Mean and Standard Deviation Calculator

Calculate sample and population mean, standard deviation, variance, median, and normal distribution 68-95-99.7 empirical confidence intervals from any raw dataset.

Data Configuration

Live statistics

Separate with commas, spaces, or new lines.

Sample Datasets:

Standard deviation is in the same units as the raw data; variance is in squared units.

Arithmetic Mean (x̄)
20.2857
Std. Dev. (s)6.0749
Variance (s²): 36.9048Count: n = 7
Median20
ModeNone
Range18
Sum (Σx)142

68-95-99.7 Empirical Rule Bands (Normal Curve)

68.27% (x̄ ± 1σ):[14.211, 26.361]
95.45% (x̄ ± 2σ):[8.136, 32.436]
99.73% (x̄ ± 3σ):[2.061, 38.511]

Mean and Standard Deviation Mathematical Formulas

The arithmetic mean (x̄ or μ) measures central tendency, while the standard deviation (s or σ) measures dispersion around that center.

1. Arithmetic Mean Formula
x̄=
Σ xᵢn
2. Sample Standard Deviation Formula (Bessel's Correction n − 1)
s=√
Σ (xᵢ − x̄)²n − 1
Step-by-Step Calculation Breakdown (Example: 10, 20, 30)
Step 1: Compute Arithmetic Mean (x̄)
• Sum Σ x = 10 + 20 + 30 = 60
• Mean x̄ = 60 ÷ 3 = 20.0000
Step 2: Calculate Squared Deviations Σ (xᵢ − x̄)²
• (10 − 20)² = (−10)² = 100
• (20 − 20)² = 0² = 0
• (30 − 20)² = 10² = 100
• Sum of Squared Deviations (SS) = 100 + 0 + 100 = 200
Step 3: Compute Sample Variance & Standard Deviation
• Sample Variance s² = 200 ÷ (3 − 1) = 200 ÷ 2 = 100.0000
• Sample Standard Deviation s = √100 = 10.0000

The 68-95-99.7 Empirical Rule Breakdown

Standard Deviation RangePercentage CoverageInterpretation
μ ± 1σ68.27%Normal typical cluster of data values
μ ± 2σ95.45%Includes ~95% of all observations
μ ± 3σ99.73%Almost all valid observations in a normal distribution
> 3σ0.27%Extreme statistical outliers

Frequently Asked Questions

What is the difference between sample and population standard deviation?
Population standard deviation (σ) divides by N (the total population size) and is used when you have every member of a group. Sample standard deviation (s) divides by (n − 1) (Bessel's correction) to provide an unbiased estimate when calculating from a smaller random sample.
How do Mean and Standard Deviation describe a dataset together?
The Mean describes the central tendency (center of gravity) of the data, while the Standard Deviation describes the average dispersion (spread) of points around that center.
What is the Empirical Rule (68-95-99.7 Rule)?
For any approximately normal (bell-shaped) distribution: ~68.27% of data falls within 1 standard deviation of the mean (μ ± 1σ), ~95.45% falls within 2 standard deviations (μ ± 2σ), and ~99.73% falls within 3 standard deviations (μ ± 3σ).
How do outliers affect Mean vs Standard Deviation?
Both the Mean and Standard Deviation are sensitive to extreme outliers because the arithmetic average includes every number and variance squares the distance from each point to the mean.

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