Sample Std Dev (s)
2.1213
Mean
6.2500
Variance
4.5000
Count
8
Need at least 2 values. Separate with commas or spaces.
Use Sample when data is a subset of a larger population (divides by n−1). Use Population when data represents the entire population (divides by N).
2.1213
2.1213
1.9843
σ = √[Σ(xᵢ − μ)² / N]Square root of the average squared deviation from the mean. Used when you have data for every member of the population.
Where:
σ= Population standard deviationxᵢ= Each individual data valueμ= Population mean (average of all values)N= Total number of values in the populations = √[Σ(xᵢ − x̄)² / (n − 1)]Same as population formula but divides by n−1 instead of N. This Bessel’s correction removes bias when estimating population variance from a sample.
Where:
s= Sample standard deviationxᵢ= Each individual data valuex̄= Sample meann= Number of values in the sampleCV = (σ / |μ|) × 100%The ratio of standard deviation to the mean, expressed as a percentage. Provides a normalized measure of dispersion.
Where:
CV= Coefficient of variation (percentage)σ= Standard deviationμ= Mean (absolute value)Inputs
Result
Mean = (4+8+6+5+3+7+8+9)/8 = 50/8 = 6.25. Squared deviations: 5.0625, 3.0625, 0.0625, 1.5625, 10.5625, 0.5625, 3.0625, 7.5625. Sum = 31.5. Sample variance = 31.5/7 = 4.5. Sample s = √4.5 = 2.1213.
Inputs
Result
Mean = (2+4+6)/3 = 4. Deviations: −2, 0, 2. Squared: 4, 0, 4. Sum = 8. Population variance = 8/3 = 2.6667. σ = √2.6667 = 1.6330.
Inputs
Result
Mean = 500/5 = 100. Deviations: 0, 10, −10, 5, −5. Squared: 0, 100, 100, 25, 25. Sum = 250. Sample variance = 250/4 = 62.5. s = √62.5 = 7.9057. CV = 7.9057/100 × 100% = 7.91%.
Standard deviation measures how spread out values are from the mean. A low standard deviation means data points cluster near the mean. A high standard deviation means data is widely spread. For example, {4, 5, 6} has a low σ of 0.82, while {1, 5, 9} has a high σ of 3.27.
| Data Set | Mean | Std Dev (σ) | Spread |
|---|---|---|---|
| 4, 5, 6 | 5 | 0.82 | Low |
| 2, 5, 8 | 5 | 2.45 | Medium |
| 1, 5, 9 | 5 | 3.27 | High |
| 5, 5, 5 | 5 | 0 | None |
Population std dev (σ) divides by N (total count) and is used when you have every member of the group. Sample std dev (s) divides by n−1 (Bessel’s correction) and is used when your data is a subset of a larger population. Sample gives a slightly larger value to correct for estimation bias.
| Type | Divisor | Symbol | When to Use |
|---|---|---|---|
| Population | N | σ | All members measured (census, full class) |
| Sample | n−1 | s | Subset of population (survey, experiment) |
Step 1: Find the mean (μ = sum/n). Step 2: Subtract the mean from each value (xᵢ − μ). Step 3: Square each deviation. Step 4: Sum the squared deviations. Step 5: Divide by N (population) or n−1 (sample). Step 6: Take the square root.
| Step | Data: {2, 4, 6} | Values |
|---|---|---|
| Mean | (2+4+6)/3 | 4 |
| Deviations | 2−4, 4−4, 6−4 | −2, 0, 2 |
| Squared | 4, 0, 4 | Sum = 8 |
| σ | √(8/3) | 1.633 |
| s | √(8/2) | 2 |
Variance is the square of standard deviation: σ² = variance, σ = √(variance). Variance measures spread in squared units, which makes it harder to interpret but mathematically convenient. Standard deviation returns to the original units by taking the square root.
| Data Set | Variance (σ²) | Std Dev (σ) | Units Example |
|---|---|---|---|
| 4, 5, 6 | 0.667 | 0.816 | ±$0.82 |
| 10, 20, 30 | 66.67 | 8.165 | ±8.17 cm |
| 100, 200, 300 | 6666.7 | 81.65 | ±$81.65 |
The coefficient of variation (CV) is the standard deviation divided by the mean, expressed as a percentage: CV = (σ/μ) × 100%. It allows comparison of variability between data sets with different units or scales. A CV below 15% indicates low variability.
| Data Set | Mean | Std Dev | CV |
|---|---|---|---|
| Heights (cm) | 170 | 6.5 | 3.8% |
| Weights (kg) | 70 | 12 | 17.1% |
| Test Scores | 85 | 8 | 9.4% |
The data sets {4,5,6} and {1,5,9} both have a mean of 5, but their standard deviations are 0.82 and 3.27 respectively. Standard deviation quantifies this difference: how far, on average, values deviate from the mean. A σ of 0 means all values are identical, while a high σ means data is widely dispersed.
In a normal distribution, the 68-95-99.7 rule maps standard deviations to data coverage: 68.27% of values fall within ±1σ of the mean, 95.45% within ±2σ, and 99.73% within ±3σ. An exam with mean 75 and σ = 10 means approximately 68% of students scored between 65 and 85. Values beyond 3σ (below 45 or above 105) are statistical outliers occurring in only 0.27% of cases.
Standard deviation uses the original units of measurement (dollars, centimeters, points), unlike variance which uses squared units. This makes σ directly interpretable: "±$8.50" is immediately meaningful whereas "72.25 dollars²" requires mental conversion. The Z-Score Calculator standardizes values using exactly this relationship: z = (x − μ) / σ.
| Data Set | Mean | Pop. σ | Sample s | CV |
|---|---|---|---|---|
| 4, 5, 6 | 5 | 0.816 | 1.000 | 16.3% |
| 2, 5, 8 | 5 | 2.449 | 3.000 | 49.0% |
| 1, 5, 9 | 5 | 3.266 | 4.000 | 65.3% |
| 5, 5, 5 | 5 | 0 | 0 | 0% |
Population standard deviation (σ) divides by N, the total count. Sample standard deviation (s) divides by n−1, known as Bessel’s correction. For {2, 4, 6}: mean = 4, squared deviations = {4, 0, 4}, sum = 8. Population: σ = √(8/3) = 1.633. Sample: s = √(8/2) = 2.000. The sample value is 22.5% larger.
Bessel’s correction compensates for a systematic bias. When you estimate the population mean from a sample, the sample mean is closer to the data points than the true population mean, making the squared deviations systematically too small. Dividing by n−1 instead of n corrects for this underestimation. For large n, the difference becomes negligible: at n = 30, the correction is 3.4%; at n = 100, it is 1.0%.
When to use which: use σ when you have measured every member of the population (all students in a class, all widgets in a batch). Use s when your data is a subset of a larger group (a survey of 500 voters from millions, a sample of 50 products from a production run). When in doubt, use s — it is the conservative choice. The Statistics Calculator can compute additional measures like median, mode, and percentiles alongside standard deviation.
The coefficient of variation CV = (σ/μ) × 100% normalizes dispersion by the mean, enabling comparison across different units and scales. Heights (mean 170 cm, σ = 6.5 cm, CV = 3.8%) are far less variable than incomes (mean $60,000, σ = $25,000, CV = 41.7%) despite income having a much larger absolute standard deviation.
Quality control uses CV thresholds: CV below 15% indicates acceptable consistency in manufacturing, 15–30% signals moderate variability worth investigating, and above 30% typically triggers process improvement. Pharmaceutical regulations often specify CV limits: USP standards for tablet content uniformity require CV below 6% for the first 10 tablets tested.
In finance, CV measures risk per unit of return. An investment with 12% expected return and 8% σ (CV = 66.7%) carries more relative risk than one with 5% return and 2% σ (CV = 40%). Portfolio theory uses standard deviation and covariance matrices to optimize risk-return tradeoffs — the foundation of modern asset allocation. The Percentage Calculator converts between absolute and relative variation measures.
| Application | Mean | σ | CV | Interpretation |
|---|---|---|---|---|
| Human heights | 170 cm | 6.5 cm | 3.8% | Very consistent |
| Test scores | 75 | 12 | 16% | Moderate spread |
| Stock returns | 8% | 15% | 187.5% | Highly volatile |
| Machine parts | 10.00 mm | 0.02 mm | 0.2% | Precision mfg |
Last Updated: Mar 26, 2026
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