Standard Deviation Calculator
Calculate population SD and sample SD simultaneously. Also shows variance, mean, quartiles, CV%, skewness, and outliers. Enter comma-separated numbers.
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How It Works
The results panel shows both simultaneously. Population SD (σ) divides by n — use for complete datasets where you have every value. Sample SD (σ̂ or s) divides by n−1 — use for samples drawn from a larger population. The difference is small for large n but significant for small datasets. For n=5: population SD underestimates true SD by about 13%.
Pop SD: σ = √(Σ(x−μ)²/N) | Sample SD: s = √(Σ(x−x̄)²/(n−1))[2,4,4,4,5,5,7,9]: Pop SD=2.00, Sample SD=2.138Variance = SD². Population variance divides by n; sample variance by n−1. Variance is additively useful: if X and Y are independent, Var(X+Y) = Var(X) + Var(Y). This property makes variance preferable in theoretical statistics even though SD is more interpretable for reporting. Both population and sample variance show in the results panel.
Pop variance: σ² = Σ(x−μ)²/N | Sample variance: s² = Σ(x−x̄)²/(n−1)[2,4,4,4,5,5,7,9]: Pop var=4.00, Sample var=4.571CV = SD / |mean| × 100%. It answers: "how large is the spread relative to the average?" This lets you compare datasets with different units. Example: comparing height variability (cm) and weight variability (kg) directly using CV%. CV below 15% = low variability; 15–30% = moderate; above 30% = high. Found in the results panel automatically.
CV% = (σ / |x̄|) × 100Mean=5, SD=2: CV = (2/5)×100 = 40% — moderately highValues more than 3 SDs from the mean are extreme outliers (outside 99.7% of a normal distribution). The calculator uses the Tukey IQR fence method (Q1−1.5×IQR, Q3+1.5×IQR) which is more robust than the SD method for non-normal data. Both approaches are shown — the outlier list appears at the bottom of the results panel.
SD method: outlier if |x − mean| > 3σ | Tukey: Q1−1.5×IQR or Q3+1.5×IQR[1,2,3,4,100]: outlier=100 (53.8 SDs above mean)Skewness > 0: right tail (mean > median). Skewness < 0: left tail (mean < median). Skewness near 0: symmetric. Excess kurtosis > 0: heavy tails (more outliers than normal). Kurtosis < 0: light tails. Both use the sample-corrected Joanes-Gill formulas, reliable for n ≥ 8. Results show in the descriptive stats panel.
Skewness = [n/((n−1)(n−2))] × Σ((xᵢ−x̄)/s)³Income data: positive skew, mean $65k vs median $45kQuick Reference
Verify these results in the calculator above.
Population SD
SD of 2,4,4,4,5,5,7,9
Pop: 2.00
Sample SD
Same data — sample SD
Samp: 2.138
Variance
Variance of same data
Pop: 4.000
SD = 0
SD of [5,5,5,5]
0
CV%
CV% when mean=5, SD=2
40%
Sample SD
SD of [1,2,3,4,5]
≈ 1.414
Variance
Variance of [1,3,5,7,9]
Pop: 8
Normal rule
68% of data within
mean ± 1 SD
Tips & Shortcuts
The results show both population and sample SD — check which your course, formula, or software expects. Excel STDEV() = sample; STDEVP() = population. Python numpy.std() defaults to population; use ddof=1 for sample.
For very small datasets (n < 10), sample SD can differ substantially from population SD. With n=3, the difference is over 22%. Always use sample SD for small research datasets.
The Frequency Table tab lets you enter grouped data efficiently — enter value 85, count 12 instead of typing 85 twelve times. The SD calculation is identical.
CV% (Coefficient of Variation) in the results lets you compare spread across datasets with different units or scales — useful when comparing variability of height (cm) vs weight (kg).
If skewness is large (>1 or <−1), the 68-95-99.7 rule does not apply — your data is not normally distributed. Use quartiles and IQR instead of SD to describe spread for skewed data.
Common Mistakes
Using population SD when the data is a sample
Sample SD (n−1 denominator) corrects for the bias of underestimating true population spread. Unless you have every member of the entire population, use sample SD. The calculator shows both — pick sample SD by default.
Interpreting SD as the maximum deviation from the mean
SD is the typical (root-mean-square) deviation, not the maximum. Individual values can deviate much more than 1 SD. For the dataset [2,4,4,4,5,5,7,9], SD=2 but the value 9 deviates 4 units from the mean of 5 — twice the SD.
Comparing SDs from datasets with different means or units
An SD of 10 kg and an SD of 10 cm are not directly comparable. Use the Coefficient of Variation (CV%) instead, which normalizes SD as a percentage of the mean for unit-free comparison.
Expecting SD to describe a skewed distribution well
SD works best for symmetric, roughly normal distributions. For heavily skewed data (income, prices, response times), use IQR and quartiles instead. The calculator provides both — check the skewness value in the results.
Entering numbers with thousand separators (1,234 instead of 1234)
Remove commas used as thousand separators before entering. The calculator interprets commas as value separators — "1,234" is parsed as two values: 1 and 234, not the number 1234.
Frequently Asked Questions
Standard deviation (SD) measures how spread out values are from the mean. Formula: SD = √(Σ(xᵢ − x̄)² / n). A low SD means values cluster tightly near the mean. A high SD means they spread out widely. SD is always non-negative and has the same units as the original data — unlike variance, which is in squared units.
Sample SD divides by (n−1) — use when your data is a subset of a larger population. The (n−1) correction (Bessel's correction) prevents underestimating the true population SD. Population SD divides by n — use only when you have every member of the entire population. In practice, sample SD is almost always correct. Excel STDEV() = sample SD; STDEVP() = population SD.
Mean = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5. Squared deviations: (2−5)²=9, (4−5)²=1 (×3)=3, (5−5)²=0 (×2)=0, (7−5)²=4, (9−5)²=16. Sum = 9+3+0+4+16 = 32. Population variance = 32/8 = 4. Population SD = √4 = 2. Sample SD = √(32/7) ≈ 2.138.
Variance = Σ(xᵢ − x̄)² / n — the average squared deviation. Standard deviation = √variance. Variance is in squared units (e.g. kg² for weight data), making it hard to interpret directly. SD is back in the original units (kg), making it immediately meaningful. The calculator shows both so you can use whichever your formula requires.
SD = 0 means every value in the dataset is identical — there is no variation whatsoever. Example: [7, 7, 7, 7, 7] has SD = 0. This is also the minimum possible SD. In practice, a very small SD (not zero) often indicates highly controlled experimental conditions or data entry errors where values were rounded aggressively.
For a normally distributed dataset: 68.3% of values fall within mean ± 1 SD, 95.4% within mean ± 2 SD, 99.7% within mean ± 3 SD. This empirical rule lets you assess how unusual a value is. A data point 2 SDs above the mean is at roughly the 97.7th percentile — rarer than 97.7% of values.
CV = (SD / |mean|) × 100%. It normalizes SD as a percentage of the mean, enabling comparison across datasets with different scales. Height SD of 10 cm on a mean of 170 cm = 5.9% CV (low variability). Investment return SD of 15% on a mean return of 5% = 300% CV (extremely high risk-to-return ratio). The calculator shows CV% automatically.
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