Python Programming for Biology: Bioinformatics and Beyond

Figure 22.4.  A normal distribution with mean and one and two standard deviations

marked. Corresponding to the example of human heights, the graph is the probability

density function for a normal (or Gaussian) distribution with a mean (μ) of 1.76 and a

standard deviation (σ) of 0.075. The values corresponding to one and two standard

deviations above and below the mean value are marked.

The  normal  distribution  is  important  because  of  the  central  limit  theorem,  which  says

that,  under  fairly  weak  assumptions,  the  distribution  of  the  average  of  a  number  of

independent  and  identically  distributed  random  variables  approaches  a  normal

distribution,  as  the  number  of  random  variables  increases.  Considering  two  random

variables, if for each point in the distribution for the first we superimpose the spread that

arises from the distribution of the second, then the summation is a ‘smoothed’ probability

distribution. The more independent random variables we add the closer the overall density

gets to the normal distribution.

This  commonly  applies  in  science  because  observed  values  are  often  complicated

combinations of multiple random variables, i.e. different factors, that all contribute to an

observed distribution of values, and data samples are often assumed to be independent and

identically  distributed.  Hence,  the  central  limit  theorem  is  often  invoked  to  justify

considering the measurement of some property to be distributed normally. For the example

of the heights of (male) humans the independent factors that contribute to the final value

may  be  things  like  multiple  genetic  factors  (each  with  probability  density  functions  for

outcomes),  nutrition,  mother’s  weight  etc.,  and  it  is  the  combination  of  all  these  random

factors that gives rise to the single statistic of height.

In the same manner as for the discrete probability distributions, we can create a simple

function to do one-tailed and two-tailed probability tests for the normal distribution using

functions from the scipy.stats module.

def normalTailTest(values, meanVal, stdDev, oneSided=True):

---

normRandVar = norm(meanVal, stdDev)

diffs = abs(values-meanVal)

result = normRandVar.cdf(meanVal-diffs) # Distrib is symmetric

if not oneSided:

result *= 2

return result

We can test this for an array of test values (i.e. human heights):

mean = 1.76

stdDev = 0.075

values = array([1.8, 1.9, 2.0])

result = normalTailTest(values, mean, stdDev, oneSided=True)

print( 'Normal one tail', result)

# Result is: [0.297, 0.03097, 0.000687]

Assuming the normal distribution and its parameters are a good model for male human

height, the results estimate that 29.7% are 1.8 metres or taller, 3.1% are 1.9 metres or taller

and 0.069 % are over 2.0 metres.