Variance and Standard Deviation: Why Squaring Won

Variance occupies a privileged position in statistics.

Why?

Not because squaring deviations is intuitively inevitable.

It is because squared deviations have extraordinarily convenient mathematics.

---

1. Population variance

For a random variable (X),

E[(X-\mu)^2].
]

Expanding,

X^2-2\mu X+\mu^2.
]

Taking expectations,

E[X^2]-2\mu E[X]+\mu^2.
]

Since

[
E[X]=\mu,
]

we obtain

[
\boxed{
\operatorname{Var}(X)=E[X^2]-E[X]^2
}
]

This identity makes variance enormously convenient computationally and theoretically.

---

2. Why deviations are measured around the mean

Consider

[
L(c)=\sum_i(x_i-c)^2.
]

Differentiate:

-2\sum_i(x_i-c).
]

Set this equal to zero:

[
\sum_i(x_i-c)=0.
]

Therefore,

[
nc=\sum_i x_i
]

and hence

[
c=\bar{x}.
]

So the arithmetic mean is precisely the value that minimizes total squared deviation.

By contrast, the median minimizes total absolute deviation.

This gives us a beautiful duality:

[
L_2 \text{ loss} \rightarrow \text{mean}
]

[
L_1 \text{ loss} \rightarrow \text{median}.
]

---

3. Sample variance and (n-1)

For a sample,

\frac{1}{n-1}
\sum_i(x_i-\bar{x})^2.
]

Why (n-1)?

Because after estimating the sample mean, only (n-1) deviations are free.

Indeed,

[
\sum_i(x_i-\bar{x})=0.
]

Once (n-1) deviations are known, the last is determined.

More formally,

\frac{n-1}{n}\sigma^2.
]

Dividing by (n-1) instead removes this downward bias for estimating population variance.

---

4. Why standard deviation exists

Variance has squared units.

If height is measured in centimeters,

[
\operatorname{Var}(X)
]

has units of

[
\text{cm}^2.
]

Taking the square root produces

[
SD=\sqrt{\operatorname{Var}(X)},
]

which returns us to centimeters.

That makes standard deviation substantially easier to communicate.

---

5. Python demonstration

import numpy as np

x = np.array([4, 5, 5, 6, 10])

print("Population variance:",
      np.var(x, ddof=0))

print("Sample variance:",
      np.var(x, ddof=1))

print("Sample SD:",
      np.std(x, ddof=1))

Verify the alternative variance formula:

var1 = np.mean(
    (x - np.mean(x))**2
)

var2 = np.mean(x**2) - np.mean(x)**2

print(var1, var2)

---

6. R demonstration

x <- c(4, 5, 5, 6, 10)

# R's var() uses denominator n - 1
var(x)
sd(x)

# Population variance
mean((x - mean(x))^2)

# Computational identity
mean(x^2) - mean(x)^2

---

7. Variances add

One of variance's greatest mathematical advantages is:

\operatorname{Var}(X)
+\operatorname{Var}(Y)
+2\operatorname{Cov}(X,Y).
]

If (X) and (Y) are independent,

[
\operatorname{Cov}(X,Y)=0
]

and therefore

\operatorname{Var}(X)
+
\operatorname{Var}(Y).
]

Standard deviations do not enjoy such a clean decomposition.

This property helped make variance foundational in experimental design, quantitative genetics, measurement theory, and signal processing.

Fisher's 1918 work made variance decomposition central to statistical thinking.

---

8. Variance's Achilles heel

Variance squares deviations.

Suppose our data are:

[
1,2,3,4,5.
]

Now replace 5 with 100.

import numpy as np

a = np.array([1,2,3,4,5])
b = np.array([1,2,3,4,100])

print(np.var(a, ddof=1))
print(np.var(b, ddof=1))

The variance explodes.

In fact, classical variance has a breakdown point effectively approaching zero: one sufficiently extreme observation can make it arbitrarily large.

This motivates an entirely different branch of statistics.

Robust statistics.