What is standard deviation?

Standard deviation measures how spread out numbers are from the mean (average). A low standard deviation means data points are close to the mean, while a high standard deviation indicates data is spread over a wide range. It is calculated as the square root of the variance.

What is the difference between population and sample standard deviation?

Population standard deviation (σ) divides by N (total number of values) and is used when you have data for an entire population. Sample standard deviation (s) divides by N-1 (Bessel's correction) and is used when your data is a sample from a larger population, providing an unbiased estimate.

Variance is the average of the squared differences from the mean. It is the square of the standard deviation. While variance measures spread, its units are squared, making standard deviation more intuitive since it shares the same units as the original data.

How do I interpret standard deviation?

In a normal distribution, about 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This is known as the 68-95-99.7 rule or the empirical rule.

Standard Deviation Calculator

Enter Numbers (comma-separated)

Separate numbers with commas, spaces, or new lines

Count (N)

Sum

150

Mean (Average)

Median

Mode

No mode

Range

Min

Max

Population Variance (σ²)

200

Population SD (σ)

14.1421

Sample Variance (s²)

250

Sample SD (s)

15.8114

Understanding Standard Deviation and Descriptive Statistics

Standard deviation is one of the most important measures in statistics. It quantifies how much individual data points differ from the mean of the dataset. A small standard deviation indicates values are clustered closely around the average, while a large standard deviation shows they are widely dispersed. It is the square root of variance and shares the same unit as the original data, making it more interpretable than variance alone.

This calculator computes both population standard deviation (σ) and sample standard deviation (s). Use population SD when your data represents the entire group you are studying. Use sample SD (which divides by N-1 instead of N, known as Bessel's correction) when your data is a subset of a larger population. The sample SD provides an unbiased estimate of the true population spread.

Beyond standard deviation, this tool computes the full suite of descriptive statistics. The mean is the arithmetic average. The median is the middle value when data is sorted. The mode is the most frequently occurring value. Together with minimum, maximum, range, and sum, these statistics give you a complete picture of your dataset's distribution. The empirical rule states that in a normal distribution, roughly 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

Formula

Population Standard Deviation:

σ = √( Σ(xᵢ − μ)² / N )

Sample Standard Deviation:

s = √( Σ(xᵢ − x̄)² / (N − 1) )

Where:

xᵢ = each individual data point
μ (or x̄) = mean of the data
N = number of data points
Σ = sum over all data points

Example Calculation

Scenario: Find the population SD of {4, 8, 6, 5, 3}

Step 1: Mean = (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.2
Step 2: Squared differences: (4−5.2)² + (8−5.2)² + (6−5.2)² + (5−5.2)² + (3−5.2)² = 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8
Step 3: Variance = 14.8 / 5 = 2.96
Result: σ = √2.96 ≈ 1.72

Standard Deviation Calculator

Understanding Standard Deviation and Descriptive Statistics

Formula

Example Calculation

Frequently Asked Questions

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