Statistics Review

This review pulls together everything from the Statistics section. Use it to test yourself across all four areas before moving on.

• Population is everyone of interest; sample is who you measured. Parameter describes a population, statistic describes a sample.
• Descriptive statistics summarizes data you have; inferential statistics generalizes beyond it with stated uncertainty.
• Bias is worse than small size. A big biased sample beats nothing but loses to a small fair one.
• Categorical data records groups; quantitative records amounts. Quantitative splits into discrete (countable) and continuous (measurable).
• Four levels: nominal (names), ordinal (ordered), interval (equal gaps, no true zero), ratio (true zero, ratios valid).
• Bar charts have gaps for categories; histogram bars touch for a continuum.

• $\bar{x} = \frac{\sum x}{n}$. The mean is the balance point and is sensitive to outliers.
• The median is positional and resistant to outliers. The mode is the only center for categorical data.
• Skew is named for the tail. Right-skewed means mean > median.
• $s = \sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}$. Deviations always sum to zero, which is why we square them.
• Divide by $n-1$ for samples (Bessel’s correction) because deviations from $\bar{x}$ run small.
• Five-number summary: min, Q1, median, Q3, max. $IQR = Q_3 - Q_1$.
• Outlier fences: $Q_1 - 1.5 \, IQR$ and $Q_3 + 1.5 \, IQR$.
• Pair mean with standard deviation, or median with IQR.

• $P(A) = \frac{n(A)}{n(S)}$ when outcomes are equally likely, and $0 \leq P(A) \leq 1$.
• $P(A') = 1 - P(A)$. Use the complement for “at least one.”
• Addition: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• Multiplication: $P(A \cap B) = P(A)P(B \mid A)$, simplifying to $P(A)P(B)$ when independent.
• Mutually exclusive means both can’t happen; independent means one doesn’t affect the other. These are nearly opposites.
• Conditional: $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$. Conditioning changes the denominator.
• $P(A \mid B) \neq P(B \mid A)$. Confusing them is the prosecutor’s fallacy.
• Bayes: $P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B)}$. Posterior ∝ likelihood × prior.
• For rare conditions the base rate dominates, so a positive test often means low probability of disease.

• $E(X) = \sum x \, P(X=x)$, the long-run average. It need not be a possible outcome.
• $\text{Var}(X) = E(X^2) - [E(X)]^2$ is the usual shortcut.
• Binomial requires BINS: binary, independent, fixed $n$, same $p$. Then $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$, with $\mu = np$ and $\sigma = \sqrt{np(1-p)}$.
• Normal distributions are symmetric with mean = median = mode. $\mu$ shifts, $\sigma$ stretches. Probability is area, so exact values have probability zero.
• $z = \frac{x - \mu}{\sigma}$, dimensionless, so it compares across scales.
• Empirical rule: 68-95-99.7 within 1, 2, 3 standard deviations. Requires approximate normality.
• Income, waiting times, and stock returns are not normal.

• CLT: sample means are approximately normal for large $n$ whatever the population shape, with $\mu_{\bar{x}} = \mu$ and $SE = \frac{\sigma}{\sqrt{n}}$.
• Quadruple $n$ to halve $SE$. Precision is expensive.
• Confidence interval: $\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$, with $z^* = 1.96$ for 95%.
• “95% confident” describes the method’s long-run success, not the probability for one interval.
• Higher confidence gives a wider interval. Use the t-distribution when $\sigma$ is unknown.
• $H_0$ carries the equality and concerns parameters, never statistics.
• $p \leq \alpha$ rejects. Otherwise fail to reject, which never means “accept.”
• p-value $= P(\text{data this extreme} \mid H_0)$. It is not $P(H_0 \mid \text{data})$.
• Type I = false positive (rate $\alpha$); Type II = false negative (rate $\beta$). Power $= 1 - \beta$.
• Significance ≠ importance. Large samples make tiny effects significant.
• $r$ measures linear association only; $r^2$ is the proportion of variation explained.
• Correlation admits reverse causation, confounding, coincidence, and selection effects.

Example 1: Center and spread.

Data: 6, 9, 9, 12, 14

Mean: $\frac{50}{5} = 10$. Median: 9. Mode: 9.

Squared deviations: $16, 1, 1, 4, 16$, summing to 38.

$$s = \sqrt{\frac{38}{4}} = \sqrt{9.5} \approx 3.08$$

Example 2: Outlier check.

Data with $Q_1 = 18$, $Q_3 = 30$. Is 50 an outlier?

$IQR = 12$, so the upper fence is $30 + 1.5(12) = 48$. Since $50 > 48$, yes.

Example 3: Complement rule.

Five dice are rolled. Find $P(\text{at least one } 4)$.

$$1 - \left(\frac{5}{6}\right)^5 = 1 - \frac{3125}{7776} \approx 0.598$$

Example 4: Conditional probability.

Of 300 people, 120 exercise and 45 of those sleep well. Find $P(\text{sleeps well} \mid \text{exercises})$.

$$\frac{45}{120} = 0.375$$

Example 5: Bayes by counts.

Prevalence 2%, sensitivity 95%, specificity 90%. Find the positive predictive value.

Per 10,000: diseased 200 → 190 true positives. Healthy 9,800 → 980 false positives.

$$\frac{190}{190 + 980} = \frac{190}{1170} \approx 0.162$$

About 16%, despite a good test, because healthy people vastly outnumber sick ones.

Example 6: Expected value.

A game pays 8 dollars with probability 0.25 and loses 2 dollars otherwise. Find $E$.

$$E = 8(0.25) + (-2)(0.75) = 2 - 1.5 = +0.50$$

Example 7: Binomial.

$n = 8$, $p = 0.5$. Find $P(X = 6)$.

$$\binom{8}{6}(0.5)^8 = 28 \times \frac{1}{256} = \frac{28}{256} \approx 0.109$$

Example 8: Z-score comparison.

Score 92 where $\mu = 80, \sigma = 6$ versus score 65 where $\mu = 50, \sigma = 10$.

$$z_1 = \frac{12}{6} = 2.0 \qquad z_2 = \frac{15}{10} = 1.5$$

The first is stronger relative to its distribution.

Example 9: Standard error.

$\sigma = 18$, $n = 81$.

$$SE = \frac{18}{9} = 2$$

Example 10: Confidence interval.

$\bar{x} = 44$, $\sigma = 12$, $n = 144$, 95% confidence.

$$SE = \frac{12}{12} = 1, \quad ME = 1.96 \quad \Rightarrow \quad (42.04, \; 45.96)$$

Example 11: Hypothesis test.

$H_0: \mu = 20$, $\bar{x} = 22.4$, $\sigma = 8$, $n = 100$, two-tailed at $\alpha = 0.05$.

$$z = \frac{22.4 - 20}{0.8} = 3.0$$

Two-tailed p-value $\approx 0.0027 \leq 0.05$, so reject $H_0$.

Example 12: Correlation.

$r = -0.7$. Direction is negative, strength is strong, and $r^2 = 0.49$, so 49% of variation is explained.