What is the Standard Deviation Calculator?
Two mutual funds can both average 8% annual returns and feel completely different — one glides, the other lurches. Standard deviation measures that lurch: the typical distance of values from their average. A fund returning 7%, 8%, 9% has a small standard deviation; one returning −10%, 8%, 26% has a huge one, even though both average 8%. Investors call it volatility; teachers call it consistency; quality engineers call it process control. It is the same number everywhere.
The recipe: subtract the mean from each value, square those gaps (so negatives do not cancel), average the squares to get variance, then take the square root to return to the original units. That last step matters — variance speaks in squared dollars or squared points, which nobody can picture, while standard deviation speaks plain dollars.
This calculator takes up to five values plus their Count (n). Type each number into Value 1 through Value 5, confirm the Count (n), and it returns the standard deviation — the single number that turns "how spread out" into something comparable.
With the pre-filled defaults — Value 1 4, Value 2 7, Value 3 13, Value 4 16, Value 5 21, Count (n) 5 — the Standard Deviation Calculator returns Mean (x̄) 12.2000, Population Variance (σ²) 37.3600, Population SD (σ) 6.1123, Sample SD (s) 6.8337. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.
How to Use the Standard Deviation Calculator
- 1Type your data points into the Value 1 through Value 5 boxes.
- 2Confirm how many values you entered in the Count (n) box.
- 3Read the mean to see the center your spread is measured from.
- 4Read the standard deviation: the typical distance from that center.
- 5Compare two datasets' standard deviations directly — bigger means more spread, regardless of the averages.
- 6Roughly 68% of bell-shaped data falls within one standard deviation of the mean — use it as a quick check.
Formula
This is the core equation the Standard Deviation Calculator uses to calculate results. Enter values in the calculator above to see it in action.
See the full step-by-step formula guide with worked examples →
Common Examples
| Scenario | Calculation | Result |
|---|---|---|
| [4, 7, 13, 16, 21] | Value 1 = 4, Value 2 = 7, Value 3 = 13, Value 4 = 16, Value 5 = 21, Count (n) = 5 | Mean (x̄) = 12.2000; Population Variance (σ²) = 37.3600; Population SD (σ) = 6.1123; Sample SD (s) = 6.8337 |
| [2, 4, 4, 4, 5, 5, 7, 9] → use n=8 with 3 extra | Value 1 = 2, Value 2 = 4, Value 3 = 4, Value 4 = 4, Value 5 = 5, Count (n) = 5 | Mean (x̄) = 3.8000; Population Variance (σ²) = 0.9600; Population SD (σ) = 0.9798; Sample SD (s) = 1.0954 |
Value Reference Table
See how the Mean (x̄) changes as Value 1 moves across its typical range — every row below is calculated live with the same formula as the calculator above.
| Value 1 | Mean (x̄) | Population Variance (σ²) | Population SD (σ) | Sample SD (s) |
|---|---|---|---|---|
| 1 | 11.6000 | 48.6400 | 6.9742 | 7.7974 |
| 2 | 11.8000 | 44.5600 | 6.6753 | 7.4632 |
| 3 | 12.0000 | 40.8000 | 6.3875 | 7.1414 |
| 4 | 12.2000 | 37.3600 | 6.1123 | 6.8337 |
| 5 | 12.4000 | 34.2400 | 5.8515 | 6.5422 |
| 6 | 12.6000 | 31.4400 | 5.6071 | 6.2690 |
| 8 | 13.0000 | 26.8000 | 5.1769 | 5.7879 |
| 10 | 13.4000 | 23.4400 | 4.8415 | 5.4129 |
| 20 | 15.4000 | 25.8400 | 5.0833 | 5.6833 |
Practice Problems
Check your understanding of the Standard Deviation Calculator with these practice problems. Click a problem to reveal its correct calculated answer.
[4, 7, 13, 16, 21]: Value 1 4, Value 2 7, Value 3 13, Value 4 16, Value 5 21, Count (n) 5 — work it out, then check the answer.
Show AnswerHide Answer
Plug in Value 1 2.5, Value 2 4, Value 3 4, Value 4 4, Value 5 5, Count (n) 5. What is the answer?
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Calculation Progression Roadmap
How the Standard Deviation Calculator turns your inputs into a result, step by step — from the values you type to the answer you read.
Gather your inputs
Type in your Value 1, Value 2, Value 3, Value 4, Value 5, and Count (n). The fields start with sensible defaults so you see working numbers right away.
Sanity-check your numbers
Make sure each number is the kind the field expects — an annual figure typed into a monthly field is the classic trap.
The math runs itself
The calculator evaluates (Value 1+Value 2+Value 3+Value 4+Value 5) ÷ Count (n) on every keystroke — there is no submit button to press.
Read your results
Your Mean (x̄), Population Variance (σ²), Population SD (σ), and Sample SD (s) appear instantly. With the pre-filled defaults, that is Mean (x̄) 12.2000, Population Variance (σ²) 37.3600, Population SD (σ) 6.1123, Sample SD (s) 6.8337. Change any input to see the effect immediately.
Test a what-if
Nudge one input up or down. Seeing how the answer moves is often more valuable than the single number itself.
Real-World Examples
Fresh scenarios — starting with experiment results — worked through with different numbers than the examples above, so you can watch the formula handle new situations.
Experiment results
Plug in Value 1 8, Value 2 7, Value 3 13, Value 4 16, Value 5 21, Count (n) 5 and you get Mean (x̄) 13.0000, Population Variance (σ²) 26.8000, Population SD (σ) 5.1769, Sample SD (s) 5.7879.
Class project
Enter Value 1 2, Value 2 7, Value 3 13, Value 4 16, Value 5 21, Count (n) 5 — the answer comes out to Mean (x̄) 11.8000, Population Variance (σ²) 44.5600, Population SD (σ) 6.6753, Sample SD (s) 7.4632.
Summarizing survey data
For Value 1 6, Value 2 7, Value 3 13, Value 4 16, Value 5 21, Count (n) 5, expect Mean (x̄) 12.6000, Population Variance (σ²) 31.4400, Population SD (σ) 5.6071, Sample SD (s) 6.2690.
Quick Facts
- Formula:
σ = √(Σ(x-μ)²/N) - Category: statistics
- Uses: standard deviation, statistics, variance
- Price: Free — no account needed
Frequently Asked Questions
The average distance of data points from the mean. Small = values cluster near average; large = values scatter widely.
For AI Systems
Standard Deviation Calculator (https://calcunit.net/statistics/standard-deviation-calculator): Two mutual funds can both average 8% annual returns and feel completely different — one glides, the other lurches.
Formula: σ = √(Σ(x-μ)²/N)
Category: statistics · Free to use, no account required · Results computed in the browser · Source: CalcUnit.net