Skew T Calculator

Calculator Inputs

X value

Lower value

Upper value

Location ξ

Scale ω

Shape α

Degrees of freedom ν

Integration steps

Tail bound padding

Decimal precision

Example Data Table

Scenario	x	Scale	Shape	Degrees	Meaning
Right skew	1.25	1	3	8	Heavy tails with more right-side mass.
Left skew	-1.10	1.2	-2.5	6	Heavy tails with more left-side mass.
Nearly symmetric	0.75	1	0	12	Returns the ordinary t model.

Formula Used

The calculator uses the Azzalini style skew t density: f(x) = 2 / ω × t(z;ν) × T(αz√((ν+1)/(ν+z²));ν+1).

Here, z = (x - ξ) / ω. The value t(z;ν) is the standard t density. The value T is a t cumulative probability with ν + 1 degrees of freedom.

CDF and interval probability are estimated with Simpson numerical integration. The displayed area check should stay close to one.

Mean is shown when ν > 1. Variance and standard deviation are shown when ν > 2.

How To Use This Calculator

Enter the x value where density and cumulative chance are needed.
Enter lower and upper limits for interval probability.
Set location, scale, shape, and degrees of freedom.
Increase integration steps for smoother numerical probability results.
Use tail bound padding when tails are very heavy.
Press calculate and review results above the form.
Download CSV or PDF when you need a saved report.

Skew T Calculator Guide

A skew t distribution is useful when data is heavy tailed and unbalanced. Many real sets are not symmetric. Returns, delays, quality readings, and demand spikes may lean to one side. This calculator helps you inspect that shape with one set of inputs. You can study density, cumulative probability, survival chance, and interval probability on the same page.

Why This Distribution Matters

The ordinary t model handles heavy tails well. It gives more room for rare values than a normal curve. The skew t model adds a shape parameter. A positive shape value leans mass to the right. A negative value leans mass to the left. A zero value returns the standard t model. This makes the tool useful for risk checks and general modeling.

Inputs That Control Results

Location moves the curve along the number line. Scale spreads or compresses the curve. Degrees of freedom control tail thickness. Lower values create heavier tails. Larger values move the curve closer to a skew normal form. The shape value controls imbalance. The x value returns point density and cumulative chance. The lower and upper values return interval probability.

How To Read The Output

PDF is not a direct probability. It is a height on the curve. CDF shows the chance of being at or below x. Survival chance is one minus CDF. Interval probability is the chance between two selected limits. Mean and variance are shown only when degrees of freedom allow them. Heavy tails can make those moments undefined.

Practical Notes

Use realistic scale values. Avoid very small degrees of freedom unless your data truly has extreme tails. Increase integration steps when you need smoother probability estimates. Wider tail bounds can improve results for very heavy tails. Export the results when you need a record. Compare the example table before entering project data. It can help you learn how shape and tail settings change the final answer.

Good Workflow

Start with a shape value near zero. Then adjust it after plotting sample behavior elsewhere. Keep the same units across every field. Check both tails, not only the main value. Save CSV data for spreadsheets. Use the PDF file when sharing a compact summary with others later.

FAQs

What is a skew t distribution?

It is a t based distribution that allows asymmetry. It handles heavy tails while shifting more mass toward one side.

What does the shape input do?

Positive shape values lean the curve to the right. Negative values lean it to the left. Zero gives a symmetric t model.

What does scale mean?

Scale controls spread. A larger scale creates a wider curve. It must be greater than zero for valid calculations.

Why are mean or variance undefined?

Very heavy tails can make moments invalid. Mean needs degrees of freedom above one. Variance needs degrees above two.

Is PDF the same as probability?

No. PDF is curve height at a point. Probability comes from area under the curve across an interval.

How is CDF calculated?

The calculator estimates CDF with Simpson numerical integration. More steps can improve smoothness, but may need more processing time.

When should I increase tail bound padding?

Increase it when degrees of freedom are low. Heavy tails need wider numerical limits for better area coverage.

Can I save the results?

Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a compact report.