Standard Deviation Calculator Calculate sample and population standard deviation, variance, and other statistical measure...

Standard Deviation Calculator

Calculate sample and population standard deviation, variance, and other statistical measures — with data visualization and interpretation.

Enter Your Data 23, 18, 27, 22, 31, 19, 25, 28, 21, 24

Decimal Places

Data Type Sample Data Population Data

Your Dataset (Sorted)

0.00

Mean (μ/x̄)

0.00

Standard Deviation (σ/s)

0.00

Variance (σ²/s²)

0

Count (n/N)

0.00

Minimum

0.00

Maximum

0.00

Range

0.00

Sum (Σx)

Standard Deviation Formulas

Population Variance

σ² = Σ(x-μ)²/N

Population Std Dev

σ = √[Σ(x-μ)²/N]

Sample Variance

s² = Σ(x-x̄)²/(n-1)

Sample Std Dev

s = √[Σ(x-x̄)²/(n-1)]

Normal Distribution Visualization

Understanding Standard Deviation

Standard deviation measures how spread out numbers are from the mean (average) value.

Key Concepts**:

• Low standard deviation**: Data points are close to the mean
• High standard deviation**: Data points are spread out over a wider range
• Population vs Sample**: Population uses N, sample uses n-1 (Bessel's correction)
• Variance**: Square of the standard deviation

Empirical Rule (68-95-99.7)**:

• 68% of data falls within 1 standard deviation of the mean
• 95% within 2 standard deviations
• 99.7% within 3 standard deviations

✅ Why It Matters**: Standard deviation helps understand data variability and make predictions about future data points.

Common Standard Deviation Mistakes

⚠️ Avoid these critical errors:

• Using population formula for sample data**: Leads to underestimation of variability
• Ignoring outliers**: Extreme values can dramatically increase standard deviation
• Assuming normality**: Standard deviation interpretation depends on distribution shape
• Small sample sizes**: Standard deviation is less reliable with small samples (n < 30)
• Confusing variance with standard deviation**: Variance is in squared units

✅ Best Practices**:

• Always specify whether you're calculating sample or population standard deviation
• Visualize your data with histograms before interpreting standard deviation
• Use confidence intervals for more robust inference with sample data
• Consider using median absolute deviation for non-normal distributions

Real-World Applications

Standard deviation is essential in:

• Finance**: Investment risk assessment, portfolio volatility
• Quality Control**: Manufacturing tolerance analysis, process capability
• Education**: Test score analysis, grade distribution
• Research**: Experimental error measurement, result reliability
• Weather**: Temperature variation analysis, climate studies

📊 Example Use Cases**:

• Investment**: Stock A (mean return 8%, σ=12%) vs Stock B (mean 8%, σ=5%) → B is less risky
• Manufacturing**: Target diameter 10mm, σ=0.1mm → 95% of parts between 9.8-10.2mm
• Education**: Test scores (mean=75, σ=10) → 68% of students scored 65-85

How to Use This Calculator

1. Enter your data** in the text area using any of these formats:

• Comma-separated: 23, 18, 27, 22
• Space-separated: 23 18 27 22
• New line-separated: 23
18
27
22
• Mixed separators: 23, 18 27
22

2. Choose data type**:

• Sample**: When your data represents a subset of a larger population
• Population**: When your data includes the entire population of interest

3. Set decimal places** for your results (0-10)

4. Click Calculate** to see all statistics with normal distribution visualization

Note: The calculator automatically handles data cleaning, sorting, and validation. Non-numeric values are ignored.