How to Calculate Standard Deviation (Step-by-Step)Standard deviation is a fundamental statistical measure that describes how spread out numbers are in a dataset. It tells you, on average, how far each value lies from the mean (average). This article walks you through the concept, formulas, step-by-step calculations for both population and sample standard deviation, worked examples, common pitfalls, and when to use each version.
What is standard deviation?
Standard deviation quantifies the amount of variation or dispersion in a set of values. A small standard deviation means the values are clustered tightly around the mean; a large standard deviation means they are more spread out.
Key terms:
- Mean (μ for population, x̄ for sample): the average of the values.
- Variance (σ² for population, s² for sample): the average of squared deviations from the mean.
- Standard deviation (σ for population, s for sample): the square root of variance, expressed in the same units as the original data.
Population vs. sample standard deviation
- Use population standard deviation when you have data for the entire population of interest.
- Use sample standard deviation when your data are a sample drawn from a larger population and you want to estimate the population standard deviation.
The difference appears in the denominator when computing variance:
- Population variance uses N (the number of observations).
- Sample variance uses N − 1 (Bessel’s correction) to correct bias in the estimation.
Formulas
Population standard deviation: σ = sqrt( (1/N) * Σ (xi − μ)² )
Sample standard deviation: s = sqrt( (1/(N − 1)) * Σ (xi − x̄)² )
Where:
- xi = each data point
- μ = population mean
- x̄ = sample mean
- N = number of observations
- Σ = sum over all observations
Step-by-step calculation (population)
- List all data points.
- Compute the mean μ = (Σ xi) / N.
- For each data point, compute the deviation from the mean: (xi − μ).
- Square each deviation: (xi − μ)².
- Sum all squared deviations: Σ (xi − μ)².
- Divide the sum by N to get the variance: σ² = (1/N) Σ (xi − μ)².
- Take the square root of variance: σ = sqrt(σ²).
Example (population): Data: 4, 8, 6, 5
- N = 4
- μ = (4 + 8 + 6 + 5) / 4 = 23 / 4 = 5.75 3–4. Deviations and squares:
- (4 − 5.75) = −1.75 → 3.0625
- (8 − 5.75) = 2.25 → 5.0625
- (6 − 5.75) = 0.25 → 0.0625
- (5 − 5.75) = −0.75 → 0.5625
- Sum squares = 3.0625 + 5.0625 + 0.0625 + 0.5625 = 8.75
- Variance σ² = 8.75 / 4 = 2.1875
- Standard deviation σ = sqrt(2.1875) ≈ 1.479
Step-by-step calculation (sample)
Follow the same steps but divide by N − 1 when computing variance.
Example (sample) — same data treated as a sample: Data: 4, 8, 6, 5
- N = 4
- x̄ = 5.75 3–5. Sum squared deviations = 8.75 (same as above)
- Sample variance s² = 8.75 / (4 − 1) = 8.75 / 3 ≈ 2.9167
- Sample standard deviation s = sqrt(2.9167) ≈ 1.708
Shortcut (computational) formula
To reduce rounding errors in manual computation, use: Variance = (1/N) * Σ xi² − (Σ xi)² / N
Variance = (1/(N − 1)) * Σ xi² − (Σ xi)² / N
This lets you compute Σ xi and Σ xi² in one pass.
Example (population) with the same data: Σ xi = 23, Σ xi² = 4² + 8² + 6² + 5² = 16 + 64 + 36 + 25 = 141 σ² = (⁄4) * [141 − (23)² / 4] = 0.25 * [141 − 529 / 4] = 0.25 * [141 − 132.25] = 0.25 * 8.75 = 2.1875
Interpreting standard deviation
- About 68% of values lie within ±1σ of the mean for a roughly normal distribution.
- About 95% of values lie within ±2σ.
- About 99.7% within ±3σ. (Empirical rule — applies well when distribution is approximately normal.)
Standard deviation is sensitive to outliers; a single extreme value can inflate it significantly.
Practical tips and common pitfalls
- Don’t mix up population and sample formulas.
- Use N − 1 for sample data to get an unbiased estimator of population variance.
- For skewed distributions or when outliers are present, consider robust measures like the interquartile range (IQR).
- For large datasets, use the computational formula or software (Excel, R, Python’s numpy) to avoid rounding error.
Examples in tools:
- Excel: population STDEV.P(range) and sample STDEV.S(range).
- Python: numpy.std(arr, ddof=0) for population, numpy.std(arr, ddof=1) for sample (or use numpy.var with sqrt).
When to use standard deviation
- Comparing variability between datasets measured in the same units.
- As a component of other statistics (z-scores, confidence intervals, control charts).
- When the mean is a meaningful measure of central tendency (not for highly skewed distributions).
Quick reference formulas
Population: σ = sqrt( (1/N) * Σ (xi − μ)² ) = sqrt( (1/N) * [ Σ xi² − (Σ xi)² / N ] )
Sample: s = sqrt( (1/(N − 1)) * Σ (xi − x̄)² ) = sqrt( (1/(N − 1)) * [ Σ xi² − (Σ xi)² / N ] )
If you want, I can provide:
- Python and Excel examples with code/formulas.
- More worked examples (including large datasets).
- A short practice quiz to test understanding.
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