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Standard Deviation Calculator

Enter your dataset, select whether it’s a sample or population, click "Calculate" to instantly see the standard deviation, variance, mean, sum, and error margin.

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Standard Deviation Calculator

This standard deviation calculator quickly finds how spread out numbers are in a dataset. It provides the mean, variance, coefficient of variation, standard error of the mean, and step-by-step calculations. Ideal for students, teachers, and professionals requiring fast, reliable results.

What Is Standard Deviation?

Definition:

Standard deviation (σ) measures how much individual data points differ from the mean. It indicates how spread out your data is.

Standard Deviation

What It Tells Us About Data Spread?

A low standard deviation indicates values are close to the mean, while a high standard deviation shows they are more widely spread. Applications include:

Finance: Portfolio volatility and investment risk
Climate Studies: Variability in temperature or rainfall
Sports: Performance consistency
Statistics: Deviation from expected outcomes

Population vs. Sample Standard Deviation:

Criterion	Sample Standard Deviation (s)	Population Standard Deviation (σ)
Formula	\(s = \sqrt{\dfrac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}\)	\(σ = \sqrt{\dfrac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}\)
Use Case	Subset of a population	Entire population data
Example	Test scores of 30 students in a class	Test scores of all students in a school
Bias Adjustment	Divide by n-1 to correct bias	Divide by N

Standard Deviation Formulas

Population Standard Deviation

\(\sigma = \sqrt{\dfrac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}\)

σ = population standard deviation
x_i = individual data point
μ = population mean
N = total number of values

Sample Standard Deviation

\(s = \sqrt{\dfrac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}\)

s = sample standard deviation
x_i = data point
\(\bar{x}\) = sample mean
n = total sample size

Step-by-Step Calculation

Calculate the Mean: Add all values and divide by total number of values.
Deviation from Mean: Subtract mean from each value.
Square Each Deviation: Square the differences from step 2.
Sum Squared Deviations: Add all squared differences.
Calculate Variance:
- Population: divide sum by N
- Sample: divide sum by (n-1)
Standard Deviation: Take square root of variance.

Example:

Dataset: (3, 4, 9, 7, 2, 5)

Step 1: Mean: \(\bar{x} = 30/6 = 5\)

Step 2 & 3: Deviations & Squared Deviations

Data (x_i)	x_i - x̄	(x_i - x̄)²
3	-2	4
4	-1	1
9	4	16
7	2	4
2	-3	9
5	0	0

Sum of squared deviations = 34

Step 4: Variance

Sample: \(s^2 = 34/(6-1) = 6.8\)

Population: \(\sigma^2 = 34/6 = 5.67\)

Step 5: Standard Deviation

Sample: \(s = \sqrt{6.8} \approx 2.61\)

Population: \(\sigma = \sqrt{5.67} \approx 2.38\)

How to Use Our Standard Deviation Calculator

Enter numbers separated by commas or spaces (decimals and negatives allowed).
Select Sample or Population.
Click Calculate to view SD, variance, mean, coefficient of variation, standard error, and step-by-step breakdown.
Use Copy/Download buttons to save results.

Applications

Statistics & research: measure variability, detect outliers, hypothesis testing
Finance: assess portfolio risk, stock volatility, build models
Data analysis: detect patterns, anomalies, forecast trends
Quality control & engineering: ensure product consistency and reliability

Related Concepts

Variance vs SD: Variance is squared units, SD is in original units
SD vs Standard Error: SD shows data spread, SE shows accuracy of sample mean
Coefficient of Variation: SD divided by mean, expressed as %