How the ttest is calculated in numpy

Question

I am conducting a t-test using stats.ttest_1samp and then I am calculating the t-test manually but each method comes up with different results. I am having some trouble figuring out how numpy is doing this calculation. Can someone tell me why I am getting different results?

Here is my code:

import numpy as np
from scipy import stats
import math

#our sample
x=[21, 33, 28, 17, 23, 30, 26, 27, 28, 31, 29, 23, 28, 27, 25]
x_bar = np.mean(x)#x_bar = 26.3999
mu0 = 25.5
n = len(x)
s = np.std(x)
se = s/math.sqrt(n)
print "t-statistic = %6.3f p-value = %6.3f" % stats.ttest_1samp(x, mu0)
t_manual = (x_bar-mu0)/se
t_manual

Here is my output:

>>> print "t-statistic = %6.3f p-value = %6.3f" % stats.ttest_1samp(x, mu0)
t-statistic =  0.850 p-value =  0.410
>>> t_manual = (x_bar-mu0)/se
>>> t_manual
0.87952082184625846
>>> 

Answer 1

If a function is implemented in python you can easily get its source code:

>>> import inspect
>>> print(inspect.getsource(ttest_1samp))

also inspect.getfile, returns the path to the file which implements the method/function. As for ttest_1samp, dropping the doc string this is the source code which should be easy to compare with yours:

def ttest_1samp(a, popmean, axis=0):
    """
    ... doc string ...
    """
    a, axis = _chk_asarray(a, axis)
    n = a.shape[axis]
    df = n - 1

    d = np.mean(a, axis) - popmean
    v = np.var(a, axis, ddof=1)
    denom = np.sqrt(v / float(n))

    t = np.divide(d, denom)
    t, prob = _ttest_finish(df, t)

    return t,prob

def _ttest_finish(df,t):
    """Common code between all 3 t-test functions."""
    prob = distributions.t.sf(np.abs(t), df) * 2  # use np.abs to get upper tail
    if t.ndim == 0:
        t = t[()]

    return t, prob

Answer 2

You should use n-1 instead of n because the number of degrees of freedom is n-1.

se = s/math.sqrt(n-1)  # use n-1 instead of n
print "t-statistic = %6.12f p-value = %6.3f" % stats.ttest_1samp(x, mu0)
t_manual = (x_bar-mu0)/se
print t_manual

Results:

t-statistic = 0.84969783903281959 p-value =  0.410
0.84969783903281959