Standard Deviation Calculator

Calculate standard deviation with step-by-step explanations and advanced statistical analysis

Data Input

Supports up to 100 numbers. Decimal values are automatically detected.

Sample datasets:

Results

Enter your data and click "Calculate" to see results

Data Points
0
Mean
0.00
Sample SD
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Population SD
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Calculation Steps

Detailed steps will appear here after calculation

Statistical Analysis

Standard Deviation Formulas

Population: σ = √[ Σ(xᵢ - μ)² / N ]

Sample: s = √[ Σ(xᵢ - x̄)² / (n - 1) ]

Where σ = population SD, s = sample SD, μ = population mean, x̄ = sample mean, N = population size, n = sample size

Calculation Process

1. Calculate the Mean

The average of all numbers in your dataset. This is the central value around which we measure variation.

mean = (sum of values) / (number of values)
2. Compute Differences

For each number, subtract the mean and square the result. This emphasizes larger deviations.

difference² = (value - mean)²
3. Sum of Squared Differences

Add up all the squared differences. This represents the total variance in your dataset.

sum of squares = Σ(value - mean)²
4. Calculate Variance

For population data: divide by number of values (N). For sample data: divide by n-1 (Bessel's correction).

variance = sum of squares / (N or n-1)
5. Square Root for SD

Take the square root of the variance to return to the original units of measurement.

standard deviation = √(variance)

Which to Use?

Population Standard Deviation (σ): When your data includes all members of the group you're studying (complete data).

Sample Standard Deviation (s): When your data is a subset representing a larger population (most common case).

Sample SD for surveys
Sample SD for experiments
Population SD for complete data

Interpretation

A smaller standard deviation indicates data points tend to be close to the mean, while a larger standard deviation indicates data points are spread out over a wider range.

Advanced: Confidence Intervals

With the standard deviation, you can estimate how much your sample mean might vary from the true population mean.

95% CI ≈ mean ± 1.96 × (s/√n)

Standard Deviation Calculator - Find Mean, Variance & SD of Data

The Standard Deviation Calculator by SiD X Tool is a powerful online tool that helps you calculate the mean, variance, and standard deviation of any data set. Whether you’re working with population or sample data, this calculator gives you instant, accurate results with clear step-by-step calculations.

📊 What is Standard Deviation?

Standard deviation (SD) measures how spread out numbers are in a data set. It tells you how far each value is from the mean (average). A low SD means the data is close to the mean, while a high SD indicates more variation.

📋 What This Calculator Shows

🔧 Features

📐 Formulas Used

Mean (μ) = (Σx) / n

Population SD: √[Σ(x – μ)² / n]

Sample SD: √[Σ(x – x̄)² / (n – 1)]

🧠 Example

👩‍🏫 Who Can Use This Tool?

FAQs – Frequently Asked Questions

Conclusion

The Standard Deviation Calculator by SiD X Tool makes statistical analysis simple and quick. Whether you’re exploring academic data or analyzing performance metrics, this tool provides everything you need — mean, variance, SD — in one clean interface. Try it now and explore the spread of your data in seconds!