Standard Deviation Calculator

Calculate standard deviation (σ) for population or sample data with full working.

📖 What is Standard Deviation?

Standard deviation is a measure of how spread out numbers in a dataset are relative to their average (mean). It tells you how much the individual data points typically deviate from the mean value. A small standard deviation means data is tightly clustered around the mean; a large one means it is widely spread.

Standard deviation is represented by the symbol σ (sigma) for population data and s for sample data. It is calculated as the square root of the variance, which is the average of the squared differences from the mean.

In practice, standard deviation is everywhere: financial analysts use it to measure investment risk (a stock with higher standard deviation is more volatile), manufacturers use it to monitor quality control (are products within acceptable tolerance?), teachers use it to understand score distribution, and scientists report it as a measure of experimental uncertainty.

The key property of standard deviation in a normal distribution is the empirical rule: approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. This is the famous "68-95-99.7 rule" or "three-sigma rule."

📐 Formula

σ = √[ Σ(xᵢ − μ)² / N ]

Population Standard Deviation (σ):

Sample Standard Deviation (s):

Where: - xᵢ = each data point - μ or x̄ = mean of the dataset - N = total population size, n = sample size - The (n−1) denominator is Bessel's correction for sample data

📖 How to Use This Calculator

Enter your numbers in the text box, separated by commas or spaces.

Select Sample if your data is a subset of a larger population (most common), or Population if you have all the data.

Click Calculate - standard deviation, variance, and a step-by-step deviation table are shown.

💡 Example Calculations

Example 1 - Sample standard deviation

Data: 10, 20, 30, 40, 50

Mean = 30, n = 5

Deviations: -20, -10, 0, +10, +20

Squared: 400, 100, 0, 100, 400 → Sum = 1000

Variance = 1000 / (5-1) = 250

Sample SD = √250 = 15.81

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Example 2 - Investment volatility

Monthly returns (%): 2.1, -1.5, 3.2, 0.8, -0.5, 1.9, 2.8, -0.9

Mean return = 0.9875%

Standard deviation = 1.72% - this tells you returns typically vary by about 1.72 percentage points from the average each month.

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Frequently Asked Questions

What is standard deviation?+

Standard deviation (σ or s) measures how spread out values in a dataset are from the mean. A small standard deviation means data points are close to the mean; a large one means they are widely spread. It is the square root of variance and the most widely used measure of statistical dispersion.

What does a standard deviation of 0 mean?+

A standard deviation of 0 means all values in the dataset are identical. There is no variation at all.

How do I use standard deviation in real life?+

Standard deviation is used in finance (measuring investment volatility), quality control (checking if products are within spec), education (normalizing test scores), and science (expressing measurement uncertainty). A stock with a higher standard deviation of returns is considered riskier.

What is the coefficient of variation?+

The coefficient of variation (CV) = (Standard Deviation / Mean) × 100%. It expresses standard deviation as a percentage of the mean, allowing comparison of variability across datasets with different units or means.

When do I use population vs sample standard deviation?+

Use population standard deviation (sigma, divides by N) when you have data for the entire population. Use sample standard deviation (s, divides by N-1) when your data is a sample drawn from a larger population and you want to estimate the population parameter. In practice, most real-world analyses use sample standard deviation because you rarely have data on the entire population. The N-1 denominator (Bessel correction) corrects for the bias introduced by using a sample.

What does a high or low standard deviation mean?+

Standard deviation measures the spread or variability of data around the mean. A low SD means the data points are clustered closely around the mean (consistent data). A high SD means the data is spread widely (variable data). Example: test scores {70, 72, 68, 71, 69} have a low SD (about 1.5). Scores {40, 60, 70, 90, 95} have a high SD (about 20). Context matters - whether an SD is high or low depends on the scale of the measurement.

What is the empirical rule (68-95-99.7 rule)?+

For a normally distributed dataset, the empirical rule states: 68% of data falls within 1 standard deviation of the mean. 95% falls within 2 standard deviations. 99.7% falls within 3 standard deviations. Example: human adult heights are approximately normally distributed with mean 170 cm and SD 10 cm. About 68% of people are between 160-180 cm. About 95% are between 150-190 cm. This rule helps quickly assess whether a value is common or unusual.

What is a good standard deviation for a dataset?+

There is no universally good standard deviation - it depends entirely on context. A low SD means values cluster tightly around the mean (high consistency). A high SD indicates high variability. In quality control, a low SD is desirable; in investment returns, a high SD signals high risk. Use the coefficient of variation (SD divided by mean) to compare variability across datasets with different scales or units.

When should I use population standard deviation vs sample standard deviation?+

Use population SD (sigma, divides by N) when you have data for the entire population. Use sample SD (s, divides by N-1) when your data is a sample drawn from a larger population and you are estimating the population SD. Most real-world analysis uses sample SD. The N-1 denominator (Bessel's correction) removes bias in the estimation.