Standard Deviation Calculator

Calculate standard deviation, variance, mean, and other statistics for your data set. Supports both population and sample calculations with step-by-step explanations.

Statistics

Standard Deviation

5.2372

s (sample)

Mean

18.0000

average

Variance27.4286
Count (n)8
Sum144.00
Min10.00
Max23.00
Range13.00
CV29.10%
Std Error1.8516

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
110-8.000064.0000
212-6.000036.0000
3235.000025.0000
4235.000025.0000
516-2.00004.0000
6235.000025.0000
7213.00009.0000
816-2.00004.0000
Sum of Squared Deviations192.0000

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

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

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.
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.
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.
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.
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.
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: 2025-01-15