Sample Standard Deviation Calculator
Calculate sample standard deviation ($s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$), sample variance ($s^2$), degrees of freedom ($df = n-1$), and SEM.
Calculate Sample Standard Deviation (s)
Enter sample dataset values separated by commas, spaces, or lines.
Calculation Results
Calculated using Bessel's corrected sample standard deviation formula: s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}
Quick Summary
The Sample Standard Deviation Calculator evaluates sample standard deviation ($s$), sample variance ($s^2$), degrees of freedom ($n-1$), and Standard Error of the Mean (SEM).
Formula Explanation
s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}
s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}How It Works
When analyzing sample data drawn from a larger population, dividing sum of squares by $n-1$ (Bessel's correction) yields an unbiased estimator of population variance. The Sample Standard Deviation Calculator computes mean $\bar{x}$, sample variance $s^2$, and sample standard deviation $s$.
Step-by-Step Worked Example
Practical Problem: Calculate sample standard deviation for a sample of 5 items: 15, 20, 25, 30, 35 ($n = 5$).
- Step 1: Calculate sample mean ($\bar{x}$): $(15+20+25+30+35)/5 = 125 / 5 = \mathbf{25.0000}$.
- Step 2: Calculate squared deviations: $(-10)^2=100, (-5)^2=25, 0^2=0, 5^2=25, 10^2=100$.
- Step 3: Sum squared deviations ($SS$): $100 + 25 + 0 + 25 + 100 = \mathbf{250.0000}$.
- Step 4: Divide by $n - 1 = 4$ for sample variance ($s^2$): $s^2 = 250 / 4 = \mathbf{62.5000}$.
- Step 5: Take square root for sample standard deviation ($s$): $s = \sqrt{62.50} = \mathbf{7.9057}$.
Real-World Calculation Examples
Scenario 1: Sample Dataset (15 to 35)
Parameters: n = 5 sample
Result: Sample SD s 7.9057 (Sample Variance 62.50).
Scenario 2: Survey Respondent Age Sample
Parameters: 22, 28, 35, 42, 50
Result: Sample SD s 11.0227 (Degrees of freedom df = 4).
Scenario 3: Quality Inspection Batch Sampling
Parameters: 10.2, 10.5, 9.8, 10.1
Result: Sample SD s 0.2944 (SEM = 0.1472).
Scenario 4: Customer Order Value Sample
Parameters: $40, $55, $70, $85
Result: Sample SD s $19.36.
Key Benefits of Using This Calculator
Bessel's Correction ($n-1$)
Applies $n-1$ degrees of freedom for unbiased sample variance estimation.
Standard Error of Mean (SEM)
Computes SEM ($s / \sqrt{n}$) to measure sample mean sampling error.
Coefficient of Variation (CV %)
Displays relative variability percentage.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is sample standard deviation ($s$)?
Sample standard deviation estimates population standard deviation based on a random subset sample.
Why does sample SD divide by n - 1?
Dividing by $n-1$ compensates for using the sample mean $\bar{x}$ instead of true population mean $\mu$.
What are degrees of freedom ($df$)?
Degrees of freedom ($df = n-1$) represent the number of independent values available to estimate variance.
How do I calculate sample SD in Excel?
Use Excel formula =STDEV.S(range).
What is symbol for sample standard deviation?
The lowercase letter $s$.
What is symbol for sample variance?
Lowercase $s^2$.
How does larger sample size $n$ affect sample SD?
Larger $n$ stabilizes sample SD closer to true population SD $\sigma$.
What is Standard Error of the Mean (SEM)?
$\text{SEM} = s / \sqrt{n}$ measures the standard deviation of the sample mean distribution.
Can sample SD be calculated for 1 observation ($n=1$)?
No — $n=1$ yields $n-1=0$, causing division by zero error.
How are confidence intervals calculated using sample SD?
A 95% confidence interval for mean is $\bar{x} \pm t^* \cdot (s / \sqrt{n})$.