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% |
Standard deviation is one of the most important measures in statistics. It quantifies the amount of variation or dispersion in a data set, telling you how much individual values typically differ from the mean.
There are two types: population standard deviation (σ), which uses all data points, and sample standard deviation (s), which adjusts for the fact that a sample underestimates population variability. The difference lies in dividing by N versus n−1.
Our calculator computes both types simultaneously, along with variance, mean, sum of squares, and the coefficient of variation. It also provides a step-by-step deviations table showing how each value contributes to the final result.
Last Updated: Mar 9, 2026
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