Standard Deviation Calculator

Standard Deviation Calculator: Sample & Population σ

This calculator finds mean, variance, and standard deviation from numbers you type or paste, with sample or population mode, a 95% margin of error, logic trace, and a bell-curve plot. Everything runs locally in your browser.

What This Calculator Does

This standard deviation calculator summarizes one numeric dataset: mean, variance, standard deviation (s or σ), margin of error (95%), and a data summary (n, sum, range, median, Q₁, Q₃, IQR). Toggle Sample (n−1) or Population (n) for the variance denominator. A Logic trace shows mean → sum of squares → variance → SD; Distribution visualization plots a normal curve with ±1σ, ±2σ, ±3σ bands. Sample mode requires n ≥ 2; population allows n ≥ 1. All math runs locally.

How the Math Works

Mean:

\bar{x} = \frac{\sum x_i}{n}

Population variance and SD:

\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n},\quad \sigma = \sqrt{\sigma^2}

Sample (Bessel): divide by n−1 instead of n. Example 2, 4, 4, 4, 5, 5, 7, 9 with sample mode: x̄ = 5, s² = 32/7, s ≈ 2.14.

How to Use This Calculator

Control reference:

Data:

Multi-line field; type or paste values.
Mode:

Sample (n−1) or Population (n).
Statistical summary:

Mean, variance, SD, MoE (95%), data summary cards.
Logic trace:

Four-step derivation; Copy logic trace button.
Distribution visualization:

Bell curve with σ-band legend when SD > 0.

Sample vs Population: When to Use n−1

Use population when the list is the entire group (every test score in one class). Use sample when the list is drawn from a larger population (survey respondents). Dividing by n−1 corrects bias from estimating the mean from the same data.

Margin of Error and the Bell Curve

Margin of error uses ±1.96 × (SD / √n) with your calculated SD — a quick interval for the mean at 95% confidence when assumptions are reasonable. The bell-curve bands illustrate the 68–95–99.7 rule for normal data; your histogram may differ.

FAQ

What is standard deviation?

Standard deviation measures how spread out values are from the mean. Population σ divides squared deviations by n; sample s divides by n−1 (Bessel’s correction). This calculator shows variance (σ² or s²), SD, and a logic trace with each step.

What is the difference between population and sample standard deviation?

Population (n) — you have every member of the group. Sample (n−1) — your list is a subset of a larger population. The Mode toggle switches the variance denominator; mean and median stay the same.

How do you calculate variance?

Find the mean, square each deviation (xᵢ − x̄)², sum them, then divide by n (population) or n−1 (sample). Standard deviation is the square root of variance. The Logic trace boxes mean, sum of squares, variance, and SD with your numbers.

What does margin of error (95%) mean?

Shown as ±1.96 × (SD / √n) using your calculated standard deviation. It is the half-width of a rough 95% confidence interval for the mean: x̄ ± MoE. Treat it as a teaching shortcut, not a substitute for full inference when assumptions fail.

What do ±1σ, ±2σ, ±3σ mean on the bell curve?

For a normal distribution, about 68% of values fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. The Distribution visualization shades those bands around your μ and SD — a reference curve, not proof your data are normal.

How many data points do I need?

Population mode needs at least 1 value. Sample mode needs at least 2. Quartiles in the Data Summary need at least 2 values. The bell curve appears when SD > 0.

Can I type numbers instead of pasting?

Yes. Enter values in the Data field — type or paste, separated by commas, spaces, or new lines. Non-numeric characters are stripped as you type.

Enter your data

Reading the statistical summary

Worked example: 2, 4, 4, 4, 5, 5, 7, 9

Mean, variance, and SD (Sample n−1)

Margin of error (95%)

Data summary

Logic trace & bell curve