Student's t-Distribution and Its Normal Approximation

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This Demonstration replaces the need for Student -distribution tables in elementary statistics courses. Critical values for confidence intervals and -value computations may be obtained using the first display, "t pdf". The checkbox, "complement" toggles between the tail-area and central-area displays.


The other displays focus on the question of how large the degrees of freedom, , need to be in order to obtain a reasonable normal approximation. Some textbooks suggest is adequate and others suggest we need . We offer a graphical and numerical approach to examine this problem.

It is fairly standard to consider a 95% confidence interval and the corresponding 5% hypothesis test. To see how well the usual normal approximation works for this case, set for the second display, "t and N(0,1) pdf". You see the exact area using the -distribution is shown in the plot label and this may be compared with the area under the normal curve, known to be 0.05. When , the correct area is 0.1449, yielding an absolute percentage error of 189.7%. As increases, the approximation is seen to improve.

The standard deviation of the -distribution is , so it is natural to ask if a distribution gives a better approximation. This question can be explored using the the third graph, " and pdf".

The fourth and fifth graphs explore these normal approximations using the Q-Q plot instead of the pdf plot. The Q-Q plot plots the quantiles of one distribution against those of another; it is better at comparing the tail behavior of the distributions. When the scales on both axes are the same, the distributions are equal if they follow the line. The checkbox is ignored for the Q-Q plots.


Contributed by: Ian McLeod (March 2011)
Open content licensed under CC BY-NC-SA



The Q-Q plot for comparing theoretical and empirical distributions is discussed with examples in [1].

[1] W. S. Cleveland, Visualizing Data, Summit, NJ: Hobart Press, 1993.

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