Standard Deviation Calculator
Sample & population σ · bell curve · deviation table
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What Is Standard Deviation?
Standard deviation is a measure of how spread out data values are from their mean. A low standard deviation means data points cluster close to the mean, while a high standard deviation means they are spread over a wider range. It is the most commonly used measure of dispersion in statistics.
Same Units as Data
Unlike variance (squared units), SD is in the same units as your data, making it directly interpretable.
Bell Curve Foundation
SD defines the shape of the normal distribution — wider curves have larger SD values.
Sample vs Population
Use s (n−1) for samples from a larger group; use σ (n) when you have the complete dataset.
The Standard Deviation Formula
Divides by n − 1 (Bessel’s correction) to give an unbiased estimate of the population variance from a sample.
Divides by n because when you have the entire population, there is no need for correction.
Step-by-Step Worked Example
Step 1: Mean = (4+8+6+5+3+7+8+1)/8 = 42/8 = 5.25
Step 2: Deviations: −1.25, 2.75, 0.75, −0.25, −2.25, 1.75, 2.75, −3.75
Step 3: Squared: 1.5625, 7.5625, 0.5625, 0.0625, 5.0625, 3.0625, 7.5625, 14.0625
Step 4: Sum of squares = 39.5
Step 5 (sample): s² = 39.5 / 7 = 5.6429
Step 6: s = √5.6429 = 2.3755
The 68–95–99.7 Rule (Empirical Rule)
For normally distributed data, the standard deviation determines how much data falls within specific ranges around the mean:
| Range | Coverage | Meaning |
|---|---|---|
| μ ± 1σ | 68.3% | About two-thirds of all values |
| μ ± 2σ | 95.4% | Nearly all values — outliers are rare |
| μ ± 3σ | 99.7% | Virtually all values — beyond is extremely rare |
Sample vs Population: When to Use Which
| Sample | Population | |
|---|---|---|
| Symbol | s | σ |
| Divisor | n − 1 | n |
| Use when | Analyzing a subset of a larger group | You have every data point in the group |
| Example | Survey of 500 voters from millions | Final grades of all 30 students in a class |
| Bias | Corrected (unbiased estimate) | Exact (no estimation needed) |
Bessel’s correction (n−1) exists because the sample mean is calculated from the same data, reducing degrees of freedom by one. This causes the sample variance to underestimate the true variance if you divide by n. Dividing by n−1 corrects this bias. For large samples (n > 30), the difference becomes negligible.
Interpreting Standard Deviation
Standard deviation alone doesn’t tell you if variability is “high” or “low” — it depends on context. Use the Coefficient of Variation (CV) to compare relative spread:
Low Variability (CV < 15%)
Data points are tightly clustered around the mean. Common in precise measurements and controlled experiments.
Moderate (15% ≤ CV ≤ 30%)
Typical spread seen in many natural and social science datasets. Generally acceptable variability.
High Variability (CV > 30%)
Data is widely spread. Common in financial returns, biological variation, and heterogeneous populations.