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
- Scope:Numeric input; non-numeric characters stripped. Sample needs n ≥ 2; population n ≥ 1. All calculations local—no data stored.
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.
Standard Deviation Formula: Mean, Variance, and σ
The mean is:
The sum of squared differences is Σ(xᵢ − x̄)². For population standard deviation:
For sample standard deviation (Bessel's correction):
Variance is σ² or s² (before the square root).
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.