Statistics
Standard Deviation
5.2372
s (sample)
Mean
18.0000
average
Enter Your Data
8 numbers entered
Data Type
Use sample when your data is a subset of a larger population (divides by n-1).
Data Distribution
Deviations from Mean
Step-by-Step Calculation
Step 1: Calculate the Mean
Mean = Sum / n = 144.00 / 8 = 18.0000
Step 2: Calculate Deviations
For each value, subtract the mean: (value - 18.00)
Step 3: Square the Deviations
Sum of squared deviations = 192.0000
Step 4: Calculate Variance
Variance = 192.0000 / 7 = 27.4286
Step 5: Take the Square Root
Standard Deviation = √27.4286 = 5.2372
Deviation Table
| # | Value (x) | Deviation (x - μ) | (x - μ)² |
|---|---|---|---|
| 1 | 10 | -8.0000 | 64.0000 |
| 2 | 12 | -6.0000 | 36.0000 |
| 3 | 23 | 5.0000 | 25.0000 |
| 4 | 23 | 5.0000 | 25.0000 |
| 5 | 16 | -2.0000 | 4.0000 |
| 6 | 23 | 5.0000 | 25.0000 |
| 7 | 21 | 3.0000 | 9.0000 |
| 8 | 16 | -2.0000 | 4.0000 |
| Sum of Squared Deviations | 192.0000 | ||
Formulas
Population Standard Deviation
σ = √[Σ(xᵢ - μ)² / N]
Sample Standard Deviation
s = √[Σ(xᵢ - x̄)² / (n-1)]
?How to Calculate Standard Deviation
Standard deviation measures how spread out data is from the mean. To calculate: 1) Find the mean, 2) Subtract mean from each value and square the result, 3) Find the average of squared differences (variance), 4) Take the square root. For sample data, divide by (n-1) instead of n. Typical notation: sigma for population, s for sample.
What is Standard Deviation?
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates values tend to be close to the mean, while a high standard deviation indicates values are spread out over a wider range. It is the square root of variance.
Key Facts About Standard Deviation
- Standard deviation (SD) measures data spread around the mean
- Population SD uses N in denominator; sample SD uses N-1 (Bessel's correction)
- Variance = standard deviation squared (SD^2)
- Formula: SD = sqrt(sum of (x - mean)^2 / n)
- About 68% of data falls within 1 SD of the mean (normal distribution)
- About 95% of data falls within 2 SD; 99.7% within 3 SD (empirical rule)
- Low SD means data points are close to the mean; high SD means more spread
- SD has same units as original data; variance has squared units
Try These Examples
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Practice Standard Deviation Problems
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Practice with 7 problems to test your understanding.
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Quick Answer
Standard deviation measures how spread out data is from the mean. To calculate: 1) Find the mean, 2) Subtract mean from each value and square the result, 3) Find the average of squared differences (variance), 4) Take the square root. For sample data, divide by (n-1) instead of n. Typical notation: sigma for population, s for sample.
Frequently Asked Questions
What is standard deviation?
Standard deviation measures the spread or dispersion of data points from the mean. A low standard deviation means data points are close to the mean, while a high standard deviation indicates data is spread out. It's calculated as the square root of variance.
What is the difference between population and sample standard deviation?
Population standard deviation (σ) uses all data points and divides by n. Sample standard deviation (s) is used when data represents a subset and divides by n-1 (Bessel's correction) to provide an unbiased estimate of the population parameter.
How do I interpret standard deviation?
For normally distributed data: about 68% falls within ±1 SD, 95% within ±2 SD, and 99.7% within ±3 SD of the mean. This is the empirical rule (68-95-99.7 rule). Values beyond 3 SD are often considered outliers.
What is variance?
Variance is the average of squared deviations from the mean. It measures how far data points spread from the mean, but in squared units. Standard deviation is the square root of variance, returning to the original units for easier interpretation.
What is coefficient of variation (CV)?
The coefficient of variation (CV) is the standard deviation divided by the mean, expressed as a percentage. It allows comparing variability between datasets with different units or means. Lower CV indicates more consistent data.
What is standard error?
Standard error (SE) measures the precision of the sample mean as an estimate of the population mean. It equals SD / √n. Larger samples have smaller standard errors, meaning more precise estimates.
Last updated: 2026-01-15
How this works
Sample vs population standard deviation: sample σ divides sum of squared deviations by (n−1) (Bessel's correction); population σ divides by n. Defaults to sample σ — that is what you almost always want when working with empirical data.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
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