📐 Standard Deviation Calculator

Calculate sample and population standard deviation, variance, mean, and count

Enter Data Set (comma or space separated):

How to Use the Standard Deviation Calculator

Standard deviation is a core statistical metric that measures the extent of variation or dispersion within a dataset. A low standard deviation indicates that data values cluster tightly around the mean (average), whereas a high standard deviation shows that values are spread out over a wide range. This tool computes both sample and population standard deviation, variance, mean, and sum of squares.

Sample vs. Population Standard Deviation

Population Standard Deviation (σ): Used when your dataset encompasses every single item in the target population (e.g. test scores of all 30 students in one class). The formula divides squared deviations by N.

Sample Standard Deviation (s): Used when your data represents a sample drawn from a larger population (e.g. 100 surveyed residents out of a city). The formula divides by N - 1 (Bessel's correction) to prevent underestimating true population variance.

Mathematical Step-by-Step Formula & Example

Dataset: 4, 8, 6, 5, 3 (N = 5)

Calculate Mean (μ): (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.2.

Squared Differences from Mean:
(4 - 5.2)² = 1.44
(8 - 5.2)² = 7.84
(6 - 5.2)² = 0.64
(5 - 5.2)² = 0.04
(3 - 5.2)² = 4.84

Sum of Squared Differences (Σ): 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.80.

Population SD (σ): √(14.80 / 5) = √2.96 = 1.7205.

Sample SD (s): √(14.80 / 4) = √3.70 = 1.9235.

Empirical Rule (68-95-99.7 Rule) for Normal Distributions

In a standard bell-curve normal distribution:

68.2% of data falls within 1 standard deviation (μ ± 1σ) of the mean.

95.4% of data falls within 2 standard deviations (μ ± 2σ) of the mean.

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

Frequently Asked Questions (FAQ)

Q: What is the relationship between standard deviation and variance?
A: Variance is the square of standard deviation (Variance = σ² or s²). Standard deviation is expressed in the original measurement units, making it easier to interpret.