Standard Deviation Calculator — Free Stats Tool
Calculate high-precision statistical metrics including mean, variance, and distribution analysis.
Std Deviation
9.8722
Mean (μ)
27.20
Variance
97.46
Count (n)
20
Median
28
Range
33
Min / Max
12 / 45
Std Error
2.2075
bar_chartFrequency Distribution
list_altStep-by-Step Calculation
Find the Mean (μ)
Add all data points and divide by the total count.
Calculate Deviations
Subtract the mean from each individual data point.
Square the Deviations
Multiply each deviation by itself to eliminate negatives.
Calculate Variance (σ²)
Find the average of squared deviations.
Final Standard Deviation
Take the square root of the variance result.
Statistical Insights
Standard Deviation: How Spread Out Is Your Data?
Reading the number
The same mean with different deviations describes completely different realities:
| Dataset (mean 70) | Std dev | Story |
|---|---|---|
| 68, 70, 71, 70, 71 | ≈ 1.1 | Remarkably consistent |
| 55, 85, 60, 80, 70 | ≈ 11.6 | Wildly variable — same average |
| Exam scores, σ = 5 | Tight class | Teaching hit one level |
| Exam scores, σ = 18 | Split class | Some flying, some drowning |
| Daily sales, low σ | Predictable | Easier inventory and staffing |
Sample vs population — the n−1 question
Dividing by n treats your data as the entire population; dividing by n−1 corrects for the fact that a sample underestimates spread. Rule: measured everyone you care about (a class's marks, this month's orders) → population. Measured a sample to infer about a larger group → sample (n−1). Exams usually specify; real analysis defaults to sample. The calculator shows both, with steps — the part coursework grades.
The 68-95-99.7 shortcut
For roughly bell-shaped data, one standard deviation around the mean covers ~68% of values, two cover ~95%, three ~99.7% — which turns σ into an instant outlier detector: a value three deviations out is rare enough to investigate. The fuller summary picture (median, quartiles, mode) assembles in the statistics calculator, and chance statements about ranges become concrete in the probability calculator.