📊 Statistics & Probability

Standard Deviation Calculator

Calculate sample and population standard deviation, variance, mean, and range from any list of numbers. Free online tool.

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What This Calculator Tells You

Standard deviation measures how spread out a set of numbers is around its average. A small standard deviation means values cluster tightly; a large one means they vary a lot. It is the basis of confidence intervals, z-scores, quality control and risk measures.

Paste your numbers and the calculator returns both the sample and population standard deviation, the mean, the count and the range.

What to Enter

Numbers — your data separated by commas, spaces or new lines. You can paste a column straight from Excel or Google Sheets.

Worked Examples

1. Daily support tickets for a week

12, 15, 11, 18, 14, 16, 13.

Mean 14.14, sample SD 2.41, population SD 2.23, range 7.
Most days fall within about 12 to 17 tickets.

2. One outlier changes everything

Delivery times in hours: 48, 52, 50, 49, 51, 95.

Sample SD jumps to 18.43 and the mean to 57.5 because of one late delivery.
Without the 95, the SD would be about 1.6. Always look at your data before trusting the summary.

3. Store revenue in $ thousands

220, 180, 260, 240, 200.

Mean 220, sample SD 31.62, population SD 28.28.

How It’s Calculated

Sample SD = √(Σ(xᵢ − x̄)² ÷ (n − 1)). Population SD = √(Σ(xᵢ − μ)² ÷ N).

SD: Standard Deviation
Σ: Summation
x: Observed value
n: Number of periods or sample size
μ: Population mean
N: Population / total count

How to Read Your Result

Sample SD (n − 1): use when your numbers are a sample of a bigger group — the usual case.
Population SD (n): use only when you have every value in the group.
Rule of thumb for bell-shaped data: about 68% of values lie within 1 SD of the mean and about 95% within 2 SD.
Coefficient of variation (SD ÷ mean) compares spread between data sets with different units or scales.

Common Mistakes to Avoid

Using population SD for a sample, which understates variability.
Ignoring outliers that inflate SD.
Applying the 68–95 rule to skewed data such as income or order values.
Mixing units, e.g. dollars and thousands of dollars in one list.

Built and maintained by GenAlpha Tools Editorial. Formula tested against known answers and edge cases · last reviewed October 2026. How we calculate · Report an error

When to Use This Calculator

Measure consistency in sales, delivery times or production.
Feed SD into confidence intervals, z-scores and sample-size plans.
Compare volatility of investments or campaigns.
Check homework and statistics assignments.

Frequently Asked Questions

What is the difference between sample and population standard deviation?

Sample SD divides by n − 1 instead of n to correct for estimating the mean from the same data. For large data sets the two are almost equal.

How is standard deviation calculated?

Find the mean, subtract it from each value, square the differences, add them up, divide by n − 1 (sample) or n (population), then take the square root.

What is a 'good' standard deviation?

There is no universal good value; it depends on the scale and purpose. Compare it with the mean, or with past periods.

What is variance?

Variance is standard deviation squared. Use the variance calculator if you need it directly.

Do you store the values I enter?

No. All calculations happen in your browser. Your data never leaves your device.

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