Python Programming for Biology: Bioinformatics and Beyond

x

):

The important difference compared to the Z-score is that the sample standard deviation (s)

is  used.  Because  the  sample  standard  deviation  is  only  an  estimate  for  the  true  standard

deviation (σ), given it is calculated from a limited number of samples, then when we wish

to perform a statistical test we will get an extra uncertainty that accounts for the sampling

process.  It  should  be  noted  that  there  are  many  different  formulations  of  T-statistics  that

can be used, depending on the question being asked. However, they all have the form of a

difference from a parameter estimate, divided by the corresponding standard error in the

sample.  For  example,  a  T-test  can  be  applied  to  determine  whether  two  sets  of  samples,

each from a normal distribution, have the same underlying mean, assuming that they have

same  variance  (which  might  not  be  known).  In  this  case  the  T-statistic  is  formulated  for

the two independent sample means   and  , with sample sizes n

x

and n

y

respectively:

Here  s

x,y

 is  the  sample  estimate  for  the  standard  deviation  that  is  shared  between  the

two  samples,  irrespective  of  whether  they  have  the  same  mean.  This  may  be  calculated

from the individual sample variances for the two distributions (  and  ) as follows:

This  is  the  standard  deviation  we  would  obtain  if  we  combined  both  samples  together,

with an adjustment because we have two sample means, one for each group. Effectively

we replace the 1 / (n − 1) fraction in the unbiased sample variance with 1 / (n

x

+ n

y

− 2) to

consider both samples.

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