Standard Deviation Calculator – Sample & Population + Mean

📊 ➜ σ

Standard Deviation Calculator

Paste your data set and get sample and population standard deviation, variance, mean, median, mode, sum, range and standard error — simultaneously, with a step-by-step walkthrough of the math. No formulas to remember, no signup, nothing uploaded.

✓ Sample + Population · ✓ Variance & SE · ✓ 10 Stats at Once · ✓ Private

📊 Descriptive Statistics Calculator

🔧 JS ⏳ · Input — · Stats — · Engine ⏳

Your data set — separated by spaces, commas or new lines

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Count (n)

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Sum

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Mean

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Median

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Mode

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Min / Max

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Sample SD (n−1)

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Population SD (n)

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Sample variance

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Std error (SD/√n)

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Population variance

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Range

🧮 Step-by-Step — How the Sample SD Was Calculated

Paste data to see the full calculation walk through.

🧮 Need other stats tools? Try our GPA Calculator or Percentage Calculator.

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How to Calculate Standard Deviation — Step by Step

Worked example using the data set 2, 4, 4, 4, 5, 5, 7, 9:

1. Find the mean: add all values (2+4+4+4+5+5+7+9 = 40) and divide by count: 40 ÷ 8 = 5.
2. Subtract the mean from each value: −3, −1, −1, −1, 0, 0, 2, 4.
3. Square each difference: 9, 1, 1, 1, 0, 0, 4, 16.
4. Sum the squares: 9+1+1+1+0+0+4+16 = 32.
5. Divide by n−1 for sample SD: 32 ÷ 7 = 4.5714 (this is the sample variance).
6. Take the square root: √4.5714 = 2.1381 — the sample standard deviation.

For population SD, divide by n instead of n−1 at step 5: 32 ÷ 8 = 4 → √4 = 2.0. Use sample (n−1) when your data is a sample of a bigger group — the usual case. Use population (n) only when your data IS the entire group. Try this exact data set above and verify every number.

The Standard Deviation Formulas

Sample SD: s = √[ Σ(x − x̄)² ÷ (n − 1) ]
Population SD: σ = √[ Σ(x − μ)² ÷ N ]
Standard error: SE = s ÷ √n

Standard deviation measures how spread out your numbers are around the mean — small SD means values cluster tightly, large SD means they're scattered. Variance is the same math without the final square root, expressed in squared units. Standard error scales the SD down by √n and estimates how precisely your sample mean estimates the true population mean.

Why Use This Standard Deviation Calculator?

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Sample + Population

Both SD types computed simultaneously — no mode-switching, no second tool.

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10 Stats at Once

SD, variance, SE, mean, median, mode, sum, range, min/max and count in one pass.

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Full Walkthrough

Step-by-step sample-SD calculation from your actual data — homework-proof.

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Messy Input OK

Paste from spreadsheets — commas, spaces, tabs and new lines all parse.

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100% Private

Research data and homework never leave your browser — nothing sent or stored.

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Instant & Offline

Pure browser math on every keystroke — unlimited use, no account.

Popular Use Cases

• 🎓 Statistics and research-methods homework — verified step-by-step
• 🔬 Lab reports — spread of measurements and standard error for error bars
• 📈 Business analysis — sales, response-time or revenue variability
• 🧪 Quality control — process consistency checks against tolerance limits
• 🏥 Understanding clinical data spread in studies and reports
• 📊 Survey analysis — how varied responses are around the average

Pro Tips for Standard Deviation

• Sample vs population in one line: measuring a subset to generalize? Sample (n−1). Measuring every member of the whole group? Population (n). When unsure, sample is almost always the right default.
• The 68-95-99.7 rule: for roughly bell-shaped data, ~68% of values fall within 1 SD of the mean, ~95% within 2 SDs, ~99.7% within 3. If your SD is 2.14, most of your data lives within ±2.14 of the mean.
• SD has the same units as your data: dollars, minutes, points — which is why it's more interpretable than variance (squared units).
• Outliers inflate SD fast: one extreme value can double your SD. Check min/max; consider whether outliers are errors or real signal before reporting.
• Standard error for error bars: SE shrinks as your sample grows — it's the right companion stat when your SD describes a sample you'll generalize from.

Frequently Asked Questions

Five steps: find the mean, subtract the mean from every value, square each difference, average the squares (÷n−1 for sample, ÷n for population), then take the square root. Paste your data above and the full walkthrough appears with your actual numbers.

The denominator. Sample SD divides by n−1 (correcting for the fact a sample underestimates spread), while population SD divides by n. Use sample when your data represents a larger group; population only when you genuinely have every member.

How spread out your data is around the average. Test scores with SD 3 are tightly clustered; the same average with SD 15 means wildly varied results. Two data sets can share a mean while telling completely different stories — SD is the difference.

Standard deviation describes spread within your data. Standard error (SD ÷ √n) estimates how precisely your sample's mean represents the true population mean — it shrinks as sample size grows. Both are shown here simultaneously.

The walkthrough shows the mean mid-calculation, so you can verify it against a given mean before the squared-differences step — useful when a problem provides the mean and asks you to continue from there.

Press STAT → Edit, enter data in L1, then STAT → CALC → 1-Var Stats. Sx is the sample SD and σx the population SD — the same two values this page computes. Verify a list here first if the calculator output looks wrong.

Sample SD is mathematically undefined for a single value (dividing by n−1 = 0), so it shows a dash — with one number there's no spread to measure. Population SD still works and correctly returns 0. All other stats (mean, median, sum) display normally.

No. All math runs in your browser with JavaScript — data sets from homework, labs or work are never sent, stored, or seen by anyone. Refreshing the page clears everything.