What does a low vs high standard deviation mean?

A low standard deviation means data points are clustered closely around the mean (low volatility). A high standard deviation means values are spread out over a wide range (high volatility or unpredictability).

What is the 68-95-99.7 Empirical Rule?

In a normal bell curve distribution, approximately 68.27% of values fall within 1 standard deviation of the mean, 95.45% within 2 standard deviations, and 99.73% within 3 standard deviations.

Can standard deviation ever be negative?

No. Because deviations are squared before summing, variance and standard deviation are always greater than or equal to zero. It equals 0 only if all numbers in the dataset are identical.

Math & StatisticsVerified: 2026-10-05

Standard Deviation Calculator

Calculate sample and population standard deviation, variance, standard error of the mean, sum of squares, and step-by-step deviations table for any dataset.

Direct Answer:Standard deviation measures the spread or dispersion of numbers from their average. For a sample, s = √[ Σ(x − x̄)² / (N − 1) ]. For example, the sample standard deviation of [10, 12, 23, 23, 16, 23, 21, 16] is 5.237, with a mean of 18.0.

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Adjust Parameters

Formula Model

Enter Data Points

Separate numbers using commas, spaces, tabs, or newlines. At least 2 numbers required.

Quick Datasets:

Sample Standard Deviation (s)

5.2372

Variance: 27.4286 | Mean: 18 (N = 8)

The sample standard deviation is 5.2372, indicating the average spread of values from the mean of 18.

Sample Variance (s²)

27.4286

Mean (x̄)

Sum of Squares (SS)

192

Standard Error (SEM)

1.8516

Coefficient of Variation

29.1%

Population Std Dev (σ)

4.899

Empirical Rule Spread (Assuming Normal Distribution)

68% of Data (±1s)

12.76 to 23.24

95% of Data (±2s)

7.53 to 28.47

99.7% of Data (±3s)

2.29 to 33.71

Calculation Breakdown

1. Calculate Mean (x̄): Sum of all values ÷ N = 144 ÷ 8 = 18
2. Calculate Deviations: Subtract mean from each data point (x_i − x̄).
3. Square Each Deviation: (x_i − x̄)², ensuring all values are positive.
4. Sum of Squared Deviations (SS): SS = 192
5. Sample Variance (s²): SS ÷ (N − 1) = 192 ÷ 7 = 27.4286
6. Sample Standard Deviation (s): √s² = √27.4286 = 5.2372

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How Standard Deviation Calculator is Calculated

Transparent mathematical formulas and assumptions used across our financial engines.

Mathematical Formula

s = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}} \quad | \quad \sigma = \sqrt{\frac{\sum_{i=1}^N (x_i - \mu)^2}{N}}

Variables & Legend

Sample Std Dev

\sigma

Population Std Dev

\bar{x}

Sample Mean

\sum (x_i - \bar{x})^2

Sum of Squares (SS)

Step-by-Step Calculation

Mean Calculation: \bar{x} = \sum x_i / n
Add all data points and divide by sample size.
Deviations: d_i = x_i - \bar{x}
Subtract the mean from each data point.
Square Deviations: d_i^2 = (x_i - \bar{x})^2
Square each deviation to remove negative signs.
Variance: s^2 = \sum d_i^2 / (n - 1)
Sum the squares and divide by degrees of freedom.
Square Root: s = \sqrt{s^2}
Take the square root to return to original units.

Conventions & Assumptions

Use Sample Standard Deviation (s) when your data represents a sample of a larger group.
Use Population Standard Deviation (σ) only when you have collected data from every single member of the population.

Frequently Asked Questions

Sample standard deviation divides by (N − 1) to correct for bias when estimating population parameters from a sample (Bessel's correction). Population standard deviation divides by N because all members of the population are known.

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