Calculator Form
Example Data Table
| Observation | Sample Value |
|---|---|
| 1 | 12 |
| 2 | 15 |
| 3 | 15 |
| 4 | 16 |
| 5 | 18 |
| 6 | 19 |
| 7 | 22 |
| 8 | 23 |
You can paste this example into the raw sample field to test the calculator quickly.
Formula Used
The calculator uses the sample standard deviation formula with Bessel’s correction:
s = √[ Σ(xi − x̄)2 / (n − 1) ]
Where x̄ is the sample mean, n is the sample size, and xi represents each observation. The sample variance is:
s2 = Σ(xi − x̄)2 / (n − 1)
Additional metrics shown by the calculator use these formulas:
- Mean: x̄ = Σx / n
- Standard Error: SE = s / √n
- Coefficient of Variation: CV = (s / |x̄|) × 100
- Mean Absolute Deviation: MAD = Σ|x − x̄| / n
For grouped input, each value is expanded according to its frequency before the sample statistics are calculated.
How to Use This Calculator
- Select either raw sample data or grouped values with frequencies.
- Enter the observations using commas, spaces, or line breaks.
- Choose the number of decimal places for displayed results.
- Select a Plotly graph type to visualize the sample spread.
- Click Calculate to generate results above the form.
- Review the summary table, chart, and calculation detail rows.
- Use the CSV and PDF buttons to save the output.
Frequently Asked Questions
1. What does sample standard deviation measure?
It measures how far sample observations typically spread around the sample mean. Larger values indicate more variability. Smaller values indicate tighter clustering of the sample data.
2. Why does this calculator use n − 1?
Using n − 1 applies Bessel’s correction. It reduces bias when estimating population variability from a sample, making sample variance and sample standard deviation better estimators.
3. What is the difference between variance and standard deviation?
Variance is the average squared spread using the sample formula. Standard deviation is the square root of variance, so it returns to the original unit of measurement.
4. Can I use grouped data here?
Yes. Switch to grouped mode, enter distinct values, then enter matching frequencies. The calculator expands the sample internally and computes the same sample statistics.
5. What happens if all values are identical?
The sample standard deviation becomes zero because every observation equals the mean. That means the sample shows no spread at all.
6. Why is coefficient of variation sometimes undefined?
Coefficient of variation divides standard deviation by the mean. If the mean is zero, the ratio is undefined, so the calculator marks it as not defined.
7. Which graph type should I choose?
Use a histogram to see distribution shape, a line plot to inspect observation order, and a box plot to view spread, quartiles, and potential outliers quickly.
8. Is this calculator suitable for population standard deviation?
No. This page uses the sample formula with n − 1. For full population data, a population standard deviation calculator using n is more appropriate.