What is the difference between population and sample standard deviation?

Population standard deviation (σ) divides squared deviations by N, the population size. Sample standard deviation (s) divides by n−1 (Bessel's correction) to correct bias when estimating a population from a sample.

How do I compute standard deviation step-by-step?

Compute the mean, subtract the mean from each value to get deviations, square the deviations, sum them, divide by N or (n−1) for population or sample variance, and then take the square root to get standard deviation.

Is standard deviation sensitive to outliers?

Yes. Standard deviation squares deviations, so outliers have a large effect. For robust alternatives use median absolute deviation (MAD) or trimmed estimators.

How do I calculate SD in Excel or Google Sheets?

In Excel/Sheets use STDEV.P(range) for population SD or STDEV.S(range) for sample SD. These functions compute the full procedure automatically.

Standard Deviation Calculator | Stats & Math Tool

Standard Deviation Calculator — Understand Variation, Compare Datasets, and Make Data-Driven Decisions

Standard deviation is the single most useful summary statistic for understanding how spread out numbers are. Whether you're analyzing classroom grades, financial returns, experimental measurements, or quality-control data, the Standard Deviation Calculator quickly computes population and sample standard deviations, shows step-by-step workings, and helps you interpret what the result means for your decisions. This guide explains the intuition, formulas, worked examples, common pitfalls, use cases across disciplines, and how to combine this calculator with related tools like the Stat Calculator, Mean & Median the GPA Calculator, and the Percentage Calculator.

What this calculator gives you

Population standard deviation (σ) when your data set represents the entire population.
Sample standard deviation (s) when your data set is a sample from a larger population (uses n−1).
Detailed step-by-step breakdown: mean, deviations, squared deviations, variance, and final square root.
Optional CSV / copy-paste input and exportable results for reports or homework.

Quick intuition

Standard deviation measures the typical distance of values from the average. A small value means the numbers cluster tightly around the mean; a large value means they are widely spread.

When to use population vs sample formulas

Population (σ): Use when your dataset contains every member of the group you're studying (e.g., test scores for all students in a class if you have every student's score).
Sample (s): Use when your dataset is a random sample drawn from a larger group and you want to estimate the population standard deviation — divide by n−1 (Bessel's correction).

Short rule

If in doubt and you plan to infer about a larger population, use the sample standard deviation (n−1).

Formulas — exact and simple

Use these formulas depending on your context:

Population variance and standard deviation

σ² = (1/N) × Σ (xi − μ)²
σ  = √σ²

Where N = population size, μ = population mean.

Sample variance and sample standard deviation

s² = (1/(n − 1)) × Σ (xi − x̄)²
s  = √s²

Where n = sample size, x̄ = sample mean. Dividing by n−1 corrects bias in the variance estimate.

Step-by-step worked example (numbers you can verify)

We'll work a full example so you can follow each step. Suppose you measured five items: 600, 470, 170, 430, 300. This classic dataset demonstrates how spread compares to the mean.

Compute the mean Sum = 600 + 470 + 170 + 430 + 300 = 1970. Mean (x̄) = 1970 / 5 = 394.
Find deviations (xi − mean): 600 − 394 = 206; 470 − 394 = 76; 170 − 394 = −224; 430 − 394 = 36; 300 − 394 = −94.
Square each deviation (to remove signs and weight larger deviations): 206² = 42,436 76² = 5,776 (−224)² = 50,176 36² = 1,296 (−94)² = 8,836
Sum squared deviations: 42,436 + 5,776 + 50,176 + 1,296 + 8,836 = 108,520.
Population variance (σ²): divide by N = 5 → 108,520 / 5 = 21,704. Population standard deviation σ = √21,704 ≈ 147.323.
Sample variance (s²): divide by n−1 = 4 → 108,520 / 4 = 27,130. Sample standard deviation s = √27,130 ≈ 164.712.

The calculator performs these steps instantly and reports both σ and s so you can choose what is appropriate for your analysis.

Interpretation: what do the numbers mean?

If the population SD is ≈147 on values that average 394, that tells you a typical value is about ±147 from the mean. Roughly 68% of values lie within one standard deviation of the mean (assuming an approximately normal distribution), so in this example about 68% are expected between 394 − 147 ≈ 247 and 394 + 147 ≈ 541.

When standard deviation is misused

Non-normal data: SD is a measure of spread but its probabilistic interpretations (68% within ±1σ, etc.) assume approximate normality. For skewed distributions, supplement SD with median and interquartile range (IQR).
Mixing populations: Don’t pool unrelated groups and report a single SD — instead compute group-specific SDs or use pooled-variance formulas if comparing similar samples.
Small samples: Sample SD with very small n is noisy — report confidence intervals or use robust statistics (bootstrapping) if appropriate.

Use cases across fields

Finance

Standard deviation of returns measures volatility. In portfolio construction, SD helps estimate risk; combined with expected return it informs Sharpe ratio and risk budgeting. Use the Compound Interest Calculator to model long-term effects and pair it with SD to stress-test scenarios.

Education & Grading

In classrooms, SD of scores shows grade dispersion. Instructors use it to determine whether curves are needed and to identify unusually wide variability that might indicate a confusing exam question.

Science & Engineering

SD quantifies measurement precision and repeatability. Report SD alongside mean values in experimental results; if you need a confidence statement, compute standard error (SE = s / √n) and build confidence intervals.

Practical tips when using the calculator

Decide population vs sample before calculating — the choice changes the denominator and the interpretation.
Check for outliers. A single extreme value can inflate SD dramatically; consider robust measures (median absolute deviation) or report both with and without obvious outliers.
For very large datasets, use the calculator’s streamed/online algorithm (if implemented) to avoid sums that overflow and to maintain numerical precision.
Round final SD sensibly in reports (usually 2 significant digits for SD is fine), but keep full precision for intermediate computations.

How to compute standard deviation in spreadsheets & scientific calculators

Common functions:

Excel / Google Sheets: =STDEV.P(range) for population, =STDEV.S(range) for sample.
Scientific calculators: Use the statistics mode: enter each xi, then request σ or s. See our Scientific Calculator guide for button-level help.
R / Python: R: sd(x) returns sample sd by default; Python (numpy): numpy.std(x, ddof=0) for population, ddof=1 for sample.

Comparing two datasets (quick guidance)

When comparing spreads, look at SD relative to the mean (coefficient of variation = SD / mean). This is useful when means differ greatly. For formal comparison of variances, use statistical tests (F-test, Levene’s test) — our Stat Calculator can help with hypothesis testing.

Workflow: How to use the Standard Deviation Calculator effectively

Collect or paste your data into the input field (CSV supported).
Choose population or sample mode depending on your situation.
Review the step-by-step table (mean, deviations, squared deviations) to spot errors and outliers.
Export results and include SD along with mean and sample size in reports.
If you need further analysis (variance tests, confidence intervals, or regression), link to the Stat Calculator or the standard deviation tool for extended features.

Frequently Asked Questions

What is the difference between variance and standard deviation?

Variance is the average of squared deviations (units squared). Standard deviation is the square root of variance and has the same units as the original data, making it easier to interpret.

Why use n−1 for samples?

Dividing by n−1 (Bessel’s correction) corrects the downward bias when estimating the population variance from a sample. It produces an unbiased estimator of the population variance under standard assumptions.

How do I handle missing values?

Remove or impute missing values before calculating SD. Our calculator ignores empty cells in pasted CSV inputs but flag data gaps so you can decide on imputation or exclusion.

Is standard deviation affected by outliers?

Yes — SD is sensitive to outliers because deviations are squared. For robust alternatives, consider median absolute deviation (MAD) or trimmed standard deviation.

When should I use coefficient of variation?

Use coefficient of variation (SD / mean) when comparing variability between datasets with different units or widely different means.

Ready to Measure Variation?

Use the Standard Deviation Calculator to compute σ or s, generate an amortized step-by-step table, and export results for reports, homework, or presentations.

Standard Deviation Calculator Stat Calculator Mean & Median GPA Calculator All Calculators

For more statistical reading, see our Student Calculator Guide and the Algebra Basics article.

Standard Deviation Calculator

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