When do I use sample vs. population standard deviation?

Use Sample Standard Deviation if your data represents a smaller subset of a larger group. Use Population Standard Deviation if your dataset contains every single member of the group you are studying.

What does a high standard deviation indicate?

A high standard deviation indicates that the data points are spread out widely over a large range of values, far from the mean. A low standard deviation means they are clustered tightly around the mean.

What is Bessel's Correction?

Bessel's correction refers to using n - 1 instead of n in the denominator of the sample variance/std dev formula. It corrects the bias in estimating population variance from a sample.

Statistics Tool

Standard Deviation Calculator

Calculate population and sample standard deviation step-by-step. Get individual squared deviations, variance, mean, and range breakdown.

Interactive Standard Deviation Calculator Workspace

Live Solver

Dataset Values (separated by commas or spaces)

Sample Standard Deviation (s)6.431692

Calculated Variance41.366667

Step-by-Step Deviations Breakdown

Dataset Mean (Average) = 23.8333

Value (x_i)	Deviation (x_i - mean)	Squared Deviation (x_i - mean)²
15	-8.8333	78.0278
20	-3.8333	14.6944
21	-2.8333	8.0278
25	1.1667	1.3611
30	6.1667	38.0278
32	8.1667	66.6944
Count: N = 6	Sum:	Σ = 206.8333

Step 1: Sum of squared deviations = 206.833333

Step 2: Divide by degrees of freedom (N - 1 = 5) to get Variance:
206.833333 / 5 = 41.366667

Step 3: Take the square root of the Variance to get Standard Deviation:
√(41.366667) = 6.431692

Mathematical Formula & Variables

Equation Models = √[ Σ(x_i - x̄)² / (n - 1) ] (Sample) or σ = √[ Σ(x_i - μ)² / N ] (Population)

Standard deviation measures the spread or dispersion of a dataset relative to its mean. The sample formula uses a denominator of n - 1 (Bessel's correction), while the population formula uses N.

Variable Definitions

Symbol	Description
s	Sample standard deviation
σ	Population standard deviation
x_i	Each individual value in the dataset
x̄ or μ	Sample average or population mean
n or N	The sample size or population count

How to Use the Standard Deviation Calculator

Input your dataset separated by commas, spaces, or lines.
Specify whether you want the Sample (default) or Population standard deviation.
Review the step-by-step table showing deviations, squared deviations, variance, and standard deviation.

Practical Example Calculation

Scenario Context: Calculating standard deviation for a sample of exam scores: 70, 80, 90.

Step 1: Calculate sample mean (x̄): (70 + 80 + 90) / 3 = 80.

Step 2: Find individual deviations (x_i - x̄): 70 - 80 = -10; 80 - 80 = 0; 90 - 80 = 10.

Step 3: Square individual deviations (x_i - x̄)²: (-10)² = 100; 0² = 0; 10² = 100.

Step 4: Sum squared deviations: 100 + 0 + 100 = 200.

Step 5: Divide by n - 1 (3 - 1 = 2): 200 / 2 = 100 (this is the Sample Variance).

Step 6: Take square root: √100 = 10.

The sample standard deviation is 10.00.

Frequently Asked Questions

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