Standard Deviation Calculator

Use Sample (n-1) when your data is a subset drawn from a larger population, such as surveying customers, testing products, or conducting experiments. This is the most common scenario.

Use Population (n) only when you have data for the entire population with no sampling, like measuring all students in a specific classroom or all employees at a company.

When in doubt, use Sample—it's the safer, more conservative choice that provides an unbiased estimate.

Low standard deviation (close to 0): Data points are tightly clustered around the mean. The dataset is homogeneous and consistent. Example: All students in a class scored between 92-98 on a test (mean=95, SD≈2).

High standard deviation (large value): Data points are scattered far from the mean. The dataset is heterogeneous with high variability. Example: Test scores ranged from 45 to 99 (mean=75, SD≈15).

The interpretation depends on your context and data scale. Always compare your SD relative to the mean (using the coefficient of variation: CV = SD/Mean).

Variance is the average of squared differences from the mean. It measures spread but is in squared units, making it harder to interpret.

Standard Deviation is the square root of variance. It's expressed in the same units as your original data, making it much more intuitive and practical for real-world interpretation.

Example: If measuring weight in pounds, variance is in (pounds)², but standard deviation is in pounds. That's why SD is preferred for communication and decision-making.

In a normal distribution:

68% of data falls within 1 standard deviation from the mean (μ ± 1σ)

95% of data falls within 2 standard deviations from the mean (μ ± 2σ)

99.7% of data falls within 3 standard deviations from the mean (μ ± 3σ)

This rule is powerful for identifying outliers. Values beyond 3 standard deviations are extremely rare and often worth investigating.

No. Standard deviation can never be negative. It's always zero or positive.

SD = 0 only when all data points are identical (no variability).

Since the formula involves squaring differences and then taking the square root, the result is always non-negative. This is by mathematical definition.

The 68-95-99.7 rule only applies to normal (bell-shaped) distributions. For skewed or non-normal data, you can use Chebyshev's inequality, which states:

At least (1 - 1/k²) of data falls within k standard deviations of the mean, for any k ≥ 1.

For k=2: At least 75% of data is within 2 standard deviations

For k=3: At least 89% of data is within 3 standard deviations

This works for any distribution shape, but it's less precise than the normal distribution rule.