%matplotlib inline

import numpy as np
from pylab import *

def de_mean(x):
    xmean = mean(x)
    return [xi - xmean for xi in x]

def covariance(x, y):
    n = len(x)
    return dot(de_mean(x), de_mean(y)) / (n-1)

pageSpeeds = np.random.normal(3.0, 1.0, 1000)
purchaseAmount = np.random.normal(50.0, 10.0, 1000)

scatter(pageSpeeds, purchaseAmount)

covariance (pageSpeeds, purchaseAmount)
-0.019528192170968867

purchaseAmount = np.random.normal(50.0, 10.0, 1000) / pageSpeeds

scatter(pageSpeeds, purchaseAmount)

covariance (pageSpeeds, purchaseAmount)
-8.8565771898786672

def correlation(x, y):
    stddevx = x.std()
    stddevy = y.std()
    return covariance(x,y) / stddevx / stddevy  #In real life you'd check for divide by zero here

correlation(pageSpeeds, purchaseAmount)
-0.62897824783314804
np.corrcoef(pageSpeeds, purchaseAmount)
array([[ 1.        , -0.62834927],
       [-0.62834927,  1.        ]])
purchaseAmount = 100 - pageSpeeds * 3

scatter(pageSpeeds, purchaseAmount)

correlation (pageSpeeds, purchaseAmount)
-1.0010010010010009