William Gosset was a brewer, but had an interest in statistics. He found the estimation of the probability for the standard deviate z unreliable if the observations were few. He derived a correction of the probability estimate according to sample size and called it t.  Gosset published his papers under the pseudonym of Student and this became known as Student's t.

Student's t allows the use of a small number of measurements to estimate what may be true of the whole population.  This forms the basis of modern inferential statistics, where a small number of observations are made, and the results are generalized to the wider population.

The t distribution curve is wider than the normal one.  Therefore, a larger area (or higher probability) of being greater than a particular deviate is obtained compared to the normal distribution. This difference varies with sample size (degrees of freedom), such that the probability of t approaches that of z when the sample size increases towards infinity. Conceptually, this is represented by the diagram to the left.

With infinite degrees of freedom (i.e., a large sample size), the one tailed t and z have the same value for a particular probability, but with fewer cases, t will be larger than z in obtaining the same probability. 

 

One and two tails

When calculating t, a one tail or two tail model needs to be specify. A one tailed t is conceptually similar to the z, and assumes all the excluded values are on one side (tail) of the normal distribution, as shown in the following diagram.

A two tailed t however, assumes the area excluded are on both sides (tails) of the t distribution, so that each side contains only half of the excluded area, as shown in the following diagram.

In calculations involving the confidence interval, the two tailed t is usually used.  

For examples :

• For a two tail model calculating 95% confidence interval using a sample of 21 cases, the t value for p=0.05 (p=0.025 excluded on each of the two tails) and degrees of freedom=20 is 2.09 is obtained. In other words, the 95% confidence interval is mean±2.09SD.
• For a one tail model, p=0.05 (all 5% excluded in one tail) the t value is 1.72. The 95% confidence interval is either -∞ to mean+1.72SD, or mean-1.72SD to +∞, depending on which tail is used.

The other panels on this page are

• Calculations: Javascript program to calculate the probability of t
• Codes: R and Python codes to calculate the probability of t
• Tables: Tables for probability of t

References

https://en.wikipedia.org/wiki/Student%27s_t-distributionWikipedia on t

Javascript algorithm adapted from Press WH, Flannery BP, Teukolsky SA, and Vetterling WT. (1994) Numerical recipes in Pascal. Cambridge University Press ISBN 0-521-37516-9. p.189

Data	Data = rows of t and df or rows of probability and df

Calculation are presented in maroon and results produced presented in navy

R Codes

TToP<-function(t,degF)          #function to calculate probability from t and df
{
  p1 = 1 - pt(t,df=degF)        #probability 1 tail
  p2 = p1 * 2                   #orobability 2 tail
  return (c(p1,p2))             #returns the 2 probabilities as a list
}
TToP(2.57,5)                    #list of values of probability, 1 and 2 tail

0.02501766 0.05

PToT<-function(p,degF)          #function to calculate t from probability and df
{
  t1 = qt(1-p, df=degF)         #t 1 tail
  t2 = qt(1-p / 2, df=degF)       #t 2 tail
  return (c(t1,t2))             #returns the 2 t values as a list
}
PToT(0.025,5)                   #values of t, 1 and 2 tail

2.570582 2.015048

Python Codes

Header and Library

import scipy.stats as st

Probability of t

def TToP(t,df):
    """ Calculate Probability from t and df """
    p1 = 1 - st.t.cdf(abs(t), df)
    p2 = p1 * 2
    return p1,p2
print(TToP(2.57,5))

(0.02501765843103365, 0.0500353168620673)

def PToT(p,df):
    """ Calculate t from probability """
    t1 = st.t.ppf((1.0 - p), df)
    t2 = st.t.ppf((1.0 - p * 2), df)
    return t1,t2
print(PToT(0.025,5))

(2.5705818366147395, 2.015048372669157)

The tables present critical t values for :

• Type I Error (α) of 0.1, 0.05, and 0.01 as major columns
• One and Two tail models as minor columns
• Degrees of freedom 1 to 250, 3 abreast, in rows, divided into 5 tables

For example, for α=0.05 with 20 degrees of freedom, the critical t values are 1.7247 (one tail) and 2.0860 (two tail)
df=1:50 df=51:100 df=101:150 df=151:200 df=201:250