Calculator
Example Data Table
| Observation | Molecular velocity (m/s) | Squared deviation |
|---|---|---|
| 1 | 422 | 30.25 |
| 2 | 435 | 42.25 |
| 3 | 417 | 110.25 |
| 4 | 440 | 110.25 |
| 5 | 429 | 0.25 |
| 6 | 433 | 12.25 |
This example uses a small molecular speed sample. The mean is 428 m/s. The sample variance equals the sum of squared deviations divided by n minus 1. The standard deviation is the square root of that variance.
Formula Used
Raw or grouped sample dispersion: σ = √[Σ(x - x̄)² / (n - 1)] for sample mode, or √[Σ(x - x̄)² / n] for population mode.
Grouped frequencies: replace each squared deviation by f(x - x̄)², then divide by the chosen denominator.
Theoretical gas model: one-dimensional velocity dispersion is σ = √(RT/M), where R is the gas constant, T is temperature in kelvin, and M is molar mass in kg/mol.
Related speeds: most probable speed = √(2RT/M), mean speed = √(8RT/πM), and RMS speed = √(3RT/M).
How to Use This Calculator
- Choose raw sample, grouped data, or the theoretical gas model.
- Enter a dataset label, preferred unit, decimal precision, and the needed values.
- Pick sample variance for experimental data or population variance for complete datasets.
- Submit the form to display results above the calculator section.
- Review the table, inspect the Plotly graph, then download CSV or PDF if needed.
Frequently Asked Questions
1. What does velocity dispersion mean in chemistry?
Velocity dispersion measures how widely molecular velocities spread around the mean. A larger value means broader motion differences across particles, while a smaller value means velocities cluster more tightly near the average.
2. When should I use sample variance instead of population variance?
Use sample variance when your velocities represent only part of a larger system. Use population variance when the dataset already contains every value you want to describe.
3. Why does the theoretical mode use temperature and molar mass?
The Maxwell-Boltzmann distribution links molecular motion to thermal energy and particle mass. Higher temperature increases dispersion, while greater molar mass reduces the spread of molecular velocities.
4. Can I paste values with commas or line breaks?
Yes. The raw sample mode accepts values separated by commas, spaces, semicolons, or new lines. That makes it easy to paste data from notebooks, spreadsheets, or lab exports.
5. What is the difference between standard deviation and standard error?
Standard deviation describes the spread of velocities themselves. Standard error estimates how precisely the sample mean represents the underlying mean, so it becomes smaller when the dataset gets larger.
6. Why are grouped midpoints used instead of class limits?
Grouped calculations approximate each class by its midpoint. This keeps the method simple and consistent, especially when original raw velocities are unavailable and only summarized frequency classes remain.
7. What does the coefficient of variation show?
The coefficient of variation expresses dispersion relative to the mean. It helps compare datasets with different average velocities because the spread is scaled into a percentage.
8. What does the graph show after calculation?
Raw samples appear as a velocity trace by observation number, grouped data appears as midpoint frequencies, and theoretical mode plots the Maxwell-Boltzmann speed distribution across the selected speed range.