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Lesson 72 — Variance and standard deviation

Statistical dispersion: how much data deviates from the mean. Population and sample variance, standard deviation, computational formula, properties of linearity and independence.

Used in: 2.º ano do EM (16-17 anos) · Equiv. Stochastik LK alemão · Equiv. Math B japonês · Equiv. H2 Statistics singapurense

\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2

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Rigorous notation, full derivation, hypotheses

Rigorous definition

Variance and standard deviation — population and sample

"Variance is more or less the mean squared distance of each data point from the mean. The unit associated with variance is in squared units. To ensure the dispersion measure has the same units as the data, we take the square root of the variance, called the standard deviation." — OpenIntro Statistics §2.1, Diez et al., CC-BY-SA.

"In statistics problems, we usually do not have access to the entire population, so we use sample data to estimate population parameters. For this, we divide by the sample degrees of freedom, $n-1$ , instead of $n$ ." — OpenStax Statistics §2.7, Illowsky & Dean, CC-BY.

Algebraic properties

Theorem· Properties of variance

For random variables $X, Y$ and constants $a, b \in \mathbb{R}$ :

\text{Var}(b) = 0

what this means · A constant has zero variance: there is no dispersion.

\text{Var}(aX + b) = a^2 \, \text{Var}(X)

what this means · Translation does not affect dispersion; scale enters as a square.

\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) \quad (X, Y \text{ independent})

what this means · For independent X and Y, variances add up — analogy to Pythagoras in L².

\text{Var}(X) \geq 0

what this means · Variance is always non-negative.

Proof of $\text{Var}(aX+b) = a^2 \text{Var}(X)$ : since $E[aX+b] = a\mu + b$ , we have $\text{Var}(aX+b) = E[(aX+b - a\mu - b)^2] = E[(a(X-\mu))^2] = a^2 E[(X-\mu)^2] = a^2 \text{Var}(X)$ . $\square$

Geometric representation — scatter plot

Two sets with the same mean but distinct dispersions. Points far from the dashed line (mean) generate high variance; grouped points generate low variance.

Solved examples

Example— 1· Direct calculation of variance and standard deviation (application)

Problem: Calculate the population variance $\sigma^2$ and standard deviation $\sigma$ for the data $\{2, 4, 4, 4, 5, 5, 7, 9\}$ .

Strategy: Apply the definition in four steps: mean, deviations, squared deviations, mean of squares.

Resolution:

Mean: $\mu = \frac{2+4+4+4+5+5+7+9}{8} = \frac{40}{8} = 5$ .
Deviations $x_i - \mu$ : $-3, -1, -1, -1, 0, 0, 2, 4$ .
Squared deviations: $9, 1, 1, 1, 0, 0, 4, 16$ . Sum $= 32$ .
Population variance: $\sigma^2 = \frac{32}{8} = 4$ .
Standard deviation: $\sigma = \sqrt{4} = 2$ .

Verification: $\sigma = 2$ has the same unit as the data. We check: the interval $[\mu - \sigma, \mu + \sigma] = [3, 7]$ contains the values 4, 4, 4, 5, 5, 7—six of the eight data points (75%), compatible with the empirical rule (at least 68% for any Bell distribution, Chebyshev guarantees at least 75% for any distribution).

Source. OpenIntro Statistics §2.1, example "Toy data for variance" — CC-BY-SA license.

Example— 2· Sample variance with Bessel's correction (application)

Problem: Eight students took a statistics test. The grades were: 72, 85, 90, 68, 78, 92, 81, 74. Calculate the sample variance $s^2$ and sample standard deviation $s$ .

Strategy: Use divisor $n-1$ because the data is a sample of a larger class. I will use the computational formula $s^2 = \frac{\sum x_i^2 - n\bar{x}^2}{n-1}$ for efficiency.

Resolution:

$n = 8$ . Sum: $72+85+90+68+78+92+81+74 = 640$ . Mean: $\bar{x} = 640/8 = 80$ .
$\sum x_i^2 = 5184 + 7225 + 8100 + 4624 + 6084 + 8464 + 6561 + 5476 = 51718$ .
$\sum x_i^2 - n\bar{x}^2 = 51718 - 8 \times 6400 = 51718 - 51200 = 518$ .
$s^2 = \frac{518}{7} \approx 74.0$ .
$s = \sqrt{74.0} \approx 8.6$ points.

Verification: Inspecting the data, all values are between 68 and 92, range 24. A standard deviation of 8.6 is reasonable—it corresponds to approximately $1/3$ of the range, a typical value.

Source. OpenStax Statistics §2.7, example "Find the Standard Deviation" — CC-BY license.

Example— 3· Variance of a discrete random variable (intermediate)

Problem: A fair 6-sided die. Let $X$ be the result. Calculate $\text{Var}(X)$ .

Strategy: Use the formula $\text{Var}(X) = E[X^2] - (E[X])^2$ with the discrete uniform distribution.

Resolution:

$P(X = k) = 1/6$ for $k \in \{1, 2, 3, 4, 5, 6\}$ .
$E[X] = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = \frac{7}{2}$ .
$E[X^2] = \frac{1^2+2^2+3^2+4^2+5^2+6^2}{6} = \frac{1+4+9+16+25+36}{6} = \frac{91}{6}$ .
$\text{Var}(X) = \frac{91}{6} - \left(\frac{7}{2}\right)^2 = \frac{91}{6} - \frac{49}{4} = \frac{182}{12} - \frac{147}{12} = \frac{35}{12}$ .
Standard deviation: $\sigma = \sqrt{35/12} \approx 1.71$ .

Verification: The general formula for $X$ uniform on $\{1, \ldots, n\}$ is $\text{Var}(X) = (n^2-1)/12$ . For $n=6$ : $(36-1)/12 = 35/12$ . It matches.

Source. Grinstead & Snell, Introduction to Probability, Ch. 6, Example 6.1 — GNU FDL license.

Example— 4· Linearity property and coefficient of variation (intermediate)

Problem: A machine fills bottles with mean volume $\mu = 500$ mL and $\sigma = 4$ mL. The company sells them in boxes of 12 bottles. (a) What is the standard deviation of the total volume of a box, assuming independent bottles? (b) What is the coefficient of variation ( $CV$ ) of an individual bottle?

Strategy: For (a), use the independence property: $\text{Var}(X_1 + \cdots + X_{12}) = 12\sigma^2$ . For (b), apply $CV = \sigma/\mu$ .

Resolution:

(a) Let $S = X_1 + X_2 + \cdots + X_{12}$ with $X_i$ independent and identical.

$\text{Var}(S) = 12 \cdot \sigma^2 = 12 \cdot 16 = 192 \text{ mL}^2$ .

$\sigma_S = \sqrt{192} = 8\sqrt{3} \approx 13.9$ mL.

(b) $CV = \frac{\sigma}{\mu} = \frac{4}{500} = 0.008 = 0.8\%$ .

A $CV$ of 0.8% is very low—the machine is consistent.

Verification: For 12 independent bottles, the standard deviation grows with $\sqrt{12}$ , not with 12. $4\sqrt{12} \approx 13.9$ . Consistent with the calculation.

Source. OpenStax Statistics §2.7, property exercise — CC-BY license.

Example— 5· Chebyshev's inequality — probability bounds (modeling)

Problem: A production line manufactures parts with mean mass $\mu = 200$ g and standard deviation $\sigma = 5$ g. Without knowing the exact distribution, determine a lower bound for the proportion of parts with mass between 185 g and 215 g.

Strategy: Chebyshev provides a universal bound: $P(|X - \mu| < k\sigma) \geq 1 - 1/k^2$ . Identify $k$ from the requested range.

Resolution:

Requested range: $[185, 215] = [\mu - 15, \mu + 15]$ .
$15 = 3\sigma$ (since $\sigma = 5$ ). Thus $k = 3$ .
Chebyshev: $P(|X - 200| < 15) \geq 1 - \frac{1}{3^2} = 1 - \frac{1}{9} = \frac{8}{9} \approx 88.9\%$ .

At least 88.9% of the parts are within tolerance, regardless of the distribution.

Verification: If production were normal, with $k=3$ we would have 99.7%—much more. Chebyshev is conservative. The value 88.9% is the worst possible bound for any distribution with $\mu = 200, \sigma = 5$ .

Source. Grinstead & Snell, Introduction to Probability, §8.1, Chebyshev Inequality — GNU FDL license.

Exercise list

40 exercises · 10 with worked solution (25%)

Application 24Understanding 3Modeling 9Proof 4

Ex. 72.1Application
Calculate the population variance and standard deviation of $\{4, 6, 8\}$ .
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Ex. 72.2Application
Calculate the sample variance $s^2$ and sample standard deviation $s$ for $\{2, 4, 4, 4, 5, 5, 7, 9\}$ .
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Ex. 72.3Application
Calculate the population standard deviation of $\{5, 6, 7, 8, 9\}$ .
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Ex. 72.4ApplicationAnswer key
What is the variance of $\{10, 10, 10, 10\}$ ? Explain geometrically.
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Ex. 72.5ApplicationAnswer key
Calculate the population variance of $\{0, 100\}$ .
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Ex. 72.6Application
Salaries (thousand R $):$ 3, 3, 4, 4, 5, 20$. Calculate the mean and sample standard deviation. Comment on the effect of the outlier.
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Ex. 72.7Application
Use the computational formula $\overline{x^2} - \bar{x}^2$ to calculate the variance of $\{1, 2, 3, 4, 5\}$ .
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Ex. 72.8Application
Waiting time (min) in 8 service calls: $5, 7, 6, 8, 4, 5, 6, 7$ . Calculate the sample standard deviation.
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Ex. 72.9ApplicationAnswer key
Weights (kg) of 6 watermelons: $8, 9, 9, 10, 11, 13$ . Calculate $s^2$ and $s$ .
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Ex. 72.10Application
$X$ assumes values $1, 2, 3$ with probabilities $\frac{1}{2}, \frac{1}{4}, \frac{1}{4}$ . Calculate $\text{Var}(X)$ .
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Ex. 72.11Application
Fair 6-sided die. Calculate $\text{Var}(X)$ .
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Ex. 72.12ApplicationAnswer key
Sum of two independent fair dice. Calculate $\text{Var}(S)$ using the independence property.
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Ex. 72.13Application
Maximum temperature (°C) over 7 days: $10, 7, 4, 9, 8, 11, 5$ . Calculate the sample variance.
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Ex. 72.14Application
Use the computational formula $E[X^2] - (E[X])^2$ to calculate the variance of $\{1, 3, 5, 7, 9\}$ .
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Ex. 72.15ApplicationAnswer key
If $\text{Var}(X) = 9$ , calculate $\text{Var}(2X + 5)$ .
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Ex. 72.16Application
If $\sigma_X = 4$ , what is the standard deviation of $3X$ ?
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Ex. 72.17ApplicationAnswer key
$\text{Var}(X) = 4$ , $\text{Var}(Y) = 9$ , $X$ and $Y$ independent. Calculate $\text{Var}(X+Y)$ and $\text{Var}(X-Y)$ .
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Ex. 72.18Application
Standardize $X = 80$ if $\mu = 70$ , $\sigma = 5$ . Calculate the z-score.
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Ex. 72.19Application
$F = 1.8C + 32$ (Celsius to Fahrenheit conversion). If $\sigma_C = 5$ °C, what is $\sigma_F$ ?
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Ex. 72.20Application
Calculate the coefficient of variation $CV = \sigma/\mu$ for heights ( $\mu = 170$ cm, $\sigma = 8$ cm) and weights ( $\mu = 70$ kg, $\sigma = 12$ kg). Which set is relatively more variable?
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Ex. 72.21Application
Standardize $\{60, 70, 80\}$ using $\mu = 70, \sigma = 10$ . What are the mean and standard deviation of the z-scores?
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Ex. 72.22Application
$\text{Var}(X) = 16$ . What is $\text{Var}(-X)$ ?
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Ex. 72.23ApplicationAnswer key
Sample mean of $n = 25$ independent observations with $\sigma = 10$ . What is the standard deviation of the mean?
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Ex. 72.24Application
Sum of 100 iid random variables with $\sigma = 1$ . What is the standard deviation of the sum?
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Ex. 72.25Understanding
Why does sample variance use the divisor $n-1$ instead of $n$ ?
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Ex. 72.26Understanding
To compare dispersion between salaries ( $) and heights (cm), is$ \sigma $or$ CV$ preferred? Why?
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Ex. 72.27Understanding
Can variance be negative?
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Ex. 72.28Modeling
Production line: mean mass 500 g, $\sigma = 5$ g. Tolerance $\pm 15$ g. How many $\sigma$ does the tolerance represent?
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Ex. 72.29ModelingAnswer key
Two funds with 8% expected return, but $\sigma_A = 5\%$ and $\sigma_B = 15\%$ . Which one to choose as risk-averse? Why?
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Ex. 72.30Modeling
You measure a resistance 10 times: $\bar{R} = 100\,\Omega$ , $s = 0.5\,\Omega$ . Estimate the standard deviation of the mean.
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Ex. 72.31Modeling
Home-to-work commute time: $\mu = 30$ min, $\sigma = 5$ min. Using Chebyshev's inequality as a conservative bound, how many minutes early should you leave to have at least a 95% chance of arriving on time?
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Ex. 72.32Modeling
Six Sigma process: $\mu = 10.00$ mm, tolerance $9.94$ to $10.06$ mm. What is the largest $\sigma$ that still satisfies the Six Sigma requirement?
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Ex. 72.33ModelingAnswer key
Stocks A: $\sigma_A = 1\%$ ; Stocks B: $\sigma_B = 2\%$ . 50-50 portfolio, zero correlation. Portfolio variance.
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Ex. 72.34Modeling
Same portfolio as the previous exercise, but with $-0.5$ correlation between stocks. Variance. Compare with the zero correlation case.
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Ex. 72.35Modeling
In machine learning, why should features with different scales be standardized before training gradient-based models?
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Ex. 72.36Modeling
ENEM Math grades: $\mu \approx 520$ , $\sigma \approx 110$ points. A student got 740. Calculate the z-score and interpret (how many standard deviations above the mean are they?).
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Ex. 72.37Proof
Prove that $\text{Var}(X) = E[X^2] - (E[X])^2$ from the definition $\text{Var}(X) = E[(X-\mu)^2]$ .
Ex. 72.38Proof
Prove that $\text{Var}(aX + b) = a^2\,\text{Var}(X)$ for any constants $a, b$ .
Ex. 72.39ProofAnswer key
Prove that $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$ when $X$ and $Y$ are independent.
Ex. 72.40Proof
Prove Chebyshev's inequality: $P(|X - \mu| \geq k\sigma) \leq \dfrac{1}{k^2}$ for $k > 0$ .

Sources

OpenIntro Statistics (4th ed.) — Diez, Çetinkaya-Rundel, Barr · CC-BY-SA. Primary source for this lesson. §2.1–§2.2 cover sample variance, standard deviation, boxplots, and applied examples.
Statistics (OpenStax) — Illowsky, Dean · CC-BY. §2.7 covers dispersion measures, computational formula, calculator exercises, and education/health data.
Introduction to Probability — Grinstead & Snell (Dartmouth) — GNU FDL. Ch. 6 covers variance of discrete random variables, algebraic properties, Chebyshev's inequality, and connection to the law of large numbers.