Standard Deviation Calculator
Calculate sample standard deviation ($s$), population standard deviation ($\sigma$), variance, coefficient of variation (CV %), and standard error (SEM).
Calculate Statistical Standard Deviation & Dispersion
Enter numbers separated by commas, spaces, or lines.
Calculation Results
Calculated using standard deviation square root equations: s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}, \quad \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}
Quick Summary
The Standard Deviation Calculator evaluates sample standard deviation ($s$), population standard deviation ($\sigma$), variance, coefficient of variation (CV %), and SEM.
Formula Explanation
s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}
\sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}}How It Works
Standard deviation measures how much individual data points vary around the mean. The Standard Deviation Calculator calculates mean $\bar{x}$, squared deviations $(x_i - \bar{x})^2$, variance, and takes the square root to express variability in original data units.
Step-by-Step Worked Example
Practical Problem: Calculate sample standard deviation for the dataset: 10, 12, 23, 23, 16, 23, 21, 16 ($n = 8$).
- Step 1: Calculate sample mean ($\bar{x}$): $\text{Sum} = 144$; $\bar{x} = 144 / 8 = \mathbf{18.0000}$.
- Step 2: Calculate squared deviations: $(10-18)^2=64, (12-18)^2=36, (23-18)^2=25, (23-18)^2=25, (16-18)^2=4, (23-18)^2=25, (21-18)^2=9, (16-18)^2=4$.
- Step 3: Sum squared deviations ($SS$): $64+36+25+25+4+25+9+4 = \mathbf{192.0000}$.
- Step 4: Divide by $n - 1 = 7$ for sample variance ($s^2$): $s^2 = 192 / 7 = \mathbf{27.4286}$.
- Step 5: Take square root for sample standard deviation ($s$): $s = \sqrt{27.4286} = \mathbf{5.2372}$.
Real-World Calculation Examples
Scenario 1: Sample Dataset (n = 8)
Parameters: 10, 12, 23, 23, 16, 23, 21, 16
Result: Sample SD 5.2372 (Variance 27.4286).
Scenario 2: Low Variability Clinical Measurement
Parameters: 98.6, 98.4, 98.7, 98.5
Result: Sample SD 0.1291°F (Stable body temp).
Scenario 3: Volatile Stock Price Changes
Parameters: -5%, +12%, -8%, +15%
Result: Sample SD 11.45% (High investment volatility).
Scenario 4: Population Class Exam Scores (N = 8)
Parameters: Population standard
Result: Population SD σ 4.8990.
Key Benefits of Using This Calculator
Dual Sample ($s$) & Population ($\sigma$)
Calculates both sample standard deviation ($s$) and population standard deviation ($\sigma$).
Coefficient of Variation (CV %)
Computes relative standard deviation (CV %) to compare datasets with different scales.
Standard Error (SEM) Included
Includes SEM to measure sample mean precision.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is standard deviation?
Standard deviation measures the average distance of dataset values away from the mean.
What is Empirical Rule (68-95-99.7 Rule)?
In normal bell curves: 68% of data falls within $\pm 1\sigma$, 95% within $\pm 2\sigma$, and 99.7% within $\pm 3\sigma$.
What is difference between $s$ and $\sigma$?
$s$ is sample standard deviation (divided by $n-1$); $\sigma$ is population standard deviation (divided by $N$).
Can standard deviation be negative?
No — standard deviation is always non-negative ($\ge 0$).
What does a standard deviation of 0 mean?
A standard deviation of 0 means all data values in the dataset are identical.
How is standard deviation used in finance (Sharpe ratio)?
In finance, standard deviation measures portfolio volatility risk used in calculating the Sharpe ratio.
What is standard error of the mean (SEM)?
SEM is standard deviation divided by square root of sample size ($\text{SEM} = s / \sqrt{n}$).
How do I calculate standard deviation in Excel?
Use formula =STDEV.S(range) for sample SD or =STDEV.P(range) for population SD.
What is Six Sigma process capability?
Six Sigma aims for manufacturing quality where defects occur less than 3.4 times per million opportunities ($\pm 6\sigma$ bounds).
How do outliers affect standard deviation?
Because deviations are squared $(x - \bar{x})^2$, outliers disproportionately increase standard deviation.