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ProbabilityConcept reference

Student’s t distributions

A standard Student t variable with nu degrees of freedom is Z/sqrt(V/nu), where Z is standard normal and V is an independent chi-squared variable with nu degrees of freedom.

2 min read1 figureSources & further reading
On this page 9 sections
  1. Overview
  2. Understand the idea
  3. The square root encloses the whole ratio V/nu
  4. The degrees of freedom belong to the chi-squared law, not to one observed value of V
  5. Using Z/(V/4) would skip the square root and produce a different random variable
  6. Key takeaway
  7. Further questions
  8. Sources & further reading
  9. Concept connections

01Understand the idea#

Student’s t distribution describes a standardized normal quantity divided by an independent estimated scale. The denominator is random, so the ratio is more variable than a standard normal. That extra uncertainty matters when a population standard deviation must be estimated from a sample.

The degrees of freedom belong to the chi-squared variable supplying the scale. Small degrees of freedom give heavier tails. This lesson identifies the construction; a later lesson uses t critical values to build an interval for a population mean.

The numerator is standard normal. Its divisor is a random scale from an independent chi-squared variable. This construction gives a Student t variable.

Independent inputs Z with standard-normal law and V with chi-square law and 4 degrees of freedom. Divide V by 4, take its square root, and divide Z by that result. The resulting ratio has Student t law with 4 degrees of freedom.Independent inputs Z with standard-normal law and V with chi-square law and 4 degrees of freedom. Divide V by 4, take its square root, and divide Z by that result. The resulting ratio has Student t law with 4 degrees of freedom.
Figure 1Independent inputs Z with standard-normal law and V with chi-square law and 4 degrees of freedom. Divide V by 4, take its square root, and divide Z by that result. The resulting ratio has Student t law with 4 degrees of freedom.
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If V∼χν2V\sim\chi^2_\nu is independent of Z∼N(0,1)Z\sim N(0,1), divide VV by its fixed degrees of freedom, then take the square root.

T=ZV/ν∼tνT=\frac{Z}{\sqrt{V/\nu}}\sim t_\nu

For Z∼N(0,1)Z\sim N(0,1) independent of V∼χ42V\sim\chi^2_4, the denominator is V/4\sqrt{V/4}. A realization Z=2,V=16Z=2,V=16 gives divisor 4=2\sqrt4=2 and ratio 1. The ratio’s distribution has 4 degrees of freedom; the observed value 16 does not change that parameter.

Check your reasoning

Independent Z∼N(0,1)Z\sim N(0,1) and V∼χ62V\sim\chi^2_6. T=Z/V/6T=Z/\sqrt{V/6}. Choose its law.

  1. Aχ62\chi^2_6
  2. BN(0,1)N(0,1)
  3. Ct6t_6
Show answer and explanation
t6t_6

Chi-squared df 66 gives t6t_6.

02The square root encloses the whole ratio V/nu#

The square root encloses the whole ratio V/νV/\nu. Using V/νV/\nu directly or forgetting the divisor ν\nu changes the random scale and breaks this standard t construction.

Check your reasoning

Independent Z∼N(0,1)Z\sim N(0,1) and V∼χ32V\sim\chi^2_3. Repair Z/(V/3)Z/(V/3) to obtain a t3t_3 variable.

  1. AZ/V/3Z/\sqrt{V/3}
  2. BZ/(V/3)Z/(V/3)
  3. CZ/VZ/\sqrt{V}
Show answer and explanation
Z/V/3Z/\sqrt{V/3}

Put V/3V/3 inside the square root.

03The degrees of freedom belong to the chi-squared law, not to one observed value of V#

The degrees of freedom belong to the chi-squared law, not to one observed value of VV. They affect the t curve’s tails: smaller values give heavier tails, while larger values approach the standard normal.

Check your reasoning

Error E∼N(0,1)E\sim N(0,1) and noise S∼χ112S\sim\chi^2_{11} are independent. Which ratio has law t11t_{11}?

  1. AE/(S/11)E/(S/11)
  2. BE/S/11E/\sqrt{S/11}
  3. CE/(11S)E/(11\sqrt{S})
Show answer and explanation
E/S/11E/\sqrt{S/11}

Divide by S/11\sqrt{S/11}.

04Using Z/(V/4) would skip the square root and produce a different random variable#

Using Z/(V/4)Z/(V/4) would skip the square root and produce a different random variable. The independence and standard-normal assumptions are part of the construction, not optional details justified by the shape of the displayed formula.

Key takeaway

Check both input laws and independence. Divide the normal input by the square root of chi-squared input over its degrees of freedom.

  • Interpret a supplied standard-normal to scaled-chi-squared ratio as a t variable under independence.

Further questions

How do these reference distributions support statistical inference?
When a statistical procedure’s assumptions hold, a reference distribution provides tail probabilities or cutoffs for assessing sampling uncertainty. The degrees of freedom and independence conditions belong to the procedure.

Sources & further reading

  1. [1]
  2. [2]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Student’s t distributions.” Math behind ML. /learn/p-student-t