What is the difference between population and sample standard deviation?

Population standard deviation uses N in the denominator because you have all values. Sample standard deviation uses N−1 to correct bias when estimating a population parameter from a sample.

When should I use standard deviation vs. variance?

Variance measures average squared deviation (units squared). Standard deviation (the square root of variance) is in the same units as the data and is more interpretable for spread.

How is standard deviation used to measure risk?

In finance, higher standard deviation of returns indicates larger volatility and therefore higher risk (both up and down swings).

Standard Deviation Calculator

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It is a Population Sample

Standard Deviation — Definition, Formula (Population & Sample) & Applications

Standard Deviation — Definition, Formula & Applications

Understand population (σ) and sample (s) standard deviation, see formulas and a worked example, and learn where to use standard deviation.

What is Standard Deviation?

Standard deviation is a statistical measure of the amount of variation or dispersion in a set of values. It quantifies how spread out values are around the mean. Standard deviation is commonly denoted by σ (sigma) for a population and s for a sample.

A low standard deviation means data points are close to the mean; a high standard deviation means they are more spread out. Standard deviation is widely used to measure risk, margin of error, and variability across many fields.

Population Standard Deviation (σ)

Use the population standard deviation formula when you have data for the entire population.

σ = sqrt( (Σ (xᵢ − μ)²) / N )

Where:
  xᵢ = each value
  μ   = population mean
  N   = total number of values

This computes the square root of the average squared deviation from the population mean.

Sample Standard Deviation (s)

If you only have a sample from the population, use the corrected sample standard deviation (dividing by N−1) to reduce bias:

s = sqrt( (Σ (xᵢ − x̄)²) / (N − 1) )

Where:
  xᵢ = sample values
  x̄ = sample mean
  N  = sample size

The N−1 denominator (Bessel's correction) helps make the sample variance an unbiased estimator of population variance.

Worked Example

Data: 1, 3, 4, 7, 8

Step 1 — Mean:

x̄ = (1 + 3 + 4 + 7 + 8) / 5 = 4.6

Step 2 — Population standard deviation:

σ = sqrt( [ (1-4.6)² + (3-4.6)² + (4-4.6)² + (7-4.6)² + (8-4.6)² ] / 5 )
  = sqrt( (12.96 + 2.56 + 0.36 + 5.76 + 11.56) / 5 )
  ≈ sqrt(33.2 / 5) ≈ sqrt(6.64) ≈ 2.577

Margin of Error & Confidence

Standard deviation can be used to compute the margin of error for a sample mean. For a 95% confidence interval, a common formula is:

Margin of error ≈ z * (s / sqrt(N))

where z ≈ 1.96 for 95% confidence and s is the sample standard deviation.

Note: For small samples or non-normal distributions, use t-scores instead of z-scores.

Common Applications

Quality Control: Set acceptable tolerance ranges for manufactured parts.
Weather & Climate: Compare temperature stability between regions.
Finance: Estimate volatility and investment risk.
Research & Experiments: Assess variability and reproducibility of results.

Quick Reference Table

Measure	Formula (summary)
Population variance	σ² = (Σ (xᵢ − μ)²) / N
Population stdev	σ = sqrt(σ²)
Sample variance	s² = (Σ (xᵢ − x̄)²) / (N − 1)
Sample stdev	s = sqrt(s²)

FAQ

What is the difference between population and sample standard deviation?: Population standard deviation uses N in the denominator because you have all values. Sample standard deviation uses N−1 to correct bias when estimating a population parameter from a sample.
When should I use standard deviation vs. variance?: Variance measures average squared deviation (units squared). Standard deviation (the square root of variance) is in the same units as the data and is more interpretable for spread.
How is standard deviation used to measure risk?: In finance, higher standard deviation of returns indicates larger volatility and therefore higher risk (both up and down swings).