σ Free Statistics Tool

Standard Deviation
Calculator

Calculate standard deviation, variance, mean, range, and more from any dataset — with step-by-step working, histogram, and bell curve visualisation.

📊 Accurate Statistical Analysis

🎓 Student & Research Friendly

🔢 Step-by-Step Solutions

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Accepts commas, spaces, or line breaks. Decimals and negative numbers supported. Paste from Excel or CSV.

Standard deviation

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σ = √(Σ(x − μ)² / N) · Mode: Population

Mean (μ)

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Variance (σ²)

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Range

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Median

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Count: —

Sum: —

Min: —

Max: —

📊 Histogram (orange = potential outliers beyond ±2σ)

Values

Mean (μ)

±1σ

🔔 Normal distribution bell curve — 68.3% / 95.4% / 99.7% rule

🟢 Within ±1σ of mean · 🔵 Within ±2σ · 🟠 Potential outliers (|z| > 2)

📐 Step-by-step calculation

💡 Statistical insights:
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ℹ️ This calculator provides statistical analysis for educational and analytical purposes. Results assume numerical input data. For research and academic use, always verify calculations independently.

Standard Deviation Calculator: Find Variance & Data Spread

Standard deviation is one of the most widely used statistical measures in science, finance, education, and everyday data analysis. It tells you how spread out values are around the average — a small standard deviation means values cluster tightly near the mean, while a large one indicates wide variability. This free standard deviation calculator computes population and sample standard deviation, variance, mean, range, median, and outlier detection from any dataset — with a step-by-step breakdown, histogram, and normal distribution bell curve.

The formulas:
Population σ = √(Σ(x − μ)² / N) — use when your dataset IS the entire population
Sample s = √(Σ(x − x̄)² / (n − 1)) — use when your dataset is a SAMPLE from a larger population
Variance = σ² (population) or s² (sample) — standard deviation squared

Population vs Sample Standard Deviation

The key difference is in the denominator of the formula. Population standard deviation divides by N (the total count), while sample standard deviation divides by n−1 (the count minus one). This adjustment — called Bessel’s correction — compensates for the fact that a sample tends to underestimate the true population variability. When your dataset contains every member of a group (all students in a class, all items in a batch), use population. When it’s a subset selected from a larger group, use sample.

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When to use population σ

Test scores for a specific class, product measurements for a complete batch, historical stock returns for a closed period, census data for an entire city. You have ALL the data points, not a sample.

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When to use sample s

Survey results from a portion of customers, laboratory measurements from a subset of samples, quality control checks on random items from production, opinion polls. Your data is drawn from a larger population.

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Standard deviation in finance

In investing, standard deviation measures volatility — how much a stock or portfolio’s returns vary around its mean return. Higher σ = higher risk and potential reward. Portfolio theory uses standard deviation to optimise risk-adjusted returns.

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Standard deviation in science

Scientific experiments report results as mean ± standard deviation (e.g. 25.4 ± 0.8 °C) to indicate precision. A small σ suggests the measurement is repeatable and consistent; large σ suggests experimental error or natural variability.

Step-by-Step Calculation Example

Step	Formula	Example (dataset: 4, 7, 13, 2, 1)
1. Mean	Σx / N	(4+7+13+2+1) / 5 = 27/5 = 5.4
2. Deviations	x − μ	−1.4, 1.6, 7.6, −3.4, −4.4
3. Squared	(x − μ)²	1.96, 2.56, 57.76, 11.56, 19.36
4. Sum of squares	Σ(x − μ)²	1.96+2.56+57.76+11.56+19.36 = 93.2
5. Variance	Σ(x−μ)² / N	93.2 / 5 = 18.64
6. SD (σ)	√variance	√18.64 = 4.317

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Frequently Asked Questions

What is standard deviation?

Standard deviation is a measure of how spread out values in a dataset are around their mean. It’s the square root of the variance — the average squared distance of each value from the mean. A standard deviation of zero means all values are identical. A small standard deviation means values cluster tightly around the mean; a large one means they’re spread widely. It’s expressed in the same units as the original data, making it directly interpretable.

What is the difference between variance and standard deviation?

Variance is the average squared deviation from the mean (σ²), while standard deviation is its square root (σ). Variance is expressed in squared units (e.g. cm²), making it harder to interpret directly. Standard deviation converts back to the original units (e.g. cm), making it much more intuitive. Both measure the same thing — data spread — but standard deviation is more commonly used in real-world communication because its units are interpretable.

What is a “good” standard deviation?

There’s no universally “good” standard deviation — it depends entirely on context and the scale of your data. A useful relative measure is the coefficient of variation (CV = σ/mean × 100%). A CV below 15% is generally considered low variability; 15–30% is moderate; above 30% is high. In manufacturing quality control, tighter standard deviations (lower variability) are better. In investment returns, higher σ indicates more volatility but potentially more return opportunity.

What does high standard deviation mean?

High standard deviation means values in the dataset are spread far from the mean — there’s significant variability. In test scores, a high SD means some students scored much higher or lower than average. In stock returns, high SD means the investment is volatile. In manufacturing, high SD indicates inconsistent production quality. Whether high SD is “bad” depends on context — high variability in investment returns can mean higher potential gains as well as losses.

What is the 68-95-99.7 rule?

For a normally distributed dataset, approximately 68.3% of values fall within ±1 standard deviation of the mean, 95.4% within ±2 standard deviations, and 99.7% within ±3 standard deviations. This is called the empirical rule or 68-95-99.7 rule. Values outside ±2 standard deviations (the outer 4.6%) are often considered potential outliers. The bell curve visualisation above shows these regions for your specific dataset.

How do you detect outliers using standard deviation?

Calculate the z-score for each value: z = (x − mean) / standard deviation. Values with |z| > 2 are potential outliers (outside 95.4% of a normal distribution); values with |z| > 3 are almost certainly outliers (outside 99.7%). This calculator detects and flags values with |z| > 2, shown in orange in the sorted data pills. Note that outlier detection assumes approximately normal distribution — skewed or non-normal datasets may show different patterns.

Can I paste large datasets into this calculator?

Yes — the input accepts any number of values separated by commas, spaces, or line breaks. You can paste directly from Excel (column → line breaks), CSV files (comma-separated), or any text source. The parser strips non-numeric tokens and handles decimals and negative numbers correctly. For very large datasets (thousands of values), calculation and rendering may take a second or two.

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