Standard Deviation Calculator

Enter a list of numbers to calculate the mean, variance, and standard deviation. Both population (σ) and sample (s) standard deviations are shown with full step-by-step working including the squared deviations table.

Enter numbers separated by commas or new lines
Enter numbers above to calculate standard deviation

How It Works

Standard deviation measures how spread out the values in a dataset are from the mean.

Step 1: Find the mean (average).
Mean = Σx ÷ n

Step 2: For each value, calculate (x − mean)².

Step 3: Sum all squared deviations: Σ(x − mean)²

Step 4a — Population standard deviation (use when your data is the entire population):
Variance σ² = Σ(x − mean)² ÷ n
Standard Deviation σ = √(σ²)

Step 4b — Sample standard deviation (use when your data is a sample from a larger population):
Variance s² = Σ(x − mean)² ÷ (n − 1)
Standard Deviation s = √(s²)

Example: Dataset [2, 4, 4, 4, 5, 5, 7, 9]
n = 8, Mean = 40 ÷ 8 = 5
Squared deviations: (2-5)²=9, (4-5)²=1, (4-5)²=1, (4-5)²=1, (5-5)²=0, (5-5)²=0, (7-5)²=4, (9-5)²=16
Sum = 32
Population variance = 32 ÷ 8 = 4, σ = √4 = 2
Sample variance = 32 ÷ 7 = 4.571, s = √4.571 ≈ 2.138

The key difference: use population (÷n) when you have ALL the data; use sample (÷n−1) when your data is a subset — the n−1 correction (Bessel's correction) removes bias.

Frequently Asked Questions

When should I use population vs sample standard deviation?

Use population standard deviation (σ, divide by n) when your dataset IS the entire population you are studying. Use sample standard deviation (s, divide by n−1) when your dataset is a sample drawn from a larger population and you are estimating the population's spread. In most real-world statistics work, sample standard deviation is correct because you rarely have the entire population.

What does a high standard deviation mean?

A high standard deviation means the values are spread widely around the mean. A low standard deviation means the values are clustered close to the mean. For example, test scores with σ=2 are very consistent; scores with σ=20 are highly variable.

What is variance?

Variance is the average of the squared deviations from the mean. Standard deviation is simply the square root of variance. Variance is in squared units (e.g., cm²), while standard deviation is in the same units as the original data (e.g., cm), making it more interpretable.

How many numbers can I enter?

The calculator handles any number of values. Enter them comma-separated or one per line. Very large datasets (thousands of values) work fine in the browser.