Measure how spread out your dataset is by calculating variance first, then standard deviation.
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Standard deviation measures how spread out values are from the mean. Calculate variance first (average of squared differences from the mean), then take the square root.
Choose sample (n−1) for data sampled from a larger population; population (n) when you have every data point. Using the wrong formula is a common error that produces slightly incorrect variance estimates.
Results
Count: 4
Mean: 5.0000
Variance: 5.0000
Standard deviation: 2.2361
Formula: sigma = sqrt(sum((x - mean)^2) / n)
Standard deviation (σ or s) quantifies how much data points vary from the average. A low SD means values cluster tightly around the mean; a high SD means they are spread widely.
**The calculation steps:**
1. Find the mean (average) of all values
2. Subtract the mean from each value (the deviation)
3. Square each deviation
4. Average the squared deviations = **variance**
5. Take the square root of the variance = **standard deviation**
**Population vs Sample:**
• **Population SD (σ)**: divide by n — use when you have data for the entire group (e.g., all students in a class)
• **Sample SD (s)**: divide by n−1 — use when your data is a sample from a larger population (e.g., a survey of 500 from millions). Dividing by n−1 (Bessel's correction) corrects for the bias in estimating the true population variance from a sample.
**Example** with dataset [2, 4, 4, 4, 5, 5, 7, 9]:
• Mean: 5
• Deviations: [−3, −1, −1, −1, 0, 0, 2, 4]
• Squared deviations: [9, 1, 1, 1, 0, 0, 4, 16]
• Variance (population): 32/8 = 4
• Standard deviation: √4 = **2**
Type or paste numbers separated by commas, spaces, or new lines. Example: 12, 15, 18, 22, 14
Select 'Sample' (n−1) if your data is a subset of a larger population — this is correct for most real-world analysis. Select 'Population' (n) only if you have data for the entire group.
The calculator shows mean, median, variance, standard deviation, min, max, and range. The step-by-step breakdown shows the squared deviations for each value.
Compare standard deviation to the mean: a low SD means values cluster tightly; a high SD means wide variation. Use ±1 SD and ±2 SD ranges to spot outliers.
Input
Monthly returns: 2%, −1%, 3%, 5%, −2%, 4%, 1%Output
Mean: 1.71% · SD: 2.36% (sample)This portfolio has average monthly return of 1.71% with typical month-to-month variation of ±2.36 percentage points — moderate volatility.
Common real-world scenarios where this tool saves time.
A class of 30 students scores with mean 74 and SD 12. Two thirds of students scored between 62–86 (±1 SD). Students scoring more than 2 SD below (below 50) are flagged for intervention.
Investment return standard deviation measures volatility. An ETF with 8% mean return and 5% SD is less volatile than one with 8% mean and 18% SD — same average return, very different risk profile.
In manufacturing, product dimensions must stay within tolerance. Calculate SD from a sample batch to check if variation exceeds the allowable range (e.g., ±3 SD = Six Sigma).
Compare SD of conversion rates across test variants. High SD means inconsistent performance; low SD with higher mean confirms a reliable improvement.
When to use each formula.
| Formula | Divides by | Use when | Excel function | Symbol |
|---|---|---|---|---|
| Population SD | n | You have all data points in the group | STDEVP() or STDEV.P() | σ (sigma) |
| Sample SD | n−1 | Your data is a sample from a larger population | STDEV() or STDEV.S() | s |
When in doubt, use sample SD (n−1). The difference is negligible for large datasets (n > 30) but matters for small samples.
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Use sample SD (÷n−1, shown as 's') when your data is a subset of a larger population — surveys, test batches, experiment samples. Use population SD (÷n, shown as 'σ') only when you have data for every member of the group. In practice, sample SD is appropriate for almost all real-world data analysis.
It tells you how spread out your data is. A SD of 2 means values typically differ from the mean by about 2 units. Small SD = data points cluster tightly around the average (consistent). Large SD = data is spread widely (variable or unpredictable).
Variance is the average of squared deviations — it is in squared units (e.g., cm²). Standard deviation is the square root of variance — it returns to the original units (cm), making it easier to interpret. Variance is mathematically convenient (used in ANOVA, regression), but SD is more intuitive for describing spread.
For a normal (bell curve) distribution: 68% of values fall within 1 standard deviation of the mean; 95% within 2 SDs; 99.7% within 3 SDs. This is also called the empirical rule. It does not apply to non-normal distributions (skewed data, bimodal data).
It depends on context. In manufacturing and testing, lower SD is better — it means consistent results. In investment returns, lower SD means lower volatility (safer). In height or test scores, you might expect a certain SD — unusually low SD suggests restricted range; unusually high suggests a heterogeneous group.
There is no minimum, but SD becomes unreliable with very small samples (n < 5). The result is technically calculable with n=2, but statistically meaningless. For reliable estimates, aim for at least 20–30 data points. Sample SD (÷n−1) corrects for small-sample bias.
In a perfect normal distribution with mean 100 and SD 15 (like IQ scores): 68% of people score 85–115 (±1 SD), 95% score 70–130 (±2 SD), 99.7% score 55–145 (±3 SD). Scores more than 3 SD from the mean are extremely rare (0.3% of cases).
Numbers you enter and all variance/standard deviation results stay in your browser—they are not uploaded to EverydayTools servers.
Choose sample or population mode correctly — using the wrong formula produces a different result.
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Reviewed on 2026-06-08.