Standard Deviation Calculator
Analyze data dispersion, variance, and central tendency with complete intermediate deviation steps and Bessel's correction.
Divisor: n - 1 = 9 (Bessel's correction)
Divisor: N = 10 (Complete population)
Pop. Variance: 68.9600
Sum: 842.00 across 10 items
Step-by-Step Calculation Table
Individual deviations from mean and squared differences.
| # | Value (x) | Deviation (x - x̄) | Squared Deviation (x - x̄)² |
|---|---|---|---|
| 1 | 88 | +3.8000 | 14.4400 |
| 2 | 92 | +7.8000 | 60.8400 |
| 3 | 79 | -5.2000 | 27.0400 |
| 4 | 85 | +0.8000 | 0.6400 |
| 5 | 94 | +9.8000 | 96.0400 |
| 6 | 67 | -17.2000 | 295.8400 |
| 7 | 73 | -11.2000 | 125.4400 |
| 8 | 91 | +6.8000 | 46.2400 |
| 9 | 84 | -0.2000 | 0.0400 |
| 10 | 89 | +4.8000 | 23.0400 |
Statistical Dispersion & Probability Distributions
Sample vs. Population (Bessel's Correction)
When analyzing a subset sample rather than an entire universe, the sample mean naturally underestimates the spread.
Dividing by n - 1 instead of n corrects for this downward bias, producing an unbiased estimator of variance.
The 68–95–99.7 Empirical Rule
In any normal bell curve distribution:
- 68.2% of data falls within ±1σ of the mean
- 95.4% falls within ±2σ
- 99.7% falls within ±3σ
Data points beyond ±3σ are statistical anomalies or outliers.
Variance vs. Standard Deviation
Variance (s²) is measured in squared units (e.g., dollars squared, seconds squared), making direct real-world interpretation difficult.
Taking the square root restores the original measurement units, giving an intuitive average distance from the mean.