Standard Deviation Calculator
Standard Deviation — A Plain-English Guide with Formulas, Examples & a Visual
1 What Is Standard Deviation?
Standard deviation (σ) quantifies spread: how tightly or loosely data crowd around the mean (μ).
Small σ → observations hug the average.
Large σ → values wander widely.
Because σ is an inverse of squaring (it is the square-root of variance), it shares units with the raw data (dollars, degrees, grams), making the number easy to interpret.
2 Population vs Sample Formulas
| Setting | Symbol | Formula | When to use |
|---|---|---|---|
| Population (measure every member) | σ | \[ \displaystyle \sigma=\sqrt{\frac{\sum_{i=1}^{N}(x_i-\mu)^2}{N}} \] | True census, complete batch |
| Sample (estimate from a subset) | s | \[ \displaystyle s=\sqrt{\frac{\sum_{i=1}^{N}(x_i-\bar{x})^2}{N-1}} \] | Surveys, test lots, experiments |
Why N − 1?
Dividing by one less than the sample size “unbiases” the variance estimate—compensating for the fact that x̄ is an imperfect stand-in for μ. This tweak is the Bessel correction.
3 Step-by-Step Example (Population σ)
Data set: 1, 3, 4, 7, 8
Mean
Squared deviations
| x | x − μ | (x − μ)² |
|---|---|---|
| 1 | −3.6 | 12.96 |
| 3 | −1.6 | 2.56 |
| 4 | −0.6 | 0.36 |
| 7 | 2.4 | 5.76 |
| 8 | 3.4 | 11.56 |
| Sum | 33.20 |
Variance
Standard deviation
4 Reading σ on a Normal Curve
(See Figure A just below.)
±1 σ captures ~68 % of values.
±2 σ captures ~95 %.
±3 σ captures ~99.7 %.
This “68-95-99.7 rule” explains why quality-control engineers scream when a widget lands beyond 3 σ.
(Fig. A was generated live; feel free to reuse it.)
5 Real-World Uses
| Field | Why σ matters |
|---|---|
| Manufacturing QC | Sets tolerance bands—anything outside ±3 σ triggers a process check. |
| Climate science | Reveals that coastal temps (~σ ≈ 10 °F) swing far less than inland temps (~σ ≈ 40 °F) even when means match. |
| Finance | A stock with 7 % mean return and σ = 50 % is riskier than one with the same mean but σ = 10 %. |
| Opinion polling | The margin of error equals ± (critical z) × standard error (σ/√N). |
| Medicine | Drug-response variability guides dosage windows and trial sizes. |
6 Quick Reference Table
| Symbol | Term | Plain meaning |
|---|---|---|
| μ | Mean | “Center” of the data |
| σ² | Variance | Average squared distance from μ |
| σ | Standard deviation | Square-root of variance (spread in original units) |
| s | Sample standard deviation | σ-estimate from a sample |
Key Takeaways
Definition: σ is the root-mean-square distance from the mean.
Population vs Sample: divide by N or (N − 1) depending on whether you measured everyone.
Interpretation: one σ gives an intuitive “typical deviation.”
Applications everywhere: quality, climate, investing, polling, medicine.
Visual cue: on a bell curve, ±1 σ already covers about two-thirds of all outcomes.
Master σ and you gain a universal ruler for comparing consistency—whether you’re baking cookies or balancing billion-dollar portfolios.