# Clear previously defined variables
%reset -f
from datascience import *
%matplotlib inline
import matplotlib.pyplot as plots
import math
import numpy as np
from scipy import stats
from ipywidgets import interact, interactive, fixed, interact_manual
import ipywidgets as widgets
import nbinteract as nbi


In this section we will develop a measure of how tightly clustered a scatter diagram is about a straight line. Formally, this is called measuring linear association.

The correlation coefficient

The correlation coefficient measures the strength of the linear relationship between two variables. Graphically, it measures how clustered the scatter diagram is around a straight line.

The term correlation coefficient isn't easy to say, so it is usually shortened to correlation and denoted by $r$.

Here are some mathematical facts about $r$ that we will just observe by simulation.

  • The correlation coefficient $r$ is a number between $-1$ and 1.
  • $r$ measures the extent to which the scatter plot clusters around a straight line.
  • $r = 1$ if the scatter diagram is a perfect straight line sloping upwards, and $r = -1$ if the scatter diagram is a perfect straight line sloping downwards.

The function r_scatter takes a value of $r$ as its argument and simulates a scatter plot with a correlation very close to $r$. Because of randomness in the simulation, the correlation is not expected to be exactly equal to $r$.

Call r_scatter a few times, with different values of $r$ as the argument, and see how the scatter plot changes.

When $r=1$ the scatter plot is perfectly linear and slopes upward. When $r=-1$, the scatter plot is perfectly linear and slopes downward. When $r=0$, the scatter plot is a formless cloud around the horizontal axis, and the variables are said to be uncorrelated.

z = np.random.normal(0, 1, 500)
def r_scatter(xs, r):
    Generate y-values for a scatter plot with correlation approximately r
    return r*xs + (np.sqrt(1-r**2))*z

corr_opts = {
    'aspect_ratio': 1,
    'xlim': (-3.5, 3.5),
    'ylim': (-3.5, 3.5),

nbi.scatter(np.random.normal(size=500), r_scatter, options=corr_opts, r=(-1, 1, 0.05))

Calculating $r$

The formula for $r$ is not apparent from our observations so far. It has a mathematical basis that is outside the scope of this class. However, as you will see, the calculation is straightforward and helps us understand several of the properties of $r$.

Formula for $r$:

$r$ is the average of the products of the two variables, when both variables are measured in standard units.

Here are the steps in the calculation. We will apply the steps to a simple table of values of $x$ and $y$.

x = np.arange(1, 7, 1)
y = make_array(2, 3, 1, 5, 2, 7)
t = Table().with_columns(
        'x', x,
        'y', y
x y
1 2
2 3
3 1
4 5
5 2
6 7

Based on the scatter diagram, we expect that $r$ will be positive but not equal to 1.

nbi.scatter(t.column(0), t.column(1), options={'aspect_ratio': 1})

Step 1. Convert each variable to standard units.

def standard_units(nums):
    return (nums - np.mean(nums)) / np.std(nums)
t_su = t.with_columns(
        'x (standard units)', standard_units(x),
        'y (standard units)', standard_units(y)
x y x (standard units) y (standard units)
1 2 -1.46385 -0.648886
2 3 -0.87831 -0.162221
3 1 -0.29277 -1.13555
4 5 0.29277 0.811107
5 2 0.87831 -0.648886
6 7 1.46385 1.78444

Step 2. Multiply each pair of standard units.

t_product = t_su.with_column('product of standard units', t_su.column(2) * t_su.column(3))
x y x (standard units) y (standard units) product of standard units
1 2 -1.46385 -0.648886 0.949871
2 3 -0.87831 -0.162221 0.142481
3 1 -0.29277 -1.13555 0.332455
4 5 0.29277 0.811107 0.237468
5 2 0.87831 -0.648886 -0.569923
6 7 1.46385 1.78444 2.61215

Step 3. $r$ is the average of the products computed in Step 2.

# r is the average of the products of standard units

r = np.mean(t_product.column(4))

As expected, $r$ is positive but not equal to 1.

Properties of $r$

The calculation shows that:

  • $r$ is a pure number. It has no units. This is because $r$ is based on standard units.
  • $r$ is unaffected by changing the units on either axis. This too is because $r$ is based on standard units.
  • $r$ is unaffected by switching the axes. Algebraically, this is because the product of standard units does not depend on which variable is called $x$ and which $y$. Geometrically, switching axes reflects the scatter plot about the line $y=x$, but does not change the amount of clustering nor the sign of the association.
nbi.scatter(t.column(1), t.column(0), options={'aspect_ratio': 1})

The correlation function

We are going to be calculating correlations repeatedly, so it will help to define a function that computes it by performing all the steps described above. Let's define a function correlation that takes a table and the labels of two columns in the table. The function returns $r$, the mean of the products of those column values in standard units.

def correlation(t, x, y):
    return np.mean(standard_units(t.column(x))*standard_units(t.column(y)))
interact(correlation, t=fixed(t),
         x=widgets.ToggleButtons(options=['x', 'y'], description='x-axis'),
         y=widgets.ToggleButtons(options=['x', 'y'], description='y-axis'))
<function __main__.correlation>

Let's call the function on the x and y columns of t. The function returns the same answer to the correlation between $x$ and $y$ as we got by direct application of the formula for $r$.

correlation(t, 'x', 'y')

As we noticed, the order in which the variables are specified doesn't matter.

correlation(t, 'y', 'x')

Calling correlation on columns of the table suv gives us the correlation between price and mileage as well as the correlation between price and acceleration.

suv = (Table.read_table('https://www.inferentialthinking.com/notebooks/hybrid.csv')
       .where('class', 'SUV'))

interact(correlation, t=fixed(suv),
         x=widgets.ToggleButtons(options=['mpg', 'msrp', 'acceleration'],
         y=widgets.ToggleButtons(options=['mpg', 'msrp', 'acceleration'],
<function __main__.correlation>
correlation(suv, 'mpg', 'msrp')
correlation(suv, 'acceleration', 'msrp')

These values confirm what we had observed:

  • There is a negative association between price and efficiency, whereas the association between price and acceleration is positive.
  • The linear relation between price and acceleration is a little weaker (correlation about 0.5) than between price and mileage (correlation about -0.67).

Correlation is a simple and powerful concept, but it is sometimes misused. Before using $r$, it is important to be aware of what correlation does and does not measure.