Standard Deviation Calculator

Standard Deviation Calculator: Sample & Population σ

How to calculate standard deviation from raw data. Sample (n−1) or population (n) variance formula. Mean, IQR, margin of error. Logic Trace and bell curve. Trusted by students and educators.

What This Standard Deviation Calculator Does

Purpose & Use Cases

This standard deviation calculator computes mean, variance, and standard deviation (σ or s) from raw data. Choose sample (n−1, Bessel's correction) or population (n). Also outputs margin of error (95%) and data summary (n, sum, range, median, Q1, Q3, IQR). Logic Trace shows each formula step; the Distribution card plots a bell curve with ±1σ, ±2σ, ±3σ bands. Ideal for how to calculate standard deviation by hand, homework verification, and quick descriptive stats.

Scope:

Numeric input; non-numeric characters stripped. Sample needs n ≥ 2; population n ≥ 1. All calculations local, no data stored.

Standard Deviation Formula: Mean, Variance, and σ

The mean is:

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

The sum of squared differences is Σ(xᵢ − x̄)². For population standard deviation:

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

For sample standard deviation (Bessel's correction):

s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}

Variance is σ² or s² (before the square root).

How to Use This Calculator

Paste or type your numbers into the data field, commas, spaces, or newlines all work as separators, and non-numeric characters are stripped automatically. Toggle between Sample (divides by n−1 using Bessel's correction for subsets drawn from a larger population) and Population (divides by n when you have every data point in the group). The dashboard updates instantly with mean, variance, standard deviation, margin of error at 95% confidence, and a data summary including count, sum, range, median, Q1, Q3, and IQR. Open the Logic Trace to follow each step: mean calculation, individual squared deviations, sum of squares, and the final variance and standard deviation derivation. The Distribution card plots a normal bell curve with ±1σ, ±2σ, and ±3σ bands centered on your calculated mean. Use Copy Results to export the full statistical summary for homework, reports, or spreadsheet workflows.

Sample vs Population: When to Use n−1

Use population when you have data for every member (e.g. all test scores in a class). Use sample when your data is a subset (e.g. a survey of 100 people). Dividing by n−1 in the sample formula corrects the bias that occurs because the sample mean x̄ is used instead of the true population mean μ.

Margin of Error and the 95% Confidence Interval

The margin of error for a 95% confidence interval of the mean is ±1.96 × (σ/√n). It quantifies uncertainty when estimating the population mean from a sample. Use it to construct an interval: mean ± margin of error.

The Normal Distribution and ±1σ, ±2σ, ±3σ

For a normal distribution N(μ, σ²): about 68% of values fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. The Distribution Visualization plots a bell curve with your calculated mean and standard deviation, highlighting these bands.

FAQ

What is standard deviation?

Standard deviation (σ or s) measures how spread out your data is from the mean. Low values cluster near the mean; high values indicate greater variability. It is the square root of variance: population σ = √[Σ(xᵢ − μ)²/n], sample s = √[Σ(xᵢ − x̄)²/(n−1)] (Bessel's correction).

What is the difference between population and sample standard deviation?

Population standard deviation (σ) divides by n (full group size). Sample standard deviation (s) divides by n−1 to correct bias when estimating from a subset. Use Population when you have every member; use Sample when your data is drawn from a larger group. This calculator has a toggle for each.

How do you calculate variance?

Variance = average of squared deviations from the mean. Population: σ² = Σ(xᵢ − μ)²/n. Sample: s² = Σ(xᵢ − x̄)²/(n−1). Standard deviation = √variance. Logic Trace shows both formulas with your numbers.

What does margin of error (95%) mean?

For a 95% confidence interval of the mean, the margin of error is ±1.96 × (σ/√n). It quantifies uncertainty when estimating the population mean from a sample. Use it with the mean to build an interval: x̄ ± MoE.

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

For a normal distribution: ±1σ ≈ 68% of data, ±2σ ≈ 95%, ±3σ ≈ 99.7%. The Distribution card plots these bands around your mean.