Chapter 92 · Correlation & Association
Covariance 🔗
Covariance is the first measure of how two variables move together: its sign tells the direction. But its magnitude depends on units, which is exactly why we standardize it into correlation. We build the intuition visually and apply it to realistic marketing data written for the book.
Statistics, Data Science and AI: A Visual Handbook · John Fisher · 2026
⚙️ Setup¶
In [1]:
import numpy as np, pandas as pd, matplotlib.pyplot as plt
from scipy import stats
import seaborn as sns # seaborn = high-level statistical plots (heatmaps, regplots, pairplots)
import statsmodels.api as sm
from statsmodels.formula.api import ols
PUR="#9333ea"; DEEP="#7e22ce"; LIGHT="#c084fc"; INK="#1a2138"; GRID="#e6e9f2"; GREEN="#059669"; RED="#ef4444"; AMBER="#d97706"; BLUE="#2563eb"
plt.rcParams.update({"figure.facecolor":"white","axes.facecolor":"white","figure.dpi":110,"font.size":11,
"axes.edgecolor":GRID,"axes.grid":True,"grid.color":GRID,"axes.axisbelow":True,"axes.spines.top":False,
"axes.spines.right":False,"axes.titlesize":12,"axes.titleweight":"bold","legend.frameon":False})
sns.set_style("whitegrid")
BASE="https://raw.githubusercontent.com/johnfisher-ai/Statistics-Data-Science-AI-Visual-Book/main/data/"
rng = np.random.default_rng(89)
DEMO 1 · WHAT COVARIANCE MEASURES
The direction two variables move together
Covariance is positive when two variables tend to rise together, negative when one rises as the other falls, and near zero when there is no linear tendency. The SIGN is what matters; we visualize all three.
In [2]:
def make(rho, n=200):
x=rng.normal(0,1,n); y=rho*x+np.sqrt(max(1-rho**2,0))*rng.normal(0,1,n); return x,y
fig,ax=plt.subplots(1,3,figsize=(11,3.2))
for a,(rho,lab) in zip(ax,[(0.8,"positive"),(-0.8,"negative"),(0.0,"none")]):
x,y=make(rho); a.scatter(x,y,s=14,color=PUR,alpha=0.6)
a.set_title(f"{lab}: cov = {np.cov(x,y)[0,1]:+.2f}"); a.axhline(0,color=GRID); a.axvline(0,color=GRID)
plt.tight_layout(); plt.show()
A positive covariance slopes up, a negative one slopes down, and an uncorrelated cloud sits near zero. Covariance turns "do these move together?" into a single signed number. np.cov(x, y) returns the 2x2 covariance matrix; the off-diagonal is the covariance.
DEMO 2 · THE FORMULA, AND THE QUADRANT INTUITION
Average of the products of deviations
Covariance is the mean of (x - x-bar)(y - y-bar). Points in the upper-right and lower-left quadrants (both above or both below their means) contribute POSITIVE products; the other two quadrants contribute negative. The balance is the covariance.
In [3]:
x,y=make(0.7, 120); xb,yb=x.mean(),y.mean()
prod=(x-xb)*(y-yb)
print(f"covariance (formula) = mean of (x-xbar)(y-ybar) = {prod.mean():.3f}")
print(f"covariance (numpy) = {np.cov(x,y,ddof=0)[0,1]:.3f} (ddof=0 matches the population mean)")
fig,ax=plt.subplots(figsize=(5.6,4))
ax.scatter(x,y,c=np.where(prod>0,PUR,AMBER),s=20,alpha=0.7)
ax.axvline(xb,color=INK,lw=1,ls="--"); ax.axhline(yb,color=INK,lw=1,ls="--")
ax.set_title("Purple = same side of both means (+), amber = opposite (-)")
ax.set_xlabel("x"); ax.set_ylabel("y"); plt.tight_layout(); plt.show()
covariance (formula) = mean of (x-xbar)(y-ybar) = 0.529 covariance (numpy) = 0.529 (ddof=0 matches the population mean)
Most points are purple (same side of both means), so the positive products win and the covariance is positive. This is the whole mechanism: covariance counts how often, and how strongly, two variables deviate from their means in the same direction.
DEMO 3 · COVARIANCE vs CORRELATION
Covariance depends on units; correlation does not
Covariance has units of x times y, so its MAGNITUDE is uninterpretable, change the units and the number changes, even though the relationship is identical. Dividing by the two standard deviations gives CORRELATION, a unitless number in [-1, 1].
In [4]:
x,y=make(0.75, 300)
print(f"cov(x, y) = {np.cov(x,y)[0,1]:.3f}")
print(f"cov(x, 1000*y) = {np.cov(x,1000*y)[0,1]:.1f} <- 1000x bigger, same relationship!")
corr = np.cov(x,y)[0,1] / (x.std(ddof=1)*y.std(ddof=1))
print(f"correlation = cov / (sx*sy) = {corr:.3f}")
print(f"corr(x, 1000*y) = {np.corrcoef(x,1000*y)[0,1]:.3f} <- unchanged (unitless)")
cov(x, y) = 0.697 cov(x, 1000*y) = 697.3 <- 1000x bigger, same relationship! correlation = cov / (sx*sy) = 0.754 corr(x, 1000*y) = 0.754 <- unchanged (unitless)
Scaling y by 1000 multiplies the covariance by 1000 but leaves the correlation untouched. That is why covariance answers "which direction?" but correlation (Chapter 93) answers "how strongly?" on a comparable scale.
REAL-WORLD EXAMPLE · AD SPEND & SALES
A covariance matrix, then a correlation heatmap
A marketing team has 160 campaigns with spend, reach, visits, and sales (covariance--ad_sales.xlsx). We explore first, then compute the covariance matrix (big, unit-dependent numbers) and the correlation matrix (a readable heatmap).
In [5]:
try: d = pd.read_excel("../../data/covariance--ad_sales.xlsx", sheet_name="Campaigns")
except FileNotFoundError: d = pd.read_excel(BASE+"covariance--ad_sales.xlsx", sheet_name="Campaigns")
num = d[["ad_spend","impressions","web_visits","sales"]]
# EXPLORE FIRST: shape, missing, summary, and a quick scatter of spend vs sales
print("shape:", d.shape, "| missing:", d.isna().sum().sum())
print(num.describe().round(0).T[["mean","std","min","max"]])
sns.lmplot(data=d, x="ad_spend", y="sales", height=3.4, aspect=1.5, scatter_kws=dict(s=16,color=PUR,alpha=0.6), line_kws=dict(color=DEEP))
plt.title("Ad spend vs sales (clear positive relationship)"); plt.tight_layout(); plt.show()
shape: (160, 6) | missing: 0
mean std min max
ad_spend 4880.0 2392.0 1038.0 8993.0
impressions 152240.0 74985.0 23951.0 297674.0
web_visits 2090.0 1021.0 26.0 3973.0
sales 27327.0 12877.0 200.0 51505.0
In [6]:
print("COVARIANCE matrix (units = product of the two columns -> huge, hard to read):")
print(num.cov().round(0))
print(f"\ncov(ad_spend, sales) = {num.ad_spend.cov(num.sales):,.0f} (a meaningless magnitude on its own)")
print("\nCORRELATION matrix (unitless, -1..1 -> readable):")
print(num.corr().round(2))
fig,ax=plt.subplots(figsize=(5.2,4))
sns.heatmap(num.corr(), annot=True, fmt=".2f", cmap="Purples", vmin=0, vmax=1, square=True, ax=ax, cbar_kws=dict(shrink=0.8))
ax.set_title("Correlation heatmap"); plt.tight_layout(); plt.show()
COVARIANCE matrix (units = product of the two columns -> huge, hard to read):
ad_spend impressions web_visits sales
ad_spend 5719734.0 1.768545e+08 2361710.0 29191117.0
impressions 176854468.0 5.622789e+09 73110416.0 898461771.0
web_visits 2361710.0 7.311042e+07 1042396.0 12001607.0
sales 29191117.0 8.984618e+08 12001607.0 165827662.0
cov(ad_spend, sales) = 29,191,117 (a meaningless magnitude on its own)
CORRELATION matrix (unitless, -1..1 -> readable):
ad_spend impressions web_visits sales
ad_spend 1.00 0.99 0.97 0.95
impressions 0.99 1.00 0.95 0.93
web_visits 0.97 0.95 1.00 0.91
sales 0.95 0.93 0.91 1.00
The covariance between ad spend and sales is about 29 million (dollars times dollars), a number you cannot interpret on its own. Standardized, that same relationship is a correlation of 0.95, instantly readable as "very strong and positive." All four marketing metrics move together, which is why the next chapter focuses on correlation, the standardized cousin built for comparing relationships.
✅ Covariance
Covariance is the mean of the products of deviations: positive when variables move together, negative when they move oppositely, near zero when unrelated. Its SIGN is meaningful but its MAGNITUDE depends on units, so we standardize by dividing by the two SDs to get correlation in [-1, 1]. On the teaching ad data, cov(spend, sales) ≈ 29 million dollars² is unreadable, while the correlation 0.95 says "very strong" at a glance. Next: correlation coefficients in depth.
Statistics, Data Science and AI: A Visual Handbook · © 2026 John Fisher