and 145
Mathematical Foundations of Standard Deviation
Standard deviation is one of the most important spread measures in statistics and forms the foundation for many statistical procedures.
Properties of Standard Deviation
Standard deviation possesses characteristic mathematical properties:
- Non-negativity: σ ≥ 0, where σ = 0 only when all values are identical
- Same unit: Has the same unit as the original data (unlike variance)
- Sensitivity: Sensitive to outliers (because of squaring)
- Linearity: σ(aX + b) = |a| · σ(X) - scales with constant factor
- Additivity: For independent variables: σ²(X+Y) = σ²(X) + σ²(Y)
Bessel's Correction Explained
Why (n-1) instead of n for samples?
In a sample, we estimate the population mean µ through the sample mean x̄. This causes the deviations (xᵢ - x̄) to be on average smaller than the true deviations (xᵢ - µ). Division by n would systematically underestimate the variance.
Friedrich Bessel discovered that dividing by (n-1) corrects this bias and provides an unbiased estimator. The denominator (n-1) is called "degrees of freedom," because after fixing x̄, only (n-1) values can vary freely.
Standard Deviation vs. Other Spread Measures
Standard Deviation
Average deviation
Same unit as data
Sensitive to outliers
Variance
Squared deviations
Squared unit
Computationally simpler
IQR (Interquartile Range)
Middle 50% of data
Same unit as data
Robust to outliers
Coefficient of Variation (CV)
The coefficient of variation is the relative standard deviation:
Usage: Compare spread of datasets with different means or units.
Example: Make weights (kg) vs. heights (cm) comparable.
Interpretation: CV < 10%: low spread, CV > 30%: high spread
Practical Guidelines
When to Use Standard Deviation?
- Normally distributed data: Optimal for symmetric distributions
- Quality control: Process capability (Cpk), Six Sigma
- Financial analysis: Volatility, risk, Sharpe ratio
- Hypothesis tests: t-test, z-test, confidence intervals
- Comparisons: When same unit as data is desired
Be Careful With
- Outliers: Squaring greatly amplifies their influence
- Skewed distributions: Median and IQR often better suited
- Small samples: n < 30 can lead to unstable estimates
- Different units: Use coefficient of variation CV
- Ordinal data: Standard deviation not meaningfully interpretable
Summary
Standard deviation is the most important spread measure in statistics. It measures the average deviation of values from the mean and has the same unit as the data itself, which facilitates interpretation. For samples, use Bessel's correction (n-1) for an unbiased estimate. The empirical rule (68-95-99.7) applies only to normal distributions and is very useful for interval estimation. In practice, standard deviation is used for quality control (Six Sigma), financial analysis (volatility), hypothesis tests, and risk analysis. With outliers or skewed distributions, consider more robust measures such as the interquartile range (IQR).
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