#### Previous topic

numpy.random.standard_normal

#### Next topic

numpy.random.triangular

# numpy.random.standard_t

`numpy.random.``standard_t`(df, size=None)

  Draw samples from a standard Student’s t distribution with df degrees of freedom.

  A special case of the hyperbolic distribution. As df gets large, the result resembles that of the standard normal distribution (`standard_normal`).

  Parameters: | df : int Degrees of freedom, should be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. Default is None, in which case a single value is returned.
  Returns: | samples : ndarray or scalar Drawn samples.

  Notes

  The probability density function for the t distribution is

  The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean.

  The derivation of the t-distribution was first published in 1908 by William Gisset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student.

  References

  [R267] | (1, 2) Dalgaard, Peter, “Introductory Statistics With R”, Springer, 2002.

  [R268] | Wikipedia, “Student’s t-distribution” http://en.wikipedia.org/wiki/Student’s_t-distribution

  Examples

  From Dalgaard page 83 [R267], suppose the daily energy intake for 11 women in Kj is:

      >>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
      ...                    7515, 8230, 8770])

  Does their energy intake deviate systematically from the recommended value of 7725 kJ?

  We have 10 degrees of freedom, so is the sample mean within 95% of the recommended value?

      >>> s = np.random.standard_t(10, size=100000)
      >>> np.mean(intake)
      6753.636363636364
      >>> intake.std(ddof=1)
      1142.1232221373727

  Calculate the t statistic, setting the ddof parameter to the unbiased value so the divisor in the standard deviation will be degrees of freedom, N-1.

      >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
      >>> import matplotlib.pyplot as plt
      >>> h = plt.hist(s, bins=100, normed=True)

  For a one-sided t-test, how far out in the distribution does the t statistic appear?

      >>> np.sum(s<t) / float(len(s))
      0.0090699999999999999  #random

  So the p-value is about 0.009, which says the null hypothesis has a probability of about 99% of being true.

  (Source code, png, pdf)