Enter distribution data
Example data table
| Mode | Example input | What it calculates |
|---|---|---|
| Raw data list | 4, 7, 9, 10, 15, 18 |
Mean, variance, deviation, range, and additional spread measures from direct observations. |
| Frequency table | 2, 3
4, 5
8, 2
10, 1 |
Weighted measures using value and frequency pairs for repeated observations. |
| Probability distribution | 0, 0.10
1, 0.20
2, 0.35
3, 0.25
4, 0.10 |
Theoretical mean and variance from a discrete probability distribution. |
Formula used
Variance measures how far distribution values spread around the mean. This tool supports raw observations, grouped frequencies, and discrete probabilities.
1) Raw values
Mean: μ = Σx / n
Population variance: σ² = Σ(x − μ)² / n
Sample variance: s² = Σ(x − x̄)² / (n − 1)
Standard deviation: σ = √variance
2) Frequency table
Mean: μ = Σ(fx) / Σf
Population variance: σ² = Σ[f(x − μ)²] / Σf
Sample variance: s² = Σ[f(x − x̄)²] / (Σf − 1)
3) Probability distribution
Expected value: E[X] = Σ[x · P(x)]
Second moment: E[X²] = Σ[x² · P(x)]
Variance: Var(X) = E[X²] − (E[X])²
Additional measures
Mean absolute deviation: average absolute distance from the mean.
Coefficient of variation: (standard deviation / |mean|) × 100%
Range: maximum value − minimum value
How to use this calculator
- Select Raw data list, Frequency table, or Probability distribution.
- Enter values in the required format shown in the input helper.
- Choose Population or Sample basis when available.
- Set your preferred decimal precision.
- Click Calculate Variance to display results above the form.
- Use Download CSV or Download PDF after calculation.
FAQs
1) What does this calculator measure?
It measures dispersion around the mean. It returns variance, standard deviation, second moment, range, mean absolute deviation, and coefficient of variation when the mean is not zero.
2) When should I use population variance?
Use population variance when your dataset represents the complete group you want to study. It divides squared deviations by the full observation count.
3) When should I use sample variance?
Use sample variance when your data is only a sample from a larger population. It uses n − 1 to reduce bias in the estimate.
4) How should I enter frequency data?
Enter one pair per line using value first and frequency second. Example: 12, 4 means value 12 appears four times.
5) How should I enter probability data?
Enter one value-probability pair per line. All probabilities must be between 0 and 1, and their total must equal exactly 1.
6) Why is coefficient of variation sometimes undefined?
It becomes undefined when the mean equals zero because the formula divides standard deviation by the absolute mean.
7) What is the second moment?
The second moment is E[X²]. It helps compute theoretical variance and shows the average squared size of values around zero.
8) Can I export the results?
Yes. After calculating, use the CSV button for spreadsheets or the PDF button for a formatted report of the displayed metrics.