σ Statistics

Standard Deviation Calculator

Enter your dataset and get full statistics — mean, median, mode, variance, SD, quartiles, and a histogram — instantly.

Enter your data
Separator:
Type
σ Std Dev (Sample)—s
μ Mean—average
σ² Variance—s²
↔ Range—max − min
Complete statistical summary
Count (N)—
Sum (Σx)—
Mean (x̄)—
Median (Q2)—
Mode—
Minimum—
Maximum—
Range—
Q1 (25th percentile)—
Q3 (75th percentile)—
IQR (Q3 − Q1)—
Standard Deviation—
Variance—
Coefficient of Variation—
Standard Error—
Frequency distribution histogram
Sorted data

How standard deviation is calculated

Standard deviation (σ or s) measures how spread out numbers are in a dataset. A small SD means values are clustered near the mean. A large SD means values are more dispersed.

Mean: μ = Σx / N
Population SD: σ = √(Σ(x−μ)² / N)
Sample SD: s = √(Σ(x−x̄)² / (N−1))
Variance = SD²  |  CV = (SD / Mean) × 100%
📊 When to use sample SD

Use sample SD (÷N−1) when your data represents a subset of a larger population. The −1 correction (Bessel's correction) makes the estimate unbiased. This is the default in most statistics software.

🌐 When to use population SD

Use population SD (÷N) only when you have data for the entire population — e.g. all students in one class, all scores in one game. If in doubt, use sample SD.

📈 Coefficient of Variation

CV = (SD/Mean)×100%. It expresses SD as a percentage of the mean, allowing comparison of variability between datasets with different units or scales.

📉 Standard Error

SE = SD/√N. It estimates how far the sample mean is likely to be from the true population mean. Smaller SE = more precise estimate of population mean.

Frequently asked questions

There's no universal "good" SD — it depends on context. A low SD relative to the mean (low CV) indicates consistency. In finance, high SD means more risk. In manufacturing, low SD means higher precision. Use CV to compare relative spread across different datasets.
In a normal distribution: 68% of data falls within ±1 SD of the mean, 95% within ±2 SD, and 99.7% within ±3 SD. This is called the empirical rule. It's useful for identifying outliers — values beyond 3 SDs from the mean are extremely unusual.
Variance (σ²) is the average squared deviation from the mean. Standard deviation (σ) is the square root of variance. SD has the same units as the original data, making it more interpretable. Variance is used in mathematical derivations; SD is used for practical interpretation.
Range = max − min (affected by outliers). IQR = Q3 − Q1 (interquartile range) measures the middle 50% of data spread and is resistant to outliers. IQR is often preferred in descriptive statistics when outliers are present.
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