2.2 Expected Value, Mean, Variance, and Descriptive Statistics

2.2 Expected Value, Mean, Variance, and Descriptive Statistics

Descriptive statistics and expectation operators quantify the central location, variability, and shape of engineering datasets. On the FE exam, you must quickly distinguish between population metrics (which divide by $N$) and sample metrics (which divide by $n - 1$), perform expectation algebra on linear combinations of random variables, and calculate standard sample metrics.

1. Expected Value (Mathematical Expectation)

The expected value $E[X]$ (or population mean $\mu$) is the first moment about the origin representing the long-run average of a random variable.

NCEES Formulas for Expectation

• Discrete: $E[X] = \sum_{i} x_i P(X = x_i)$
• Continuous: $E[X] = \int_{-\infty}^{\infty} x f(x) dx$
• Expectation of a Function $g(X)$: $E[g(X)] = \sum_{i} g(x_i) P(x_i) \quad \text{or} \quad \int_{-\infty}^{\infty} g(x) f(x) dx$

Algebraic Properties of Expectation

For any constants $a, b, c$ and random variables $X, Y$:

1. $E[c] = c$
2. $E[aX + b] = a E[X] + b$
3. $E[X + Y] = E[X] + E[Y]$ (holds whether $X$ and $Y$ are independent or dependent)
4. If $X$ and $Y$ are independent: $E[X Y] = E[X] E[Y]$

2. Variance, Covariance, and Linear Combinations

Variance measures the second moment about the mean—the expected squared deviation of a random variable from its mean.

NCEES Formulas for Variance and Covariance

$\text{Variance: } \sigma^2 = Var(X) = E[(X - \mu)^2] = E[X^2] - (E[X])^2$ $\text{Standard Deviation: } \sigma = \sqrt{Var(X)}$ $\text{Covariance: } Cov(X, Y) = \sigma_{XY} = E[(X - \mu_X)(Y - \mu_Y)] = E[XY] - E[X]E[Y]$ $\text{Correlation Coefficient: } \rho_{XY} = \frac{Cov(X, Y)}{\sigma_X \sigma_Y}, \quad -1 \le \rho_{XY} \le 1$

Properties of Variance

1. $Var(c) = 0$
2. $Var(aX + b) = a^2 Var(X)$
3. $Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X, Y)$
4. $Var(aX - bY) = a^2 Var(X) + b^2 Var(Y) - 2ab Cov(X, Y)$

CRITICAL NCEES RULE: If $X$ and $Y$ are independent, $Cov(X, Y) = 0$. Therefore: $Var(aX + bY) = a^2 Var(X) + b^2 Var(Y)$ $Var(aX - bY) = a^2 Var(X) + b^2 Var(Y)$ Notice that for independent variables, variances ALWAYS ADD, even when taking the difference of two variables!

Worked Engineering Example: Mechanical Assembly Tolerance Stack-Up

Problem: An assembly shaft diameter $D$ is normally distributed with $\mu_D = 25.00\text{ mm}$ and variance $\sigma_D^2 = 0.04\text{ mm}^2$. The housing bore $B$ is normally distributed with $\mu_B = 25.35\text{ mm}$ and variance $\sigma_B^2 = 0.09\text{ mm}^2$. Assuming $D$ and $B$ are independent, determine the mean, variance, and standard deviation of the clearance gap $C = B - D$.

Solution: Step 1: Compute Expected Clearance Mean $E[C]$: $E[C] = E[B - D] = E[B] - E[D] = 25.35 - 25.00 = 0.35\text{ mm}$

Step 2: Compute Clearance Variance $Var(C)$: Since $B$ and $D$ are independent, $Cov(B, D) = 0$: $Var(C) = Var(B - D) = (1)^2 Var(B) + (-1)^2 Var(D) = Var(B) + Var(D)$ $Var(C) = 0.09 + 0.04 = 0.13\text{ mm}^2$

Step 3: Compute Clearance Standard Deviation $\sigma_C$: $\sigma_C = \sqrt{0.13} = 0.3606\text{ mm} \approx 0.361\text{ mm}$

3. Sample Descriptive Statistics

When working with experimental sample data of size $n$, sample statistics estimate the true population parameters.

NCEES Formulas for Sample Statistics

• Sample Mean (Unbiased Estimator of $\mu$): $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$
• Sample Variance (Unbiased Estimator of $\sigma^2$): $s^2 = \frac{1}{n - 1} \sum_{i=1}^n (x_i - \bar{x})^2 = \frac{\sum_{i=1}^n x_i^2 - \frac{(\sum_{i=1}^n x_i)^2}{n}}{n - 1}$ The denominator uses $(n - 1)$ degrees of freedom (Bessel's correction) to prevent underestimating population variance.
• Sample Standard Deviation: $s = \sqrt{s^2}$
• Standard Error of the Mean (SE): $\sigma_{\bar{x}} = \frac{s}{\sqrt{n}}$ Quantifies the variability of sample means drawn from the population.
• Coefficient of Variation (CV): $CV = \frac{s}{\bar{x}} \times 100\%$ A dimensionless measure of relative dispersion, ideal for comparing variability across datasets with different units or scales.

4. Worked Engineering Example: Complete Sample Data Analysis

Problem: Five steel core samples are subjected to destructive compression testing. The recorded failure loads (in kN) are: $x = \\{30.0, 33.0, 36.0, 39.0, 42.0\\}$ Calculate the sample mean $\bar{x}$, sample variance $s^2$, sample standard deviation $s$, standard error of the mean $SE$, and coefficient of variation $CV$.

Solution: Step 1: Compute Sample Mean $\bar{x}$: $\sum x_i = 30.0 + 33.0 + 36.0 + 39.0 + 42.0 = 180.0\text{ kN}$ $\bar{x} = \frac{180.0}{5} = 36.0\text{ kN}$

Step 2: Compute Sample Variance $s^2$ using squared deviations:

$x_i$	Deviation $(x_i - \bar{x})$	Squared Deviation $(x_i - \bar{x})^2$
30.0	$30.0 - 36.0 = -6.0$	36.0
33.0	$33.0 - 36.0 = -3.0$	9.0
36.0	$36.0 - 36.0 = 0.0$	0.0
39.0	$39.0 - 36.0 = 3.0$	9.0
42.0	$42.0 - 36.0 = 6.0$	36.0
Sum	0.0	90.0

Apply Bessel's correction with $n - 1 = 4$: $s^2 = \frac{90.0}{5 - 1} = \frac{90.0}{4} = 22.50\text{ kN}^2$

Step 3: Compute Sample Standard Deviation $s$: $s = \sqrt{22.50} = 4.7434\text{ kN}$

Step 4: Compute Standard Error of the Mean $SE$: $SE = \frac{s}{\sqrt{n}} = \frac{4.7434}{\sqrt{5}} = \frac{4.7434}{2.23607} = 2.1213\text{ kN}$

Step 5: Compute Coefficient of Variation $CV$: $CV = \frac{s}{\bar{x}} \times 100\% = \frac{4.7434}{36.0} \times 100\% = 13.176\% \approx 13.18\%$

Comparison Note: If population variance $\sigma^2$ were computed, the divisor would be $N = 5$, giving $\sigma^2 = 90.0 / 5 = 18.00\text{ kN}^2$. Using $n-1 = 4$ correctly adjusts for sample bias.

5. Measures of Position and Summary Table

In addition to mean and variance, descriptive statistics includes measures of central tendency and dispersion:

• Median: The middle value when data is sorted in ascending order (50th percentile).
• Mode: The most frequently occurring data value.
• Range: $R = x_{max} - x_{min}$.
• Interquartile Range (IQR): $IQR = Q_3 - Q_1$ (difference between 75th and 25th percentiles).

Population vs. Sample Statistics Summary Table