Calculator Inputs
Example Data Table
| Distribution | x | Parameter One | Parameter Two | Sample Data | Use Case |
|---|---|---|---|---|---|
| Normal | 2 | 0 | 1 | 2, 4, 5, 7, 9 | Standard score density |
| Uniform | 6 | 2 | 10 | 3, 5, 8, 11, 13 | Equal interval model |
| Binomial | 3 | 10 | 0.40 | 1, 2, 3, 3, 5 | Success count probability |
| Poisson | 4 | 3.2 | Not used | 1, 3, 4, 7, 12 | Event count probability |
Formula Used
Normal Probability Density
f(x) = 1 / (sigma sqrt(2 pi)) × exp(-0.5 × ((x - mean) / sigma)^2)
Uniform Probability Density
f(x) = 1 / (b - a), when x is between a and b.
Exponential Probability Density
f(x) = lambda × exp(-lambda × x), when x is at least zero.
Binomial Probability Mass
P(X = k) = C(n, k) × p^k × (1 - p)^(n - k)
Poisson Probability Mass
P(X = k) = exp(-lambda) × lambda^k / k!
Kurtosis
Kurtosis = fourth central moment / variance squared.
Excess kurtosis = kurtosis - 3.
How to Use This Calculator
- Select the probability distribution that matches your data model.
- Enter the x value where probability density or mass is needed.
- Enter parameter one and parameter two where required.
- Add sample data to calculate mean, variance, skewness, and kurtosis.
- Press calculate to show results above the form.
- Use CSV or PDF export for reports and records.
Advanced Probability Distribution and Kurtosis Guide
Purpose of the Calculator
This calculator helps compare theoretical probability values with sample shape. It supports common models used in statistics. You can test a point value against a selected distribution. You can also measure kurtosis from raw sample data. The result gives a useful bridge between theory and observation.
Understanding Distribution Values
A probability density function describes relative concentration. It does not always give direct probability at one point. Continuous models need intervals for exact probability. Discrete models use probability mass. This tool labels the result by model type. Normal, uniform, and exponential models return density. Binomial and Poisson models return mass.
Why Kurtosis Matters
Kurtosis measures tail weight and peak behavior. A high value may show heavy tails. A low value may show lighter tails. Excess kurtosis compares the sample shape with normal shape. A normal curve has excess kurtosis near zero. Positive excess suggests heavier tails. Negative excess suggests flatter tail behavior.
Using Sample Moments
The calculator uses central moments for shape analysis. The second moment supports variance. The third moment supports skewness. The fourth moment supports kurtosis. These values help describe spread and unusual observations. They should be reviewed with sample size. Very small samples can give unstable shape values.
Model Selection Tips
Choose normal data for balanced measurements. Choose uniform data for equal chance intervals. Choose exponential data for waiting time studies. Choose binomial data for fixed trials. Choose Poisson data for event counts. Check whether your assumptions match the process. A good model should fit the data source.
Interpreting Results
Use the distribution result for point evaluation. Use the sample table for descriptive statistics. Compare kurtosis with excess kurtosis. Review skewness before judging tail behavior. Outliers can strongly affect kurtosis. Always inspect the original data. Export your report when results need sharing.
Practical Value
This tool is useful for coursework, audits, simulations, and research. It gives quick numerical checks. It also documents formulas beside results. That makes the output easier to verify. Use it as a guide for statistical review. For formal decisions, pair it with visual plots.
FAQs
What does this calculator measure?
It measures distribution density or mass at a chosen x value. It also calculates sample mean, variance, skewness, and kurtosis from entered data.
Is PDF value the same as probability?
Not always. For continuous distributions, density at one point is not direct probability. Probability needs an interval. Discrete models return probability mass.
What is kurtosis?
Kurtosis describes tail weight and peak behavior. It uses the fourth central moment compared with variance squared.
What is excess kurtosis?
Excess kurtosis subtracts 3 from kurtosis. It compares the sample shape with a normal distribution baseline.
Which distributions are supported?
The calculator supports normal, uniform, exponential, binomial, and Poisson models for common statistical analysis tasks.
How should I enter sample data?
Enter numbers separated by commas, spaces, or semicolons. At least two valid values are needed for sample statistics.
Why is adjusted excess kurtosis unavailable?
Adjusted excess kurtosis needs more than three observations. It may also be unavailable when all values are identical.
Can I export the results?
Yes. Use the CSV button for spreadsheet records. Use the PDF button for a simple printable report.