Statistics & Data

Free Standard Deviation & Variance Calculator

Understanding variability in data is fundamental to statistics, financial risk analysis, scientific research, and quality control. Our free standard deviation calculator enables you to calculate sample standard deviation and population standard deviation formula metrics instantly. Whether you are analyzing exam scores, stock returns, or experimental trials, this variance calculator online statistics tool provides step-by-step numerical breakdowns.

Standard Deviation & Statistics Calculator

Sample (s) & Population (σ) Standard Deviation, Variance, Mean, Median, Range & Step Table

Enter Numbers (separated by commas or spaces)

Calculated Statistical Summary (sample)

Sample Std Dev (s)

15.811388

Variance (s²)

250.000000

Mean (x̄): 30.0000

Median: 30.0000

Range: 40.0000

Std Error (SE): 7.0711

Step-by-Step Deviation Table (N = 5, Σ(x - x̄)² = 1000.0000)

#	Value (xᵢ)	Deviation (xᵢ - x̄)	Squared Dev (xᵢ - x̄)²
1	10	-20.0000	400.0000
2	20	-10.0000	100.0000
3	30	0.0000	0.0000
4	40	10.0000	100.0000
5	50	20.0000	400.0000

Mathematical Formulas & Calculation Methods

Sample Standard Deviation Formula (s)

Measures dispersion in a sample subset by dividing by (n - 1) degrees of freedom using the calculate sample standard deviation formula.

s = √[ Σ(x_i - x̄)² ÷ (n - 1) ]

Population Standard Deviation Formula (σ)

Measures dispersion across an entire population by dividing by total count N using the population standard deviation formula.

σ = √[ Σ(x_i - μ)² ÷ N ]

Variance Formula (s² or σ²)

The square of standard deviation representing average squared deviation using our sample vs population variance solver.

Variance = (Standard Deviation)²

Worked Step-by-Step Examples

Example 1: Calculating Sample Standard Deviation for Test Scores

"Find the sample standard deviation for student test scores: 70, 80, 90, 100. <strong>calculate standard deviation step by step</strong>."

Steps:

Calculate mean: (70 + 80 + 90 + 100) ÷ 4 = 85.
Find squared deviations: (70-85)² = 225, (80-85)² = 25, (90-85)² = 25, (100-85)² = 225.
Sum squared deviations: 225 + 25 + 25 + 225 = 500.
Divide by n - 1 = 3: 500 ÷ 3 = 166.67 (Variance).
Take square root: √166.67 = 12.91.

Result: The sample standard deviation is 12.91.

Example 2: Population Variance in Quality Control

"A factory measures bolt diameters: 10mm, 12mm, 14mm. Compute population variance."

Steps:

Mean: (10 + 12 + 14) ÷ 3 = 12.
Squared deviations: (10-12)² = 4, (12-12)² = 0, (14-12)² = 4.
Sum: 4 + 0 + 4 = 8.
Divide by N = 3: 8 ÷ 3 = 2.67.

Result: The population variance is 2.67 mm².

What is Standard Deviation & Variance Calculator?

A standard deviation calculator measures how spread out numbers are from their average mean value.

A low standard deviation indicates that data points cluster tightly around the mean, while a high standard deviation shows wider data dispersion.

How to Use Standard Deviation & Variance Calculator

1Enter your raw data numbers separated by commas or spaces.
2Select whether your data represents a sample or an entire population.
3View calculated mean, variance, and standard deviation instantly.
4Copy results directly for statistics reports or research papers.

Key Features

Dual support for sample standard deviation and population metrics.
Real-time mean, variance, and sum of squares computation.
Step-by-step formula breakdown for statistics homework verification.
Flexible input parsing for comma-separated or space-separated numbers.
100% free web execution with complete data privacy.

Benefits of Using Standard Deviation & Variance Calculator

Saves time solving statistics homework and research data sets.

Eliminates manual squaring and square root errors.

Helps financial analysts evaluate investment portfolio volatility.

Runs instantly on mobile phones and desktop computers.

Practical Use Cases

Statistics Students

Verify class homework and master mean and standard deviation tool principles.

Financial Analysts

Measure asset return volatility and risk distribution in stock portfolios.

Quality Control Engineers

Monitor manufacturing tolerance variations across product batches.

Researchers & Scientists

Analyze experimental trial data and report standard error margins.

Frequently Asked Questions

What is the difference between sample and population standard deviation?

Sample standard deviation uses (n - 1) in the denominator to adjust for sample bias, whereas population standard deviation divides by total population count N using the population standard deviation formula.

Can standard deviation be negative?

No. Standard deviation is always zero or positive because it is the square root of squared deviations.

How do you calculate variance from standard deviation?

Variance is simply standard deviation squared using our variance calculator online statistics tool.

Is this statistics calculator free?

Yes, this standard deviation calculator is 100% free with unlimited computations.

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