The Central Idea
Let \(X\) be a random variable with mean
\[ \mu=\mathbb{E}[X]. \]The variance of \(X\) is the expected squared distance from the mean:
\[ \boxed{\operatorname{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right]}. \]The standard deviation is the positive square root of the variance:
\[ \boxed{\sigma_X=\sqrt{\operatorname{Var}(X)}}. \]Why We Need a Measure of Spread
Two random variables can have the same expected value while behaving very differently.
Suppose
\[ P(X=10)=1 \]while
\[ P(Y=0)=\frac12,\qquad P(Y=20)=\frac12. \]Both have expected value \(10\), but \(X\) never moves away from the mean while \(Y\) is always \(10\) units away.
Variance distinguishes these distributions.
Deviation from the Mean
The quantity
\[ X-\mu \]is the deviation from the mean.
If \(X>\mu\), the deviation is positive. If \(X<\mu\), the deviation is negative.
Why Ordinary Deviations Do Not Work
The expected deviation from the mean is always zero:
\[ \mathbb{E}[X-\mu] = \mathbb{E}[X]-\mu = \mu-\mu = 0. \]Positive and negative deviations cancel, so they cannot directly measure spread.
Why We Square the Deviations
Squaring makes every deviation nonnegative:
\[ (X-\mu)^2\ge0. \]Larger deviations also receive greater weight because squaring magnifies their contribution.
Discrete Variance Formula
For a discrete random variable,
\[ \boxed{ \operatorname{Var}(X) = \sum_x (x-\mu)^2P(X=x) }. \]Example: A Simple Discrete Distribution
Suppose
\[ P(X=0)=0.25,\qquad P(X=2)=0.50,\qquad P(X=4)=0.25. \]First find the mean:
\[ \mu = 0(0.25)+2(0.50)+4(0.25) = 2. \]Compute the Squared Deviations
\[ (0-2)^2=4,\qquad (2-2)^2=0,\qquad (4-2)^2=4. \]Weight them by their probabilities:
\[ \operatorname{Var}(X) = 4(0.25)+0(0.50)+4(0.25) = \boxed{2}. \]Standard Deviation of the Example
Since \(\operatorname{Var}(X)=2\),
\[ \boxed{\sigma_X=\sqrt2\approx1.414}. \]The Computational Variance Formula
Expanding the squared deviation leads to a second useful formula:
\[ \boxed{ \operatorname{Var}(X) = \mathbb{E}[X^2]-\bigl(\mathbb{E}[X]\bigr)^2 }. \]Why the Computational Formula Is Useful
Instead of calculating every deviation from the mean directly, compute \(\mathbb{E}[X]\) and \(\mathbb{E}[X^2]\), then subtract the square of the mean.
Example Using \(\mathbb{E}[X^2]\)
For the preceding distribution,
\[ \mathbb{E}[X^2] = 0^2(0.25)+2^2(0.50)+4^2(0.25) = 6. \]Therefore,
\[ \operatorname{Var}(X)=6-2^2=\boxed{2}. \]Variance Is Always Nonnegative
Because variance is the expectation of a squared quantity,
\[ \boxed{\operatorname{Var}(X)\ge0}. \]A negative variance is impossible.
When Variance Is Zero
Variance is zero exactly when the random variable is constant with probability \(1\).
If \(P(X=c)=1\), then
\[ \boxed{\operatorname{Var}(X)=0}. \]Variance and Units
Variance is measured in the square of the original units.
- If \(X\) is measured in dollars, variance is measured in dollars squared.
- If \(X\) is measured in hours, variance is measured in hours squared.
- If \(X\) is measured in meters, variance is measured in square meters.
Why Standard Deviation Is Easier to Interpret
Standard deviation removes the squared units:
\[ \sigma_X=\sqrt{\operatorname{Var}(X)}. \]It therefore has the same units as the random variable and provides a natural scale for spread.
Variance for Continuous Random Variables
If \(X\) is continuous with density \(f_X(x)\), then
\[ \boxed{ \operatorname{Var}(X) = \int_{-\infty}^{\infty}(x-\mu)^2f_X(x)\,dx }. \]Equivalently,
\[ \boxed{ \operatorname{Var}(X) = \int_{-\infty}^{\infty}x^2f_X(x)\,dx-\mu^2 }. \]Example: Uniform Distribution on \([0,4]\)
Suppose
\[ f_X(x)= \begin{cases} \frac14, & 0\le x\le4,\\ 0, & \text{otherwise}. \end{cases} \]We know \(\mu=2\). Also,
\[ \mathbb{E}[X^2] = \int_0^4x^2\left(\frac14\right)\,dx = \frac{16}{3}. \]Therefore,
\[ \boxed{\operatorname{Var}(X)=\frac43} \]and
\[ \boxed{\sigma_X=\sqrt{\frac43}=\frac{2}{\sqrt3}}. \]Effect of Adding a Constant
If \(Y=X+b\), every value and the mean shift by the same amount, so the deviations do not change.
\[ \boxed{\operatorname{Var}(X+b)=\operatorname{Var}(X)}. \]Effect of Multiplying by a Constant
If \(Y=aX\), every deviation is multiplied by \(a\). Squaring multiplies variance by \(a^2\):
\[ \boxed{\operatorname{Var}(aX)=a^2\operatorname{Var}(X)}. \]Standard Deviation Under Scaling
\[ \boxed{\sigma_{aX}=|a|\sigma_X}. \]The absolute value appears because standard deviation cannot be negative.
General Linear Transformation
For \(Y=aX+b\),
\[ \boxed{\operatorname{Var}(Y)=a^2\operatorname{Var}(X)} \]and
\[ \boxed{\sigma_Y=|a|\sigma_X}. \]The shift \(b\) affects center but not spread.
Example: Linear Transformation
Suppose \(\operatorname{Var}(X)=9\) and \(Y=4X-7\). Then
\[ \operatorname{Var}(Y)=4^2(9)=\boxed{144} \]and
\[ \boxed{\sigma_Y=12}. \]Variance of a Sum
Variance is not generally linear. For two random variables,
\[ \operatorname{Var}(X+Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2\operatorname{Cov}(X,Y). \]The covariance term measures how the variables vary together.
Variance of Sums of Independent Random Variables
If \(X\) and \(Y\) are independent with finite variances, then
\[ \boxed{ \operatorname{Var}(X+Y) = \operatorname{Var}(X)+\operatorname{Var}(Y) }. \]Variance of a Difference
If \(X\) and \(Y\) are independent, then
\[ \boxed{ \operatorname{Var}(X-Y) = \operatorname{Var}(X)+\operatorname{Var}(Y) }. \]The minus sign disappears after squaring.
Bernoulli Variance
If \(X\) is Bernoulli with success probability \(p\), then \(\mathbb{E}[X]=p\) and \(X^2=X\).
Therefore,
\[ \boxed{\operatorname{Var}(X)=p(1-p)}. \]Binomial Variance
If
\[ X\sim\operatorname{Binomial}(n,p), \]then
\[ \boxed{\operatorname{Var}(X)=np(1-p)} \]and
\[ \boxed{\sigma_X=\sqrt{np(1-p)}}. \]Variance Measures Dispersion Around the Mean
Larger variance generally means more probability is located farther from the mean. Smaller variance indicates tighter concentration near the mean.
Standard Deviation as a Characteristic Scale
Standard deviation provides a characteristic scale for spread around the mean.
It does not mean that every outcome lies exactly one standard deviation from the mean.
Comparing Spread Across Distributions
When two random variables use the same units, their standard deviations provide a direct comparison of the scale of variation.
Variance Does Not Determine the Distribution
Different distributions can have the same mean and variance.
Mean and variance are important summaries, but they do not completely specify a probability distribution.
Variance and Tail Behavior
Variance is sensitive to large deviations because those deviations are squared.
Rare outcomes far from the mean can therefore make substantial contributions to variance.
Finite Mean Does Not Guarantee Finite Variance
Some distributions have finite expected value but infinite variance.
This can happen when the tails are heavy enough that \(\mathbb{E}[X^2]\) diverges.
Common Error: Averaging Unsquared Deviations
The expected ordinary deviation from the mean is zero:
\[ \mathbb{E}[X-\mu]=0. \]Variance uses squared deviations to prevent cancellation.
Common Error: Forgetting the Probability Weights
For a discrete random variable, use
\[ \operatorname{Var}(X)=\sum_x(x-\mu)^2P(X=x). \]Squared deviations should not be averaged equally unless the outcomes are equally likely.
Common Error: Using \(\mathbb{E}[X^2]\) as the Variance
The correct formula is
\[ \operatorname{Var}(X) = \mathbb{E}[X^2]-\bigl(\mathbb{E}[X]\bigr)^2. \]Common Error: Confusing \(\mathbb{E}[X^2]\) with \((\mathbb{E}[X])^2\)
These are generally different quantities and should be computed separately.
Common Error: Letting Variance Be Negative
Variance cannot be negative. A negative result signals an arithmetic or algebraic error.
Common Error: Treating Standard Deviation as Variance
Variance and standard deviation are related by
\[ \sigma_X=\sqrt{\operatorname{Var}(X)}. \]Variance uses squared units; standard deviation uses original units.
Common Error: Adding Standard Deviations
Even for independent variables, standard deviations generally do not add.
Independent variances add instead.
Common Error: Forgetting the Square on a Scale Factor
If \(Y=aX+b\), then
\[ \operatorname{Var}(Y)=a^2\operatorname{Var}(X), \]not \(a\operatorname{Var}(X)\).
A Reliable Procedure Using Squared Deviations
- Find \(\mu=\mathbb{E}[X]\).
- Compute each deviation \(x-\mu\).
- Square each deviation.
- Weight by the corresponding probability or density.
- Add or integrate across the full distribution.
- Take the square root if standard deviation is required.
- Interpret variance in squared units and standard deviation in original units.
A Reliable Procedure Using the Computational Formula
- Compute \(\mathbb{E}[X]\).
- Compute \(\mathbb{E}[X^2]\).
- Square the mean.
- Subtract \((\mathbb{E}[X])^2\) from \(\mathbb{E}[X^2]\).
- Check that the result is nonnegative.
- Take the positive square root for standard deviation.
Practice 1: Two-Point Distribution
Suppose
\[ P(X=0)=\frac12,\qquad P(X=4)=\frac12. \]Find \(\mathbb{E}[X]\), \(\operatorname{Var}(X)\), and \(\sigma_X\).
Practice 2: Use the Computational Formula
Suppose
\[ \mathbb{E}[X]=3,\qquad \mathbb{E}[X^2]=13. \]Find \(\operatorname{Var}(X)\).
Practice 3: Standard Deviation
Suppose \(\operatorname{Var}(X)=25\). Find \(\sigma_X\).
Practice 4: Constant Random Variable
Suppose \(P(X=7)=1\). Find the variance and standard deviation.
Practice 5: Linear Transformation
Suppose \(\operatorname{Var}(X)=4\) and \(Y=3X+10\). Find \(\operatorname{Var}(Y)\) and \(\sigma_Y\).
Practice 6: Independent Sum
Suppose independent random variables satisfy
\[ \operatorname{Var}(X)=5,\qquad \operatorname{Var}(Y)=7. \]Find \(\operatorname{Var}(X+Y)\).
Practice 7: Bernoulli Variance
Let \(X\) be Bernoulli with \(p=0.4\). Find \(\operatorname{Var}(X)\).
Practice 8: Binomial Variance
Let
\[ X\sim\operatorname{Binomial}(20,0.3). \]Find the mean, variance, and standard deviation.
Practice 9: Continuous Variance
Let \(X\) be uniform on \([0,2]\), so
\[ f_X(x)= \begin{cases} \frac12, & 0\le x\le2,\\ 0, & \text{otherwise}. \end{cases} \]Find \(\operatorname{Var}(X)\).
Practice 10: Identify the Error
A student calculates \(\operatorname{Var}(X)=-2\) and reports the answer.
Explain why the result cannot be correct.
Practice Answers
Practice 1
\[ \mathbb{E}[X] = 0\left(\frac12\right) + 4\left(\frac12\right) = 2. \] \[ \operatorname{Var}(X) = (0-2)^2\left(\frac12\right) + (4-2)^2\left(\frac12\right) = \boxed{4}. \] \[ \boxed{\sigma_X=2}. \]Practice 2
\[ \operatorname{Var}(X) = 13-3^2 = \boxed{4}. \]Practice 3
\[ \boxed{\sigma_X=5}. \]Practice 4
\[ \boxed{\operatorname{Var}(X)=0},\qquad \boxed{\sigma_X=0}. \]Practice 5
\[ \operatorname{Var}(Y) = 3^2(4) = \boxed{36}. \] \[ \boxed{\sigma_Y=6}. \]Practice 6
\[ \operatorname{Var}(X+Y)=5+7=\boxed{12}. \]Practice 7
\[ \operatorname{Var}(X)=0.4(0.6)=\boxed{0.24}. \]Practice 8
\[ \mathbb{E}[X]=np=20(0.3)=6. \] \[ \operatorname{Var}(X)=np(1-p)=20(0.3)(0.7)=4.2. \] \[ \sigma_X=\sqrt{4.2}\approx2.049. \]Practice 9
For the uniform distribution on \([0,2]\), \(\mathbb{E}[X]=1\). Also,
\[ \mathbb{E}[X^2] = \int_0^2x^2\left(\frac12\right)\,dx = \frac43. \]Therefore,
\[ \boxed{\operatorname{Var}(X)=\frac13}. \]Practice 10
Variance is an expected squared deviation and therefore cannot be negative. A value of \(-2\) indicates an arithmetic or algebraic error.
What to Remember
Variance measures average squared distance from the mean:
\[ \boxed{ \operatorname{Var}(X) = \mathbb{E}\left[(X-\mu)^2\right] }. \]The computational formula is
\[ \boxed{ \operatorname{Var}(X) = \mathbb{E}[X^2]-\bigl(\mathbb{E}[X]\bigr)^2 }. \]Standard deviation is
\[ \boxed{\sigma_X=\sqrt{\operatorname{Var}(X)}}. \]Under a linear transformation,
\[ \boxed{\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)} \]and
\[ \boxed{\sigma_{aX+b}=|a|\sigma_X}. \]The central interpretation is: variance measures squared spread around the mean, while standard deviation expresses that spread on the original scale of the random variable.
Where We Go Next
Mean and variance summarize a probability distribution, but many important applications depend on recognizable distribution families with specific structural assumptions.
The next lesson, Bernoulli and Binomial Distributions, develops the basic probability models for single successes and repeated independent trials.