Some Python code I did for a blog article.

fun_with_gaussians.py

import scipy.stats as st
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

def z_score(x, m, s):
    return (x - m) / s

def p(x, m, s):
    z = z_score(x, m, s)
    return st.norm.cdf(z)

def main():
    
    # z-scores
    print(z_score(95, 100, 15), z_score(130, 100, 15), z_score(7, 100, 15))
    # We should see -0.3333333333333333 2.0 -6.2 or 1/3 deviation below average, 2 above and 6.2 below.

    # probabilities
    scores = np.arange(60, 161, 20)
    z_scores = list(map(lambda x: z_score(x, 100, 15), scores))
    less_intelligent = list(map(lambda x: p(x, 100, 15), scores))
    df = pd.DataFrame()
    df['test_score'] = scores
    df['z_score'] = z_scores
    df['less_intelligent'] = less_intelligent
    print(df)
    
    # visualization
    get_ipython().magic('matplotlib inline')
    mu, sigma = 100, 15. # mean and standard deviation
    s = sorted(np.random.normal(mu, sigma, 1000))
    count, bins, ignored = plt.hist(s, 30, normed=True)
    plt.style.use('ggplot')
    plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
                np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
                linewidth=2)
    plt.show()
    
    # IQ Score of 7
    z = z_score(7, 100, 15)
    prob = p(7, 100, 15)
    rounded_prob = round(prob, 15)
    print("The z-score {0} and probability {1} of a test score of 7.".format(z, rounded_prob))

    instances_per_billion = round((1/prob) / 1000000000, 2)
    people_on_the_planet = 7.125 # billion
    instances_on_the_planet = people_on_the_planet / instances_per_billion
    instances_on_the_planet
    
    # Likely Voters
    votes = pd.Series([46.3, 45.3, 46.3, 46.3, 49.4, 47.8, 42.7, 43.3, 49.0, 47.7, 48.3, 46.5, 46.5, 49.0, 48.0])
    # I thought it was easier to read percentages as 46.3, but I'm converting those numbers here to fit
    # in the set [0,1] as well-behaved probabilities do.
    votes = votes.apply(lambda x: x / 100) 
    votes.describe()
    
    hillary_z_score = st.norm.ppf(votes.mean())
    hillary_z_score
    iq = 15 * hillary_z_score + 100
    iq

main()